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+{"text":"\\section{Introduction}\n\\label{intro}\n\nIt is well known that Type II plateau supernovae (SNe IIP hereafter) are explosive events showing hydrogen (and metal) lines with P-Cygni profiles in their spectra, and an extended period (typically the early $\\sim$ 100 days of post-explosion evolution) during which the bolometric light curve remains constant to within $\\sim$ 0.5 magnitudes \\citep*[e.g.][] {filip97,turatto03,turatto07,smartt09,pastorello12}. After this plateau or photospheric phase, the bolometric light curve shows a rapid decay followed by a transition to a linear decline of $\\sim$ 0.98 magnitudes every 100 days \\citep[e.g.][]{sollerman02}. During this post-explosion period (radioactive-decay phase), the continuum electromagnetic emission is thought to be powered by the energy released from the radioactive decay of \\chem{56}{Ni} through the nuclear decay chain \\chem{56}{Ni} $\\rightarrow$ \\chem{56}{Co} $\\rightarrow$ \\chem{56}{Fe} \\citep[e.g.][]{PZ11}.\\par\n\n``Typical'' SNe IIP (e.g.~SNe 2004et, 1999em and 1999gi) have P-Cygni line profiles with widths of several thousand km s$^{-1}$ (from $\\sim$ 3000 to $\\sim$ 15000 km s$^{-1}$) and bolometric luminosities at the plateau from $\\sim$ 10$^{42}$ to $\\sim$ 10$^{43}$ erg s$^{-1}$ \\citep[e.g.][]{sahu06,misra07,maguire10,hamuy01,elmhamdi03,leonard02a,leonard02b}. The \\chem{56}{Ni} masses powering their light curve during the radioactive-decay phase are in the range $\\sim$ 0.06-0.10$M_\\odot$ \\citep[e.g.][]{turatto90,sollerman02}. From a theoretical point of view, these features are explained in terms of core-collapse explosions with an energy of the order of 1 foe ($\\equiv$10$^{51}$\\,ergs), occurring in sufficiently massive progenitors (i.e.~stars with mass on main sequence larger than $\\sim$ 8-10$M_\\odot$) that retain a substantial part (greater than $\\sim$ 3-5 $M_\\odot$) of their hydrogen-rich envelope at the time of collapse \\citep[e.g.][]{woosley86,hamuy03a,heger03,pumo09,PZ13}. \nSpecifically the progenitors of SNe IIP are thought to be stars with initial (i.e.~at the zero age main sequence; ZAMS hereafter) mass up to $\\sim$ 25-30$M_\\odot$\\, that explode during the so-called red supergiant phase, even if the exact upper limit for the initial mass of the SNe IIP's progenitors is uncertain from both the theoretical and the observational point of view \\citep*[e.g.][and references therein]{smartt09,we12,kochanek12}.\\par\n\nThe discovery of SN 1997D and SN 1997D-like events (e.g.~SNe 1999br, 1999eu, 1999gn, 1994N, 2001dc, 2002gd, 2003Z, 2004eg, 2005cs, 2006ov, 2008bk and SN 2009md) has revealed the existence of a sub-group of SNe IIP with the following peculiar observational properties \\citep*[e.g.][]{turatto98,benetti01,zampieri03,pasto04,pasto09,spiro14}: an under-luminous (at least a factor $\\sim$ 5-10 lower than in normal SNe IIP) bolometric light curve at all epochs, a long lasting ($\\gtrsim$ 100 days) plateau, and spectra with relatively narrow P-Cygni lines which are indicative of low expansion velocities  ($\\lesssim$ 10$^3$ km s$^{-1}$ from the end of the plateau onwards and, in general, at least a factor $\\sim$ 2-3 lower than in typical SN IIP explosions at any epoch) of the ejected material. \nThese properties can be explained by Ni-poor (less than $\\sim$ 10$^{-2}$$M_\\odot$), low-energy (of the order of tenths of foe) explosive events that seem to form the tail of a rather smooth distribution of SNe IIP. In fact, there is no evidence of a significant jump between the observed properties of these low-luminosity (LL) SNe IIP and the population of standard SNe IIP (see the early work of \\citealt[][]{zampieri07} and \\citealt[][]{spiro14} as well as the statistical studies of \\citealt[][]{anderson14}, \\citealt[][]{faran14} and \\citealt[][]{sanders15}).\\par\n\nIn contrast with other underluminous transients (e.g.~SN 2008S-like events, the archetypal SN impostor SN 1997bs and similar transients such as SNe 2002kg, 2003gm and 2007sv) whose own nature of actual sub-luminous SNe or non-terminal eruptions is still widely debated \\citep*[e.g.][]{vandyk00,maund06,smith11,kochanek12b,AK15,tartaglia15,adams16}, LL SNe IIP seem to be genuine SN explosions with well-established general features. However, there are still uncertainties primarily linked to the real nature of their progenitors. \nIndeed theoretical arguments indicate the following three scenarios for the progenitors \\citep[see][and references therein]{spiro14}: a) stars forming neon-oxygen degenerate cores (also known as super-asymptotic giant branch stars) sufficiently massive to evolve into a so-called electron-capture SN, b) low-energy explosions of red supergiant stars at the end of their quiescent evolution having initial masses below $\\sim$ 15$M_\\odot$, and c) terminal explosions from more massive stars (i.e.~with initial masses $\\gtrsim$ 20$M_\\odot$) where a non negligible fraction of the ejecta falls back onto the compact remnant (also known as fall-back SNe).\\par\n\nThere are three approaches usually adopted to constrain the progenitors mass of observed LL SNe IIP: 1) the detection of the progenitor stars in pre-SN images  that allows a direct estimate of their masses, 2) the hydrodynamical modelling of the SN observables (i.e.~bolometric light curve, evolution of line velocities and continuum temperature at the photosphere), and 3) the modelling of the observed nebular spectra \\citep[e.g.][]{jerkstrand12,jerkstrand14}. \nThe first two methods, used more frequently, often produce discrepant results. Direct progenitor detections of LL SNe IIP provide ZAMS masses estimates in the range 8-15$M_\\odot$ \\citep*[see e.g.][]{maund05,li06,eldridge07,mattila08,smarttetal09,crockett11,fraser11,vandyk12,maund14a}, while the range of masses estimated from the hydrodynamical modelling is typically wider, including in some cases more massive progenitors \\citep*[up to $\\sim$ 20$M_\\odot$; see e.g.~][]{zampieri03,utrobin07,uc08}. \nFor several events, however, the results from hydrodynamical modelling are in agreement with those obtained with the direct progenitor detection method \\citep[see e.g.][]{zampieri07,pasto09, spiro14, takats14}. They best agree for the low-to-intermediate mass progenitors, pointing to low-energy explosions of red supergiant stars or super-asymptotic giant branch stars \\citep[although the occurrence of electron-capture SNe from these stars is questioned; see e.g.][]{eldridge07}.\nNevertheless, both approaches used to constraining progenitor masses have their uncertainties and caveats. For the direct progenitor detection method, the main problems are: 1) uncertainties in the stellar evolutionary models (e.g.~treatment of mixing processes, rotation and mass-loss) used to infer the mass; 2) uncertainties in the extinction estimates \\citep[][]{we12}, although \\citet[][]{kochanek12} substantially reduce the relevance of this problem; and 3) possible selection effects because the method can be applied only to relatively close (within $\\sim$ 30 Mpc) LL SNe IIP. For hydrodynamical modelling, the main caveats are a possible overestimate of the progenitor mass due to the one-dimensional approximation \\citep[][]{uc09} and the poorly known pre-SN structure \\citep[][]{dessart13}. \nAll of this makes it difficult to progress in our knowledge on the LL SNe IIP's progenitors and, in particular, on the parameters describing the progenitor star at the time of the explosion, such as the progenitor radius at shock breakout, the ejected mass, and the explosion energy.\\par\n\nWith the aim of clarifying the real nature of the LL SNe IIP's progenitors, we present the radiation-hydrodynamical modelling of three LL SNe IIP (namely, SNe 2003Z, 2008bk and 2009md). For these three well-observed SNe it is possible to derive reliable measurement of the physical parameters describing their progenitors at the time of the explosion. Together with the radiation-hydrodynamical modelling of other well-sampled LL SNe IIP presented in previous works \\citep[SNe 2005cs, 2008in, 2009N, 2009ib and 2012A; see][]{spiro14,takats14,takats15,tomasella13}, we now have a sample of LL SNe IIP for which it is possible to carry out a comparative study based on the same modelling approach, enabling us to also identify possible systematic trends. A preliminary analysis of this type was carried out by \\citet[][]{zampieri07} on a more limited sample of SNe IIP.\\par\n\nThe plan of the paper is the following. We illustrate the radiation-hydrodynamical modelling procedure in Sect.~\\ref{modelling} and shortly review the observational data in Sect.~\\ref{sample}. In Sect.~\\ref{results} we present and discuss our results, devoting Sect.~\\ref{single_sne} to the three new objects and Sect.~\\ref{systematics} to a comparative study of LL SNe IIP. A summary with final comments is presented in Sect.~\\ref{summary}.\\par\n\n\n\n\n\\section{Radiation-hydrodynamical modelling}\n\\label{modelling}\n\nTo perform the radiation-hydrodynamical modelling we use the same well-tested approach applied to other observed SNe (e.g.~2007od, 2009bw, 2009E, 2012aw, 2012ec and 2013ab; see \\citealt{inserra11,inserra12}, \\citealt{pasto12}; \\citealt{dallora14}; \\citealt{barbarino15}; and \\citealt{bose15}, respectively). In this approach, the SN progenitor's physical properties at the explosion (namely the ejected mass $M_{ej}$, the progenitor radius at the explosion $R$ and the total explosion energy $E$) are constrained through the hydrodynamical modelling of all the main SN observables (i.e.~bolometric light curve, evolution of line velocities and the temperature at the photosphere), using a simultaneous $\\chi^2$ fit of the observables against model calculations. \nIt is well known that almost identical light curves can be obtained for more than one set of the parameters describing the SN progenitor's physical properties at the time of the explosion \\citep[e.g.][]{arnett80,iwamoto98,nagy14}. This problem can, in turn, affect the search for possible correlations among the parameters, as correlations can be induced by covariance rather than by true physical effects \\citep[][]{PP15}. For this reason, we try to reduce the ``degeneracy'' in the best-fitting model parameters by fitting simultaneously the evolution of the line velocity, the continuum temperature and the light curve.\\par\n\nTwo codes are employed for computing the models. The first is a semi-analytic code that solves the energy balance equation for ejecta of constant density in homologous expansion \\citep[see][for details]{zampieri03}. The second is a general-relativistic, radiation-hydrodynamics Lagrangian code that is\nspecifically designed to simulate the behavior of the main SN observables and the evolution of the physical properties of the ejected material at the time of the explosion from the breakout of the shock wave at the stellar surface up to the radioactive-decay phase (see \\citealt*{pumo10} and \\citealt{PZ11}, for details).\nThe distinctive features of this code are \\citep[cf.~also][]{PZ13}: a) a fully general relativistic approach; b) an accurate treatment of radiative transfer coupled to hydrodynamics at all optical depth regimes; c) the coupling of the radiation moment equations with the equations of relativistic hydrodynamics during all the post-explosive evolution, adopting a fully implicit Lagrangian finite difference scheme; and d) a description of the evolution of the ejected material which takes into account both the gravitational effects of the compact remnant and the heating effects linked to the decays of the radioactive isotopes synthesized during the SN explosion.\\par\n\nThe semi-analytic code is employed to carry out a preparatory analysis aimed at determining the parameter space describing the SN progenitor at the explosion. This guides the simulations performed with the general-relativistic, radiation-hydrodynamics code that are more realistic but time consuming.\\par\n\nWe point out that the models are appropriate when the SN emission is dominated by the expanding ejecta with no significant contamination from interaction. In performing the $\\chi^{2}$ fit, the observational data taken at the earliest phases (i.e.~within the first $\\sim$ 20 days after explosion) are not included. This choice is made because the models could not accurately reproduce the early evolution of the main observables since the initial conditions used in the simulations are not able to precisely mimic the outermost high-velocity shell of the ejecta that forms after the shock breakout at the stellar surface \\citep[see][for details]{PZ11}.\\par\n\nAs for the comparison between the observations and the simulated SN observables, we remind the reader that the observed bolometric light curve is reconstructed from multi-color photometry and reddening measurements, whereas the photospheric velocity and temperature are estimated from the observed spectra \\citep[see Sections 2.6, 3.4, and 5.1 of][for details on these procedures]{inserra11}. In particular, to estimate the photospheric velocity from the spectra, we use the minima of the profile of the Sc lines or, when they are not available, the Fe lines.\\par\n\nIn the radiation-hydrodynamical modelling procedure the \\chem{56}{Ni} mass initially present in the ejecta of the models is held fixed and its value is set so as to reproduce the observed bolometric luminosity during the radioactive decay phase. To this end, the initial \\chem{56}{Ni} mass of the semi-analytic models is held fixed to that inferred from the observed late-time light curve. In the models computed with the general-relativistic, radiation-hydrodynamics code, the initial amount of \\chem{56}{Ni} is in general larger, since the code accounts also for the material (including \\chem{56}{Ni}) which falls back onto the compact remnant during the post-explosive evolution (see \\citealt{PZ11} and the results of the modelling of SN 2009E in \\citealt{pasto12}). \nHowever, in all the models presented here, the fall-back is negligible (a few hundredths of a solar mass) and, consequently, the initial \\chem{56}{Ni} mass essentially coincides (within the errors) with that inferred from the observations. \nOther quantities held fixed in the radiation-hydrodynamical modelling procedure are the explosion epoch and the distance modulus. They are both used for determining the observed bolometric light curve and the evolution of the observed photospheric velocity and temperature as a function of phase.\\par\n\n\n\n\\subsection{Uncertainties on the best-fitting model parameters}\n\\label{error}\n\nThe free model parameters of the fit are the ejected mass $M_{ej}$, the progenitor radius at the time of the explosion $R$ and the total explosion energy $E$. Although the evaluation of their uncertainties is not straightforward \\citep[see also][]{zampieri03,uc09}, we estimate that the typical error due to the $\\chi^{2}$ fitting procedure is $\\sim$ 10-15\\% for $M_{ej}$ and $R$ and $\\sim$ 20-30\\% for $E$. These errors are the 2-$\\sigma$ confidence intervals for one parameter based on the $\\chi^{2}$ distributions produced by the semi-analytical models. \nWe used these models to determine the confidence intervals because it is needed a sufficiently high coverage of the parameter space, which is obtained through the calculation of thousands of models. The computation of such an extensive grid of models with the general-relativistic, radiation-hydrodynamics code is too expensive in terms of CPU time (e.g., running a single general-relativistic, radiation-hydrodynamics model takes up to $\\sim$ 4-6 days).\\par\n\nThe inferred uncertainties on the best-fitting model parameters do not include possible systematic errors related to the input physics (e.g.~opacity treatment, degree of \\chem{56}{Ni} mixing and He\/H ratio in the ejecta of the models) nor uncertainties on the assumptions made in evaluating the observational quantities (e.g.~the adopted reddening, explosion epoch and distance modulus). A discussion of the approximations on the input physics and the ensuing systematic errors can be found in \\citet[][]{zampieri03}, while a detailed study of the effects of different opacity treatments on our radiation-hydrodynamical modelling will be presented in Pumo et al. (in prep.). \nHere, we recall that variations of the degree of \\chem{56}{Ni} mixing and the He\/H ratio mainly affect the plateau length of the models and not the simulated plateau luminosity or expansion velocity \\citep[see e.g.][]{PZ13}. As a consequence, uncertainties related to these quantities should lead mainly to errors in the value of $M_{ej}$ as the plateau length depends mostly on it. Similar conclusions are also valid for uncertainties on quantities (e.g.~the adopted explosion epoch) that mainly affect the observed plateau length and, only to a secondary extent, the behaviour of the observed photospheric velocity and temperature.\\par\n\nUncertainties related to the distance modulus or the adopted reddening basically produce a systematic variation in the brightness of the observed bolometric light curve at all phases and affect all three best-fitting model parameters. For example, the uncertainties on the reddening adopted for SN 2012aw modify the best-fitting model parameters obtained with our radiation-hydrodynamical modelling procedure by $\\sim$ 15-30\\% \\citep[specifically $M_{ej}$, $R$ and $E$ vary up to $\\sim$ 20\\%, 30\\% and 15\\%, respectively; see][for details]{dallora14}. \nSomewhat larger changes in the best-fitting model parameters (specifically $M_{ej}$, $R$ and $E$ vary up to $\\sim$ 25\\%, 30\\% and 35\\%, respectively) are found after modifying the distance modulus of SN 2005cs from the value of 29.26 assumed by \\citet[][]{pasto09} to the one of 29.46 adopted in \\citet[][]{spiro14}. Although in these two test cases the variations of the best-fitting model parameters are significant, they do not have a dramatic impact on the overall results and the progenitor scenario.\\par\n\n\n\\section{Sample of modelled underluminous IIP SNe}\n\\label{sample}\n\nWe model the well-studied LL SNe IIP 2003Z, 2008bk and 2009md. All the observational data used in the present work are taken from \\citet[][and references therein]{spiro14}, where the observational features of these LL SNe IIP as well as a detailed description of the data reduction techniques have been extensively presented and discussed.\\par\n\n\n\\begin{table}\n   \\centering\n   \\caption{Basic parameters (see text for details) for the sample of modelled LL SNe IIP.}\n   \\begin{tabular}{l ccc}\n   \\hline\\hline\n   SN                                      & 2003Z   & 2008bk   & 2009md \\\\\n   \\hline\n   Adopted explosion epoch [JD]            & 52665   & 54550    & 55162  \\\\\n   Adopted distance modulus                & 31.70   & 27.68    & 31.64  \\\\\n   Estimated mass of \\chem{56}{Ni} [$M_\\odot$] & 0.005   & 0.007    & 0.004  \\\\\n  \\hline\\hline\n \\end{tabular}\n \\label{tab_sample}\n  All reported quantities are taken from Table 13 of \\citet[][]{spiro14} that, in turn, used data from \\citet[][]{mattila08} and \\citet[][]{vandyk12} for SN 2008bk, and from \\citet[][]{fraser11} for SN 2009md.\n\\end{table}\n \n\nTo be thorough, here (see Table \\ref{tab_sample}) we recall the main assumptions made for evaluating the bolometric light curve and obtaining the behavior of the photospheric velocity and temperature as a function of the phase (i.e.~the adopted explosion epoch and distance modulus) as well as the amount of \\chem{56}{Ni} estimated through a comparison with the late-time luminosity of SN 1987A. In Figures \\ref{fig:03Z}, \\ref{fig:08bk} and \\ref{fig:09md} we also show the modelled observables (see the green squares) for SNe 2003Z, 2008bk and 2009md, respectively.\\par\n\n\n\n\\section{Results and discussion}\n\\label{results}\n\n\\subsection{Individual objects}\n\\label{single_sne}\n\nThe best-fitting models for SNe 2003Z, 2008bk and 2009md are shown in Figures \\ref{fig:03Z}, \\ref{fig:08bk} and \\ref{fig:09md}, respectively. The estimated uncertainties on the best-fitting model parameters are $\\sim$ 10-15\\% for $M_{ej}$ and $R$ and $\\sim$ 20-30\\% for $E$ (cf.~Section \\ref{error}). To estimate the total stellar mass at the time of the explosion we consider a mass of $\\sim$ 1.3-2$M_\\odot$\\, for the compact remnant \\citep[e.g.][]{demorest10,sukhbold16}. To evaluate the ZAMS mass, we add to the inferred value of the total stellar mass at the time of the explosion an estimate of the mass lost during the pre-SN evolution based on the non-rotating stellar models reported in the recent literature \\citep*[e.g.][]{heger00,hirschi04,pumo09,CL13}.\\par\n\nFor SN 2003Z, the  $\\chi^2$ fit procedure returns a best-fit model with a total (kinetic plus thermal) energy of 0.16 foe, a radius at the time of the explosion of 1.8 $\\times$ 10$^{13}$ cm ($\\sim$ 260$R_\\odot$) and an ejected mass of 11.3$M_\\odot$. Adding the mass of the compact remnant to that of the ejected material, we obtain a total stellar mass at the time of the explosion of $\\sim$ 12.6-13.3$M_\\odot$. Such a mass and the other best-fit model parameters $R$ and $E$ are consistent with a low-energy explosion of a low-mass red supergiant star. The inferred radius could also indicate a yellow supergiant star as the progenitor of SN 2003Z, as it has been hypothesized for other SNe IIP \\citep[e.g.~SNe 2004et and 2008cn;][]{li05,EliasRosa09} including some underluminous events \\citep[e.g.~SN 2009N;][]{takats14}.\nAssuming that $\\sim$ 0.6-1.8$M_\\odot$\\, are lost during the whole (i.e.~main sequence plus red\/yellow supergiant phase) pre-SN evolution (see models with pre-SN mass close to $\\sim$ 13$M_\\odot$), the progenitor mass of SN 2003Z on the ZAMS is in the range $\\sim$ 13.2-15.1$M_\\odot$. \nUnfortunately, there is no independent estimate of the SN 2003Z progenitor's initial mass, so it is not possible to make a direct comparison with our results. However, the value we infer for the progenitor mass of SN 2003Z on the ZAMS is comparable to the low progenitor masses found from observations of the progenitors of other LL SNe IIP (cf.~Sect.~\\ref{intro}). Interestingly, with their hydrodynamical model, \\citet[][]{utrobin07} derive a ZAMS mass of 14.4-17.4$M_\\odot$\\, for the progenitor of SN 2003Z, consistent with our estimate, although they find a wider overall range. The values of all our other  derived parameters are also close to those of \\citet[][]{utrobin07}, including the radius at explosion that led \\citet[][]{utrobin07} to hypothesize a yellow supergiant star as the progenitor of SN 2003Z.\\par \n                                                        \nFor SN 2008bk, the inferred best-fit model has a total energy of $0.18$ foe, a radius at explosion of $3.5 \\times 10^{13}$ cm ($\\sim$ 500$R_\\odot$) and an ejected mass of 10$M_\\odot$. Adding the mass of the compact remnant, we obtain a total stellar mass at explosion of $\\sim$ 11.3-12$M_\\odot$. These values are fully consistent with the explosion of a low-mass red supergiant star, even if they may be also marginally consistent (within the errors) with an explosion of a super-asymptotic giant branch star with an initial mass close to the upper limit of the mass range typical of this class of stars, M$_{mas}$ \\citep[see][and references therein]{pumo09}. \nConsidering that the mass lost during the pre-SN evolution is $\\sim$ 0.6-0.9$M_\\odot$\\, for an exploding low-mass red supergiant star (see models with pre-SN mass close to $\\sim$ 11.5$M_\\odot$) or $\\sim$ 0.1-0.3$M_\\odot$\\, for a super-asymptotic giant branch star with an initial mass close to M$_{mas}$, the progenitor mass of SN 2008bk on the ZAMS is in the range $\\sim$ 11.4-12.9$M_\\odot$\\, fully in agreement with the estimate of 11.1 to 14.5$M_\\odot$\\, from the direct progenitor detection method by \\citet[][]{maund14b}.\\par\n\n\n\n\\begin{figure}\n \\includegraphics[angle=-90,width=85mm]{.\/P15_fig01.ps} \n \\caption{Comparison of the evolution of the main observables of SN 2003Z with the best-fit model computed with the general-relativistic, radiation-hydrodynamics code. The best-fit model parameters are: total energy $0.16$ foe, radius at explosion $1.8 \\times 10^{13}$ cm, and ejected mass $11.3$ $M_\\odot$. Top, middle, and bottom panels show the bolometric light curve, the photospheric velocity, and the photospheric temperature as a function of time. Blue triangles mark ``early'' observations not considered in the fitting procedure (see Sect.~\\ref{modelling} for details).\n \\label{fig:03Z}}\n\\end{figure}\n\\begin{figure}\n \\includegraphics[angle=-90,width=85mm]{.\/P15_fig02.ps} \n \\caption{Same as Fig.~\\ref{fig:03Z}, but for SN 2008bk. The best-fit model parameters are: total energy $0.18$ foe, radius at explosion $3.5 \\times 10^{13}$ cm, and ejected mass $10$ $M_\\odot$.\n \\label{fig:08bk}}\n\\end{figure}\n\\begin{figure}\n \\includegraphics[angle=-90,width=85mm]{.\/P15_fig03.ps} \n \\caption{Same as Fig.~\\ref{fig:03Z}, but for SN 2009md. The best-fit model parameters are: total energy $0.17$ foe, radius at explosion $2 \\times 10^{13}$ cm, and ejected mass $10$ $M_\\odot$.\n \\label{fig:09md}}\n\\end{figure}\n\n\nFor SN 2009md, the best-fit model has a total energy of $0.17$ foe, a radius at explosion of $2 \\times 10^{13}$ cm ($\\sim$ 290$R_\\odot$) and an ejected mass of 10$M_\\odot$. The estimated total stellar mass at the time of the explosion is $\\sim$ 11.3-12$M_\\odot$, consistent with the explosion of a low-mass red supergiant star. \nHowever, as in SN 2003Z, the  estimated radius at the time of the explosion could also suggest a yellow supergiant star as progenitor. Moreover, the best-fit model parameters may be also marginally consistent (within the errors) with the explosion of a super-asymptotic giant branch star with an initial mass close to M$_{mas}$. Using the same values of mass loss adopted for SN 2008bk, we find that the progenitor mass of SN 2009md on the ZAMS is in the range $\\sim$ 11.4-12.9$M_\\odot$\\, once again in agreement with the value (ranging from 7 to 15$M_\\odot$) inferred from modelling the progenitor \\citep[][]{fraser11}, although the identification of the real progenitor star of SN 2009md in the pre-SN images should be taken with caution \\citep[][]{maund15}.\\par\n\n\n\n\\subsection{Underluminous IIP SNe: a comparative analysis}\n\\label{systematics}\n\nSNe 2005cs, 2008in, 2009N, 2009ib and 2012A \\citep[whose radiation-hydrodynamical models were presented in previous works; see][]{spiro14,takats14,takats15,tomasella13} along with SNe 2003Z, 2008bk and 2009md (whose radiation-hydrodynamical models were presented in Sect.~\\ref{single_sne}) form a sample of well-observed LL SNe IIP modelled in the same way. For them it has been thus possible to derive reliable and homogeneous estimates of the physical parameters describing the progenitors at the time of the explosion (see Table \\ref{tab_systematics}). \nNote also that the sample is composed of both faint SNe IIP (namely, SNe 2003Z, 2005cs, 2008bk and 2009md) with bolometric luminosity at the plateau less than $\\sim$~3~$\\times 10^{41}$~erg~s$^{-1}$ and so-called ``intermediate-luminosity'' objects (namely, SNe 2008in, 2009N, 2009ib and 2012A) that are located between faint and standard SNe IIP (see Figure \\ref{fig:bol}). \nAll of this enables us to compare these LL SNe IIP, making possible the identification of potential systematic trends inside this sub-group of SNe IIP.\\par\n\n\n\\begin{figure}\n \\includegraphics[angle=-90,width=85mm]{.\/P15_fig04.ps}\n \\caption{(Pseudo-)Bolometric luminosities during the first 250 days after the explosion for the SNe IIP reported in Table \\ref{tab_systematics}. Data are taken from \\citet[][]{spiro14} for SNe 2003Z, 2008bk and 2009md (cf.~Sect.~\\ref{sample}) and from the papers reported in the last column of Table \\ref{tab_systematics} for the remaining SNe.\n \\label{fig:bol}}\n\\end{figure}\n\n\n\\begin{table*}\n   \\centering\n   \\caption{Best-fitting model parameters and selected observational quantities.}\n   \\begin{tabular}{l ccccccc}\n   \\hline\\hline\n   \\multicolumn{8}{c}{\\normalsize LL SNe IIP}\\\\\n   \\multicolumn{8}{c}{Faint objects}\\\\\n   SN     & $E$   & $M_{ej}$ & $R$    & $E\/M_{ej}$  & \\chem{56}{Ni}       & L$_{50}$               & Ref.               \\\\\n          & [foe] & [$M_\\odot$]  & [cm]   & [foe\/$M_\\odot$] & [$M_\\odot$]             & [erg sec$^{-1}$]       &                    \\\\\n   2003Z  & 0.16  & 11.3     & 1.8e13 & 0.014       & 0.005  ($\\pm$0.003) & 1.36e41 ($\\pm$7.58e40) & this work          \\\\   \n   2008bk & 0.18  & 10.0     & 3.5e13 & 0.018       & 0.007  ($\\pm$0.001) & 2.76e41 ($\\pm$3.45e40) & this work          \\\\\n   2009md & 0.17  & 10.0     & 2.0e13 & 0.017       & 0.004  ($\\pm$0.001) & 1.88e41 ($\\pm$3.70e40) & this work          \\\\\n   2005cs & 0.16  &  9.5     & 2.5e13 & 0.017       & 0.006  ($\\pm$0.003) & 2.43e41 ($\\pm$1.35e41) & \\citet{spiro14}    \\\\\n   \\multicolumn{8}{c}{Intermediate-luminosity objects}\\\\  \n   SN     & $E$   & $M_{ej}$ & $R$    & $E\/M_{ej}$  & \\chem{56}{Ni}       & L$_{50}$               & Ref.               \\\\\n          & [foe] & [$M_\\odot$]  & [cm]   & [foe\/$M_\\odot$] & [$M_\\odot$]             & [erg sec$^{-1}$]       &                    \\\\\n   2008in & 0.49  & 13.0     & 1.5e13 & 0.038       & 0.012  ($\\pm$0.005) & 3.71e41 ($\\pm$1.36e41) & \\citet{spiro14}    \\\\\n   2009N  & 0.48  & 11.5     & 2.0e13 & 0.042       & 0.020  ($\\pm$0.004) & 5.40e41 ($\\pm$1.58e41) & \\citet{takats14}   \\\\\n   2009ib & 0.55  & 15.0     & 2.8e13 & 0.037       & 0.046  ($\\pm$0.015) & 4.67e41 ($\\pm$1.14e41) & \\citet{takats15}   \\\\\n   2012A  & 0.48  & 12.5     & 1.8e13 & 0.038       & 0.011  ($\\pm$0.004) & 4.93e41 ($\\pm$6.97e40) & \\citet{tomasella13}\\\\\n  \\hline\n  \\multicolumn{8}{c}{\\normalsize Standard-luminosity SNe IIP}\\\\\n   SN     & $E$   & $M_{ej}$ & $R$    & $E\/M_{ej}$  & \\chem{56}{Ni}       & L$_{50}$               & Ref.               \\\\\n          & [foe] & [$M_\\odot$]  & [cm]   & [foe\/$M_\\odot$] & [$M_\\odot$]             & [erg sec$^{-1}$]       &                    \\\\\n   2013ab & 0.35  & 7.0      & 4.2e13 & 0.050       & 0.06   ($\\pm$0.003) & 1.30e42 ($\\pm$2.60e41) & \\citet{bose15}     \\\\\n   2013ej & 0.70  & 10.6     & 4.2e13 & 0.066       & 0.02   ($\\pm$0.01)  & 1.04e42 ($\\pm$2.08e41) & \\citet{huang15}    \\\\\n   2012aw & 1.50  & 19.6     & 3.0e13 & 0.079       & 0.06   ($\\pm$0.013) & 1.29e42 ($\\pm$2.58e41) & \\citet{dallora14}  \\\\\n  \\hline\n  \\hline\n  \\end{tabular}\n \\label{tab_systematics}\n \n The quantities shown are (from left to right): the SN name, the best-fitting model parameters $E$, $M_{ej}$ and $R$ (cf.~Sect.~\\ref{modelling}), the ratio of $E$ to $M_{ej}$, the fixed \\chem{56}{Ni} mass inferred from the observations (see Sect.~\\ref{modelling}), the plateau (pseudo-)bolometric luminosity (measured at 50 days after explosion and derived by interpolating the data shown in Figure \\ref{fig:bol}), and the reference to the paper where the radiation-hydrodynamical models are presented in detail. Estimated uncertainties on the values inferred from the observations are in brackets. Top and bottom panels refer respectively to the sample of LL SNe IIP and to three SNe IIP of standard luminosity (namely, SNe 2013ab, 2013ej and 2012aw) modelled in the same way (i.e.~using the approach described in Sect.~\\ref{modelling}), which are reported for the sake of comparison.\n\\end{table*}\n\n\nAs it can be seen in the top panel of Table \\ref{tab_systematics}, all the model parameters (with the only exception of $R$) as well as the ratio $E$\/$M_{ej}$ and the \\chem{56}{Ni} mass of the faint SNe IIP are systematically lower than those for the intermediate-luminosity objects, showing that the progenitors of the faint SNe IIP are slightly less massive and experience less energetic explosions than the progenitors of the intermediate-luminosity objects.\\par\n\nMoreover, our radiation-hydrodynamical models suggests that the present sample of faint SNe IIP and intermediate-luminosity objects originate from stars of low-to-intermediate mass, in agreement with the results found for some of them by modelling their progenitors. In particular we find that the best-fit model parameters of all the modelled LL SNe IIP are consistent with low-energy explosions of red (or yellow) supergiant stars and, for some faint objects \\citep[SNe 2009md and 2008bk as well as SN 2005cs, whose model is presented in][]{spiro14}, they can also be consistent with explosions of massive, super-asymptotic giant branch stars as electron-capture SNe.\\par\n\nThe data reported in Table \\ref{tab_systematics} and Figures \\ref{fig:bol} to \\ref{fig:correl2} also confirm that LL SNe IIP form the underluminous tail of the family of SNe IIP \\citep[see also][]{hamuy03b,pasto04,zampieri07,spiro14,anderson14,faran14,sanders15}. With the warning that our sample could be still too small to draw final conclusions, the main parameter ``guiding'' the distribution seems to be the ratio of $E$ to $M_{ej}$, not just the explosion energy $E$. Indeed, Figures \\ref{fig:correl1} and \\ref{fig:correl2} reveal a relationship between the observed quantities such as the plateau luminosity and the \\chem{56}{Ni} mass and the ratio $E\/M_{ej}$, which monotonously increases from $\\sim$ 0.015-0.02 to $\\sim$ 0.04 up to values $\\gtrsim$ 0.05, when passing from faint to intermediate-luminosity up to ``standard-luminosity'' events, respectively.\\par\n\nIt is also clear that the data, in particular the correlation of \\chem{56}{Ni} with $E\/M_{ej}$ in Figure \\ref{fig:correl2}, show significant scatter. Indeed, although the explosion energy as well as the other model parameters and the \\chem{56}{Ni} mass tend to increase when moving from LL SNe IIP to SNe IIP of standard luminosity, there are several exceptions (see Table \\ref{tab_systematics}). \nFor example, SN 2013ab, a SN IIP with standard luminosity, is characterized by an explosion energy of 0.35 foe, significantly lower than the explosion energies of standard SNe IIP and between those of faint SNe and the ones of intermediate-luminosity objects. Also the ejected mass is small and has the lowest value of the sample of SNe IIP reported in Table \\ref{tab_systematics}. On the other hand, the ratio $E\/M_{ej}$, the radius at the explosion and the \\chem{56}{Ni} mass are similar to those of standard SNe IIP, explaining the normal luminosity of this object. \nAnother exception is SN 2009ib \\citep[see also][]{takats15}, an intermediate-luminosity object with \\chem{56}{Ni} mass closer to that of normal SNe IIP, explosion energy slightly higher than the one of the other intermediate-luminosity objects and ejected mass among the highest of the sample of SNe IIP reported in Table \\ref{tab_systematics}. However the ratio $E\/M_{ej}$ is very close to that of the other intermediate-luminosity objects, explaining the intermediate luminosity of this SN. Other minor outliers are: SN 2008bk with its relatively large radius at explosion and SN 2013ej with a value of \\chem{56}{Ni} mass closer to that of intermediate-luminosity objects and an ejected mass similar to that of faint SNe.\\par\n\n\n\\begin{figure}\n \\includegraphics[angle=-90,width=85mm]{.\/P15_fig05.ps}\n \\caption{Correlation between the plateau (pseudo-)bolometric luminosity and the ratio $E\/M_{ej}$ for the SNe IIP reported in Table \\ref{tab_systematics}. The plateau luminosity is measured at 50 days after the explosion and derived by interpolating the data shown in Figure \\ref{fig:bol}. Its errorbars are coincident with the values reported in Table \\ref{tab_systematics} which were inferred from the observations. The errorbars on the $E\/M_{ej}$ ratios are estimated by propagating the uncertainties on $E$ and $M_{ej}$, adopting a value of 30\\% for the relative error of $E$ and 15\\% for that of $M_{ej}$ (cf.~Sections \\ref{error} and \\ref{single_sne}).\n\\label{fig:correl1}}\n\\end{figure}\n\\begin{figure}\n \\includegraphics[angle=-90,width=85mm]{.\/P15_fig06.ps}\n \\caption{Same as Fig.~\\ref{fig:correl1}, but for the correlation between the \\chem{56}{Ni} mass and the ratio $E\/M_{ej}$. The errorbars on the \\chem{56}{Ni} masses are the value reported in Table \\ref{tab_systematics}, inferred from the observations. The errorbars on the $E\/M_{ej}$ ratios are evaluated as described in the caption of Fig.~\\ref{fig:correl1}.\n \\label{fig:correl2}}\n\\end{figure}\n\n\n\n\\section{Summary and  further comments}\n\\label{summary}\n\nIn order to improve our knowledge of the real nature of the progenitors of LL SNe IIP, we made radiation-hydrodynamical models of the well-studied SNe 2003Z, 2008bk and 2009md members of this sub-group of explosive events. We used the same well-tested approach applied to several other observed SNe (e.g.~2007od, 2009bw, 2009E, 2009N, 2009ib, 2012A, 2012aw, 2012ec and 2013ab; see \\citealt{inserra11,inserra12}, \\citealt{pasto12}; \\citealt{takats14,takats15}; \\citealt{tomasella13}; \\citealt{dallora14}; \\citealt{barbarino15}; and \\citealt{bose15}, respectively). In this approach, the SN progenitor's physical properties at explosion (namely the ejected mass $M_{ej}$, the progenitor radius at the time of the explosion $R$ and the total explosion energy $E$) are constrained by modelling the bolometric light curve, the evolution of line velocities and the temperature of the photosphere, and performing a simultaneous $\\chi^2$ fit of the model calculations to these observables.\\par\n\nThe inferred parameters describing the SN progenitors and their ejecta for SNe 2003Z, 2008bk and 2009md are fully consistent with low-energy explosions of red supergiant stars with relatively low mass, although the value of $R$ could also suggest yellow supergiant stars as the progenitors of SNe 2003Z and 2009md. The best-fitting model parameters inferred for SNe 2008bk and 2009md may also be consistent with the explosion of super-asymptotic giant branch stars with initial masses close to the upper limit of the mass range typical of this class of stars.\\par\n\nAssuming a mass of $\\sim$ 1.3-2 $M_\\odot$\\, for the compact remnant and a ``standard'' (i.e.~not enhanced by rotation) mass loss during the pre-SN evolution, we estimate that the progenitor masses on the ZAMS are in the range  $\\sim 13.2$-$15.1$$M_\\odot$\\, for SN 2003Z and in the range $\\sim 11.4$-$12.9$$M_\\odot$\\, for SNe 2008bk and 2009md. The latter two estimates agree with those based on direct observation of the progenitors.\\par\n\nSince these results were obtained in the same way as those for SNe 2005cs, 2008in, 2009N, 2009ib and 2012A, we can conduct a comparative study on this sub-group of SNe IIP. The main findings of this comparative analysis can be summarized as follows:\n\\begin{itemize}\n \\item the progenitors of faint SNe IIP are slightly less massive and experience less energetic explosions than the progenitors of the intermediate-luminosity objects, even though both faint SNe IIP and intermediate-luminosity objects originate from low-energy explosions of red (or yellow) supergiant stars of low-to-intermediate mass;\n \\item some faint SNe IIP may also be explained as electron-capture SNe involving massive super-asymptotic giant branch stars, although the existence of such explosive events is still not completely proven;\n \\item LL SNe IIP form the underluminous tail of the family of SNe IIP where the main parameter ``guiding'' the distribution seems to be the ratio $E\/M_{ej}$. \n\\end{itemize}\n\nAdmittedly, our sample is still too small to draw final conclusions. For this reason, other studies based on a larger sample of LL SNe IIP, including also more extreme events as SNe 1999br-like, are needed to confirm these findings. Moreover it should be useful to further check the results performing the radiation-hydrodynamical modelling in a ``self-consistent'' way, that is to say using numerical calculations which include the SN explosion and the explosive nucleosynthesis, and that start from pre-SN models evaluated by means of stellar evolution codes \\citep[see][for further details]{PZ11,PZ12}.\\par\n\nAlthough none of the LL SN IIP in our sample appears to be well modelled with massive ejecta and\/or explained in terms of a fall-back SN, at present we cannot rule out that a minor fraction of their progenitors may be more massive than $\\sim$ 15$M_\\odot$\\, and\/or undergo significant fall-back. While a larger sample of LL SNe IIP is necessary to draw any firm conclusion, recent one-dimensional hydrodynamical simulations of neutrino-driven SNe \\citep[][]{ugliano12,ertl16,sukhbold16} indicate that there is no monotonic progenitor mass dependence of the properties of core-collapse SNe. More specifically, for certain progenitor structures at explosion, not even particularly massive stars (15$M_\\odot$ $\\lesssim$ initial mass $\\lesssim$ 40$M_\\odot$) could lead to direct collapse and the formation of a black hole \\citep[see][]{OO13,PT15}. \nThus, it is possible that extreme and comparatively rarer events with very low ejected \\chem{56}{Ni} and explosion energy may hide among LL SNe IIP perhaps with more somewhat different observational properties and be explained in terms of almost failed explosions of not-particularly-massive stars undergoing significant fall-back, as suggested earlier by \\citet[][]{zampieri03}. \nFor example, in the simulations of \\citet[][]{ugliano12} and \\citet[][]{sukhbold16}, fall-back SNe occur in only a few cases for progenitor stars with initial masses in the range $\\sim$ 25-40$M_\\odot$\\, and, in these cases, the explosion properties (i.e.~explosion energy, ejected mass, and amount of \\chem{56}{Ni}) seem to be qualitatively similar to those of observed LL SNe IIP, albeit somewhat more extreme. However, as observed by the same authors \\citep[see e.g.][]{ugliano12}, the results of such simulations must be used with caution for a direct comparison with the observed properties of LL SNe IIP, because 1) they are based on sets of progenitor models which, even for similar initial masses, exhibit large structural variations that may not be completely realistic, and 2) the simulations do not consider multidimensional effects that can play a critical role in the core-collapse SN mechanism \\citep[see also][]{ertl16}. \nSo additional multidimensional hydrodynamical studies focused on the progenitor-explosion and progenitor-remnant connections should be performed and compared to a sufficiently extended sample of LL SNe IIP before drawing any final conclusion on the possible occurrence of fall-back in some LL SNe IIP. \nObservationally, a statistically sound test on the existence of progenitors having significant fall-back will come by searching for failed SNe \\citep*[see e.g.~][]{kochanek08,gerke15} and from larger numbers of direct progenitor detections. If a stringent mass progenitor upper limit of $\\sim$ 15$M_\\odot$\\, will be established, then this would severely limit the occurrence of such a process. In such a case, either successful and completely failed explosions form two ``final states'' separated by a sharp transition, the outcome of which depending on fine details of the internal structure of their progenitors, or almost failed explosions are not seen because they are typically too weak to be detected in present surveys.\n\n\n\n\\section*{Acknowledgments}\nM.L.P.~acknowledges the financial support from INAF-OAPA and CSFNSM. AP, SB, EC, MT are partially supported by the PRIN-INAF 2014 project ``Transient Universe: unveiling new types of stellar explosions with PESSTO''. We thank an anonymous referee for his\/her valuable comments and suggestions that improved our manuscript.\n\n\\bibliographystyle{aa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\nIn many applications such as rocket engines, imperfectly-expanded supersonic\njets are often accompanied by a powerful emission of tonal sounds. Such tones\nare commonly referred to as jet screech. In addition to the intensified noise\nemission, it may also lead to disastrous structure damage because of sonic\nfatigue. It is, therefore, of practical importance to understand the mechanism of\nsupersonic jet screech and to devise effective ways to suppress these screech\ntones. \n\nScreech occurs in imperfectly-expanded supersonic jets, and such jets are often\ncharacterized by quasi-periodic shock cells and are complex in nature. Jet screech was first\ndiscovered in the experiment conducted by Powell in the 1950s. In his pioneering\nwork,~\\citet{19533Powell} proposed the well-established feedback loop\nwhich consists of four stages, i.e. the instability growth in jets, the\ninteractions between shock and instability waves, the acoustic waves\npropagating upstream, and the receptivity of the shear layer at the nozzle lip.\nPowell proposed the phase and gain conditions which must be satisfied to sustain the feedback loop. For the phase condition, the frequency of the fluctuation $f$ is supposed to close the feedback loop as\n\\begin{equation}\n    \\frac{N}{f}=\\frac{d}{U_{c}}+\\frac{d}{c^{*}}+\\Psi,\n    \\label{phase_condition}\n\\end{equation}\nwhere $d$ denotes the distance between the nozzle lip and the sound source, \n$U_{c}$ is the average velocity of the instability waves travelling downstream, $c^{*}$ is the speed of sound propagating upstream,\n$\\Psi$ represents an additional phase delay, and $N$ is an integer. For the gain condition, the gain from each of the four stages must satisfy\n\\begin{equation}\n    Q\\eta_{s}\\eta_{u}\\eta_{r}\\geq1,\n\\end{equation}\nwhere $Q$ denotes the gain associated with the growth of the instability waves, and $\\eta_{s}$, $\\eta_{u}$ and $\\eta_{r}$ represent the efficiencies of energy transmission in the last three stages, respectively. The resonance conditions were then reconsidered by examining the energy exchange between the instability and acoustic waves at the sound source location and the nozzle exit~\\citep{NewC_3}. In the recent work, these conditions~\\citep{NewC_3} have been rewritten in terms of magnitude and phase conditions, details of which can be found in~\\citet{NewC_2} and~\\citet{NewC_1}.\n\nTwo important characteristics of jet screech are widely studied over the past few decades. The first is the screech frequency. Considering the feedback enhancement during the\nloop, Powell proposed that the screech frequency $f$ can be calculated via \n\\begin{equation}\nf=\\frac{U_{c}}{s(1+M_{c})},\n\\label{equ_screech f}\n\\end{equation}\nwhere $s$, $U_{c}$, and $M_{c}$ are the shock cell spacing, the convection velocity and the \\adddd{convective} Mach number of the instability waves, respectively. Subsequently,~\\citet{1986Tam_Proposed} proposed the weakest link theory suggesting the screech as the limit of the broadband shock-associated noise~(BBSAN) when the observer angle approaches $180^\\circ$. In 1999, \\citet{Panda_standingwave} discussed the link between screech and hydrodynamic-acoustic~(HA) standing waves, and a new formula was developed. In early measurements~\\citep{19533Powell}, it was found that the screech frequency experienced abrupt changes as the inlet pressure increased.\nThis frequency jumping phenomenon is commonly referred to as mode staging, and four different stages, i.e. stages A, B, C, and D were observed by Powell, among which the stage A can be further divided into two stages named $\\rm{A}_{1}$ and $\\rm{A}_{2}$~\\citep{M.Merle}. \n\\add{It was found that the azimuthal mode of both sound and instability waves changes as the mode staging occurs,} which shows a strong sensibility to the facility and initial conditions~\\citep{1969V.M.ANUFRIEV, E.GUTMARK1990, Panda_standingwave_2}, and the switch from one mode to another is nearly immediate~\\citep{Nagel_modestaging}. Despite different stages show different characteristics, it is interesting to note that they can appear simultaneously in one jet flow~\\citep{1996AIAA_Raman}.\nTo interpret mode staging, \\citet{Tam_and_shen} suggested that it was the neutral acoustic waves, rather than free acoustic waves, that complete the feedback loop in $\\rm{A}_{2}$ \nand B modes. \\add{Recent works~\\citep{Gojon_modeStaging,edgington_modestaging,Xiang-Ru_2020,NewC_1} showed that both the $\\rm{A}_{1}$ and $\\rm{A}_{2}$ modes were closed by the\nneutral acoustic waves~(or guided-jet modes).}\n\\citet{X.D.Li} used the original phase condition shown in~(\\ref{phase_condition}) and inferred the value of $N$ in their numerical study. They showed that $N$ differed across various stages, and when  this difference was considered the prediction  was in very good agreement with the experimental data. However, the mechanism behind this mode transition and its high sensitivity to the initial conditions are yet to be clarified. \\add{In addition, it was found that nonlinearity could arise during the mode staging in circular jets~\\citep{nonlinear_in_staging} and  screech tones were not independent but were instead nonlinearly phased locked to each other in rectangular jets~\\citep{1997_POF_SHWalker}, which increases difficulties for its modeling.}\n\nIn addition to the screech frequency and mode staging, noise directivity is the second characteristic that has been widely studied. \nIt was found that the\nacoustic radiation at the fundamental frequency appeared strongest in the upstream\ndirection, whereas at the harmonic frequency there was a strong beaming to the\nside of the jet~\\citep{19533Powell}. To explain this, Powell proposed the monopole array theory,\nwhich was generally in good agreement with experiment results. The\ndirectivity of the fundamental tone and its harmonics in supersonic round nozzles were measured by~\\citet{1983Norm}, the results of which were compared with the monopole\narray theory when nine monopoles with a parabolic intensity\ndistribution were considered. One particularly interesting observation was that the strongest emission appeared somewhere near $150^\\circ$, not $180^\\circ$ to the downstream jet axis. A quick decay\noccurred when the observer angle approached $180^\\circ$, which could not be\npredicted by Powell's model. As a matter of fact, if all the sound sources were\nmonopoles and the frequency of the fundamental tone was obtained by~(\\ref{equ_screech f}), the acoustic radiation would be the strongest at $180^\\circ$.\nFollowing Norum's idea, the directivity pattern of three equal-spaced monopoles of various intensities was studied by~\\citet{M.Kandula}. It seemed that this variation did not affect the location of the main directivity lobe. The temperature influence on the directivity pattern~\\citep{heateffect} was also investigated in round jets, but no significant difference was found from the unheated one. In the case of rectangular nozzles, the screech problem may become more complicated compared to axisymmetric round jets. However, this problem can be greatly simplified when the rectangular nozzle is of high aspect ratios, in which case it reduces to a two-dimensional problem. Numerical simulations~\\citep{numerical_directivity_of_rectangular} and experiments~\\citep{1997_POF_SHWalker} were conducted to study the directivity patterns of the screech and its harmonics in rectangular jets of high aspect ratios and the results were similar to that of~\\citet{19533Powell}. Recently, \\citet{2014TAM} considered another nonlinear interaction\nmechanism between shock, instability, and acoustic waves. They then proposed a model to predict the lobe\nposition in the directivity patterns.\nThe result was in good agreement with the experimental data~\\citep{1983Norm} at harmonic frequencies, but appeared less so for the fundamental tone, in particular for the lobe position.\n\nAs argued by Powell, four stages were involved in the screech cycle. Among the four stages, it is believed that the interaction between the shock and instability waves plays a critical role in understanding the physics \\adddd{of screech}. Not only because this interaction produces sound that is directly measurable, but also because it is the key to understand the noise generation mechanism. Despite its importance, not many theoretical models are proposed to predict the interaction. The reason is in part due to the complex flow nature present in the interaction~\\citep{TAmaning}, especially when the shock waves are intense in highly underexpanded and overexpanded jet flows. ~\\citet{1973Harper} used Powell's phased-array model to study the interaction between the disturbance in jet shear\nlayers and the shock cells. The frequency of the emitting sound was obtained. Subsequently,~\\citet{1988Tam} developed a shock cell model composed \nof time-independent waveguide modes~\\citep{1985Tam}. The turbulence structures were modeled using a  noise initiated at the shear layer at the nozzle lip. The weak interaction between these two components then gave rise to the sound field. \\add{Although the sound generated by shock-vortex interaction was analytically studied in the first part of the work, it was remarked by the author that~\\citep{1988Tam} \"enormous amounts of numerical computations are\nrequired\" for practical calculations. Thus, \"a model source function\" was used instead in light of the extreme complexity of the practical evaluation of the formulation to calculate the directivity patterns of BBSAN}.~\\citet{2005Lele} further developed Tam's theory. Based on the method proposed by~\\citet{1952lighthill}, he used the wavepacket model to describe the instability waves initialized by the white noise, and a vortex sheet model was utilized to calculate the shock cell structure. These two components were inserted into the Helmholz equation as the source term. The sound field was obtained by integration. This model was used in subsequent numerical simulations~\\citep{2019Wong}. However, the sound sources in the above-mentioned models were obtained by a simple combination of the shock and instability waves. A correct source term directly from governing equations would be more desirable.~\\add{A somewhat different approach to model the sound generation is the so-called shock leakage mechanism. It was proposed by~\\citet{SKLele_TAManning_shockleakage, TAmaning}, and theoretically developed by~\\citet{2003Suzeki} and~\\citet{KS_TAManning_shockleakage}. Recently, it was experimentally \nobserved by~\\citet{edgington_shockleakage}. In addition, a very recent work~\\citep{absolute_instability} numerically studied the linear stability characteristics of shock-containing jet flows, where the shock was assumed to be of small amplitude and a sinusoidal form. It was found that the characteristics of the instability waves in shock-containing jets are different from those in shock-free supersonic jets, and a new interpretation of screech was proposed based on this observation.}\n\nIn 1994,~\\citet{1995Kerchen} developed \\adddd{an analytical} shock\ninstability-wave interaction model for 2D planar vortex sheet flows. The source term was obtained directly from the governing equations. One shock cell was considered to interact with the instability waves near the vortex sheet. However, it\nwas found that the radiation field peaked at $48^\\circ$ to the downstream jet axis, which contradicted experimental observations. Despite of numerous attempts, an analytical and quantitative study of the interaction\nbetween the shock and instability waves, which is capable of predicting not only screech\nfrequencies, but also directivity patterns of screech tones,\nappears yet to be seen. This paper aims to\ndevelop such a model to predict the sound arising from the interaction\nbetween shock and instability waves. The model follows the\nasymptotic expansion method proposed by~\\citet{1995Kerchen}, but a more realistic jet and shock cell structures are considered. \nThis paper is structured as follows. Section~\\ref{sec:types_paper} presents a detailed analytical derivation of the model, while section~\\ref{section:results and discussion} shows the prediction of the screech frequency and the directivity patterns of the fundamental tone and its\nharmonics. The near-field pressure and noise generation mechanism are subsequently discussed. \nConclusions are presented in section~\\ref{section:conclusion}.\n\n\\section{Analytical formulation}\n\\label{sec:types_paper}\n\\subsection{The interaction model}\n\\label{subsection:base assumptions}\nTo enable analytical progression, we start with a vortex sheet model. As shown\nin figure~\\ref{fig:example1}, the coordinate axes $(x^\\prime,y^\\prime)$ are\nchosen to be parallel and perpendicular to the nozzle centreline, respectively. Here $D$ is the jet height~(note that $D$ is generally not equal to the height of the nozzle, and we take the height of the fully-expanded jet as the base flow height~\\citep{tam_diameter}). $U_{1}$ is the jet velocity at the nozzle exit plane, while $U$ is\nthe velocity of the fully-expanded jet after exiting from the nozzle. The fully-expanded base flow described in figure~\\ref{fig:example1} takes the form\n\\begin{equation}\n  \\boldsymbol{u}_{0}= \n    \\begin{cases}\n      0, & |y^\\prime|>D\/2 \\\\\n      U \\mathbf{e}_{x^\\prime}, &  |y^\\prime|\\leq D\/2,\n  \\end{cases} \n\\end{equation}\nwhere $\\mathbf{e}_{x^\\prime}$ is the unit vector in the $x^\\prime$ direction. We assume that the shock and\ninstability waves are  of small amplitudes and can be linearized around the base flow and described by\nlinear theories. Of course, it is known that the interaction between the shock\nand instability waves primarily occurs several shock cells downstream from the\nnozzle exit~\\citep{1994Suda,S.kaji,Sources}. At these locations, instability waves are\nlikely to grow to a significant amplitude where nonlinear effects become\nimportant and the instability waves may start to saturate and even decay.\nHowever, it is known that linear theories can predict the wavelength of these\nlarge coherent structures well beyond the linear stage~\\citep{Crow_Champion, 2013_annual_rev,2021Edington}. We may, therefore, use linear stability analysis to determine the\nwavelength (hence convection velocity) of instability waves, which are\nparticularly important for the generation of screech tones. The linear growth\nrate calculated describes the early evolving of the instability wave and its role is\ndiscussed separately in subsequent modelling. The interaction between the shock and instability waves several shock-cells downstream are likely to be nonlinear in strongly underexpanded jets. However, a linear interaction model may suffice to describe the interaction between weak shock and instability waves. Besides, similar to the successful prediction of the wavelength and convection velocities of large coherent structures by linear theories, a linear theory may still possess the many essential features of a nonlinear jet screech. We therefore start with a linear interaction between the shock and instability waves. With these\nassumptions we start to seek an analytical model describing noise generation\ndue to the interaction between shock and instability waves.\n\n\\begin{figure}\n   \\centering\n   \\includegraphics[width = 0.65\\textwidth]{figure1.eps}\n   \\caption{The schematic of the vortex-sheet flow configuration and Cartesian\n   coordinates. The origin is fixed at the centre of the nozzle while $x^\\prime$ and\n   $y^\\prime$ represent the streamwise and cross-flow coordinates, respectively.}\n\\label{fig:example1}\n\\end{figure}\n\nGiven the fact that the Reynolds number is high and the instability waves are\nessentially inviscid, we start from the Euler equations shown as follows\n\\begin{equation}\n    \\frac{\\mathrm{D} \\rho}{\\mathrm{D} t^\\prime } + \\add{\\rho}\\bnabla \\cdot \\boldsymbol{u}=0,\n\\end{equation}\n\\begin{equation}\n    \\rho \\frac{\\mathrm{D}\\boldsymbol{u}}{\\mathrm{D} t^\\prime}= -\\bnabla p,\n\\end{equation}\n\\begin{equation}\n    \\frac{\\mathrm{D} s}{\\mathrm{D} t^\\prime} = 0,\n\\end{equation}\nwhere $t^{\\prime}$ denotes time, $\\boldsymbol{u}=(u,v)$ the velocity, $p$ the\npressure, $\\rho$ the density, $s$ the entropy, and \\adddd{$\\mathrm{D}\/\\mathrm{D} t^\\prime=\\partial\/\\partial t^\\prime+ \\boldsymbol{u}\\cdot\\bnabla$}.  \nBecause the entropy increase\nacross a weak shock is a high-order small term~\\citep{1995Kerchen}, the\nisentropic condition is used here. \n\nTo determine the solution, both kinematic and dynamic boundary conditions need to be satisfied across the vortex sheet. The dynamic boundary conditions read\n\\begin{equation}\n    \\label{equP}\n\\left.p_{+}\\right|_{y^\\prime = h^{'}} = \\left.p_{-}\\right|_{y^\\prime = h^\\prime},\n\\end{equation}\nwhile the kinematic boundary condition requires\n\\begin{equation}\\label{equB1}\n\\left. v_{+}\\right|_{y^\\prime = h^\\prime} =\\left(\\frac{\\partial h^\\prime}{\\partial t^\\prime}+\\left. u_{+}\\right|_{y^\\prime = h^\\prime}\\frac{\\partial h^\\prime}{\\partial x^\\prime}\\right),\n\\end {equation}\n\\begin{equation}\\label{equB2}\\left.\nv_{-}\\right|_{y^\\prime = h^\\prime} =\\left(\\frac{\\partial h^\\prime}{\\partial t^\\prime}+\\left. u_{-}\\right|_{y^\\prime = h^\\prime}\\frac{\\partial h^\\prime}{\\partial x^\\prime}\\right),\n\\end{equation}\nwhere $(\\cdot)_+$ and $(\\cdot)_-$ represent the quantities outside and within the jet flow respectively, and $h^\\prime$ denotes the displaced $y^\\prime$ of the vortex sheet.\n\nFollowing \\citet{1995Kerchen}, we use $\\delta$ and $\\epsilon$ to denote the strength of the shock and instability waves, respectively. These two parameters are assumed to be of the small magnitude representing small perturbation compared to the mean jet flow. \\add{Note that although shock waves can be intense in strongly imperfectly-expanded jets, they could also be infinitesimally weak when the wave angle \\adddd{approaches the Mach angle}~\\citep{Anderson}. In this model, we consider a slightly imperfectly expanded jet, in which case the shock-associated perturbations can be relatively weak, thus a linear model may be developed~\\citep{tam_machwave,1985Tam,  1995Kerchen}}. The velocity field can be expanded\nusing these two parameters as \\citep{1995Kerchen}\n\\begin{equation}\n   \\boldsymbol{u}=\\boldsymbol{u}_{0}\n   +\\delta \\boldsymbol{u}_{m}\n   +\\epsilon\\boldsymbol{u}_{v}\n   +\\delta^{2}\\boldsymbol{u}_{m2}\n   +\\epsilon^{2}\\boldsymbol{u}_{v2}\n   +\\delta\\epsilon\\boldsymbol{u}_{i}+...\\ ,\n\\end{equation}\nwhere $\\boldsymbol{u}_{0}$ is the mean \\adddd{velocity}, $\\boldsymbol{u}_{m}$\nrepresents the linear perturbation due to shock waves, and $\\boldsymbol{u}_{v}$\nis the linear unsteady perturbation due to instability waves. For higher\norders, the $\\delta^2$ term represents nonlinear steady modification of the\nshock waves and is independent of time. As we are interested in sound\ngeneration, this term can be neglected. The second-order term\n$\\epsilon^2$ represents the nonlinear correction to the linear stability waves, and\nwe neglect it by only considering the leading-order contribution from the\n$\\epsilon$ term. The omission of this term follows immediately if $1 \\gg \\delta\n\\gg \\epsilon$ is assumed, which implies $\\epsilon^2$ is the highest-order term\nof the expansion. The $\\delta \\epsilon$ term represents the interaction due to\nshock and stability waves, as a result of which sound is generated. \\add{Physically, this entails that the shock and instability waves that interact to produce sound in realistic flows may be approximated by the linear shock and instability solutions to the base flow.} The\npressure, density, and vortex sheet displacement have similar expansions, i.e.\n\\begin{equation}\n   p=p_{0}+ \\delta p_{m} +\\epsilon p_{v}+\\delta^{2}p_{m2}+\\epsilon^{2}p_{v2}+\\delta\\epsilon p_{i}+...\\ ,\n\\end{equation}\n\\begin{equation}\n   \\rho=\\rho_{0}+ \\delta\\rho_{m} +\\epsilon\\rho_{v}+\\delta^{2}\\rho_{m2}+\\epsilon^{2}\\rho_{v2}+\\delta\\epsilon\\rho_{i}+...\\ ,\n\\end{equation}\n\\begin{equation}\n   h^\\prime=h^\\prime_{0}+ \\delta h^\\prime_{m} +\\epsilon h^\\prime_{v}+\\delta^{2}h^\\prime_{m2}+\\epsilon^{2}h^\\prime_{v2}+\\delta\\epsilon h^\\prime_{i}+...\\ .\n\\end{equation}\n\nThe \\adddd{mean} temperature across the jet flow is\ndifferent, while the \\adddd{mean} pressure $p_{0}$ \\adddd{remains identical} in both regions. If a perfect gas is assumed, then $c_0=\\sqrt{\\gamma p_0\/\\rho_0}$ can be used to calculate the \\adddd{mean} speed of\nsound inside and outside the jet, and it is straightforward to show that\n$\\rho_{0-}c_{0-}^{2}=\\rho_{0+}c_{0+}^{2}$. We define $M_{-}=U\/c_{0-}$ and\n$M_{+}=U\/c_{0+}$ to denote the \\adddd{mean} Mach numbers inside and outside the jet,\nrespectively. Substituting all the expansions to the Euler equation and\nboundary conditions, and collecting the terms $O(\\delta)$, $O(\\epsilon)$ and\n$O(\\delta\\epsilon)$, we obtain the equations governing the shock, instability waves and\ntheir interaction, respectively.\n\n\\subsection{The shock model}\nThe $O(\\delta)$ terms in the governing equations representing the linear perturbation induced by the shock wave satisfy\n\\begin{equation}\n    \\frac{\\mathrm{D}_{0}p_{m}}{\\mathrm{D} t^{'}}+\\rho_{0}c_{0}^{2}\\bnabla\\cdot\\boldsymbol{u}_{m}=0,\n\\end{equation}\n\\begin{equation}\n    \\rho_{0}\\frac{\\mathrm{D}_{0}\\boldsymbol{u}_{m}}{\\mathrm{D} t^{'}}=-\\bnabla p_{m},\n    \\label{equ_shock_momentum}\n\\end{equation}\nwhere \\adddd{$\\mathrm{D_0}\/\\mathrm{D} t^\\prime$ denotes $\\partial\/\\partial t^\\prime+ \\boldsymbol{u}_0\\cdot\\bnabla$}. These two equations can be combined to yield\n\\begin{equation}\n    \\bnabla^{2}p_{m}-M^{2}\\frac{\\partial^{2}p_{m}}{\\partial x^{\\prime2}}=0.\n    \\label{equ:waveequation}\n\\end{equation}\n\\adddd{Equation (\\ref{equ:waveequation}) reduces to the Laplace equation outside the jet, since the mean flow velocity is zero there. Within the jet, by defining $\\beta=\\sqrt{M_{-}^{2}-1}$, we can rewrite~(\\ref{equ:waveequation})\nto be}\n\\begin{equation}\n    \\frac{\\partial^{2}p_{m-}}{\\partial y^{\\prime2}}-\\beta^{2}\\frac{\\partial^{2}p_{m-}}{\\partial x^{\\prime2}}=0.\n    \\label{equ:waveequation2}\n\\end{equation}\n\nWe see that the linear pressure field \\adddd{within the jet} induced by weak shock waves satisfies the\nwave equation. Such an equation can admit many solutions subject to different\nboundary conditions; for illustration purposes, \\citet{1995Kerchen} used a step\nfunction for his single planar vortex sheet. In order to have a much more realistic shock cell structure, we use\nPack's model~\\citep{1950Pack} in this paper. Note that we use the jet height $D$ here, instead of the nozzle height, to nondimensionalize the streamwise and cross-flow coordinates, i.e. $x=x^\\prime\/D$, $y=y^\\prime\/D$. The velocity potential perturbation \\adddd{within the jet} induced by the shock waves can be\nwritten as~\\citep{1950Pack, tam_diameter}\n\\begin{equation}\n    \\phi=\\sum_{j=1}A_{j}\\cos(\\beta a_{j} y)\\sin(a_{j} x),\n    \\label{Pack's model}\n\\end{equation}\nwhere $j$ represents the $j$th mode of the shock wave. Two parameters $A_{j}$\nand $a_{j}$ in the equation above \\adddd{are}\n\\begin{equation}\n    A_{j}=(-1)^{j}\\frac{4\\beta}{\\pi^{2}}\\frac{\\mathcal{U}}{(2j-1)^2},\n    \\label{equ:A}\n\\end{equation}\n\\begin{equation}\n    a_{j}=\\frac{(2j-1)\\pi}{\\beta}.\n\\end{equation}\nThe constant $\\mathcal{U}$ in the above equation is defined by  \\add{$\\mathcal{U}=U_{1}-U.$} We see from~(\\ref{equ:A}) that the amplitude of this potential function decreases quickly as the\nmode number increases. In light of the linearity of the model, it is convenient\nto consider each mode separately. In the present study, we focus on the\nleading-order mode. Higher-order terms can be easily included at a later stage\nshould necessity \\add{arise}. For the leading-order mode, the shock wave is\nperiodically distributed along the streamwise direction with a shock spacing \\adddd{$s=2\\pi\/a_{1}.$} In what follows, the subscripts in parameters $A_{1}$ and $a_{1}$ are omitted for\nclarity.\n\n\\add{The corresponding velocity, pressure, and the vortex sheet deflection at the boundary of the jet flow are shown in Appendix~\\ref{appA}.}\n\\subsection{The instability waves} \n\\label{sec:filetypes} \nSimilar to the derivation of the shock equations, the governing equations for\nthe instability waves can be obtained by collecting the $O(\\epsilon)$ terms,\ni.e.\n\\begin{equation}\n    \\frac{\\mathrm{D}_{0}p_{v}}{\\mathrm{D} t^\\prime}+\\rho_{0}c_{0}^{2}\\bnabla\\cdot\\boldsymbol{u}_{v}=0,\n    \\label{equ_v_continuty}\n\\end{equation}\n\\begin{equation}\n    \\rho_{0}\\frac{\\mathrm{D}_{0}\\boldsymbol{u}_{v}}{\\mathrm{D} t^\\prime}=-\\bnabla p_{v}.\n    \\label{equ_v_momentum}\n\\end{equation}\nConsidering the $O(\\epsilon)$ terms in the boundary conditions shown in\n(\\ref{equP}), (\\ref{equB1}) and (\\ref{equB2}), we see that the two matching\nconditions can be linearised to the dynamic and kinematic conditions on $y=\\pm\nD\/2$, i.e.\n\\begin{equation}\n    p_{v+}=p_{v-}, \n\\end{equation}\n\\begin{equation}\n        v_{v+}=\\frac{\\partial h^\\prime_{v}}{\\partial t^\\prime},\\quad v_{v-}=\\frac{\\partial h^\\prime_{v}}{\\partial t^\\prime}+U\\frac{\\partial h^\\prime_{v}}{\\partial x^\\prime},\n        \\label{equ_v_continuity of displacement}\n\\end{equation}\nwhere $h^\\prime_{v}$ denotes the disturbed height of the vortex sheets due to the\ninstability waves. Since the initial base flow is irrotational and\ninviscid both inside and outside of the vortex sheet, the linear perturbation\ncan be expressed as a velocity potential $\\phi_{v}$. The continuity\nequation~(\\ref{equ_v_continuty}), and the momentum equation~(\\ref{equ_v_momentum}), can\nbe combined to yield \n\\begin{equation}\n    \\bnabla^{2}\\phi_{v}-\\frac{1}{c_{0}^{2}}\\frac{\\mathrm{D}_{0}^{2}\\phi_{v}}{\\mathrm{D} t^{\\prime2}}=0.\n\\end{equation}\nSimilarly the dynamic boundary condition reduces to\n\\begin{equation}\n\\rho_{0+}\\frac{\\partial \\phi_{v+}}{\\partial t^\\prime}=\\rho_{0-}\\left[\\frac{\\partial \\phi_{v-}}{\\partial t^\\prime}+U\\frac{\\partial \\phi_{v-}}{\\partial x^\\prime} \\right].\n\\end{equation}\nFor the kinematic boundary condition, the two equations shown in\n(\\ref{equ_v_continuity of displacement}) can be combined to yield\n\\begin{equation}\n    \\left(\\frac{\\partial}{\\partial t^\\prime}+U\\frac{\\partial}{\\partial x^\\prime}\\right)\\frac{\\partial \\phi_{v+}}{\\partial y^\\prime}=\\frac{\\partial^{2}\\phi_{v-}}{\\partial t^{'} \\partial y^\\prime}.\n\\end{equation}\n\nWith temporal and spatial harmonic assumptions, the perturbations induced by the\ninstability waves have the form of\n   \\add{$ \\mathrm{e}^{{\\mathrm i}(\\alpha^\\prime x^\\prime-\\omega^\\prime t^\\prime)},$} where the spatial wavenumber $\\alpha^\\prime$ is complex and the eigenvalue with a \nnegative imaginary part represents instability. We use the velocity of the fully-expanded jet flow $U$ to nondimensionalize\nother variables, for instance the nondimensional time, frequency and wavenumber\nare \\add{$t=t^\\prime U\/D, \\omega=\\omega^\\prime D\/U, \\alpha=D\\alpha^\\prime$}\\adddd{, respectively.} \n\n\nCombining the governing equations and boundary conditions, and noticing that the base flow outside the jet is 0, we find that the velocity potential can be expressed as\n\\begin{equation}\n    \\phi=U D \\mathrm{e}^{i(\\alpha x-\\omega t)}\\times\n    \\begin{cases}\n      \\frac{1}{M_{+}^{2}}\\mathrm{e}^{-m_{+}y}, & y>\\frac{1}{2} \\\\[2pt]\n      \\frac{1}{M_{-}^{2}}\\frac{\\omega}{\\omega-\\alpha}(k_{1}\\mathrm{e}^{m_{-}y}+k_{2}\\mathrm{e}^{-m_{-}y}),& y\\leq |\\frac{1}{2}|\\\\\n     \\frac{1}{M_{+}^{2}}k_{3}\\mathrm{e}^{m_{+}y},& y<-\\frac{1}{2},\n    \\end{cases} \n    \\label{equ:instabilty_potential}\n\\end{equation}\nwhere $ m_{+}=\\sqrt{\\alpha^{2}-\\omega^{2}M_{+}^{2}},\\\nm_{-}=\\sqrt{\\alpha^{2}-M_{-}^{2}(\\omega-\\alpha)^{2}}$. The branch cut is chosen such that\nthe real part of $m_{+}$ is positive. $k_{1}, k_{2}, k_{3}$ are undetermined\ncoefficients. It can be seen that both antisymmetric and symmetric modes can\nexist, corresponding to $k_{3}=- 1$ and $k_{3}=1$, respectively. Using the dynamic and kinematic boundary conditions, the dispersion\nrelations can be found to be\n\\begin{equation}\n    \\mathrm{e}^{2m_{-}}=\\frac{(\\omega^{2}m_{-}\/M_{-}^{2}-(\\omega-\\alpha)^2m_{+}\/M_{+}^{2})^{2}}{(\\omega^{2}m_{-}\/M_{-}^{2}+(\\omega-\\alpha)^2m_{+}\/M_{+}^{2})^{2}}.\n    \\label{equ:dispersion relationship}\n\\end{equation}\nFor the symmetric mode, this reduces to\n\\begin{equation}\n    {\\rm tanh}(\\frac{m_{-}}{2})\\frac{\\omega^{2} m_{-}}{M_{-}^{2}(\\omega-\\alpha)^{2}}+\\frac{m_{+}}{M_{+}^{2}}=0,\n    \\label{dispersion_sym}\n\\end{equation}\nwhere the parameters attain the following values\n\\begin{equation}\n    k_{1}=k_{2}=\\frac{\\mathrm{e}^{-\\frac{1}{2}m_{+}}}{2{\\rm cosh}(\\frac{1}{2}m_{-})}, \\quad k_{3}=1.\n\\end{equation}\nFor the antisymmetric mode, the dispersion relationship reduces to\n\\begin{equation}\n    \\frac{\\omega^{2} m_{-}}{M_{-}^{2}(\\omega-\\alpha)^{2}}+{\\rm tanh}(\\frac{m_{-}}{2})\\frac{m_{+}}{M_{+}^{2}}=0,\n    \\label{dispersion_anti}\n\\end{equation}\nwhere the three parameters take the values of\n\\begin{equation}\n    k_{1}=-k_{2}=\\frac{\\mathrm{e}^{-\\frac{1}{2}m_{+}}}{2{\\rm sinh}(\\frac{1}{2}m_{-})},\\quad k_{3}=-1.\n\\end{equation}\n\n\\add{ The corresponding pressure, velocity, and the deflection of the jet boundary due to the instability waves are shown in Appendix~\\ref{appB}}. We can see that the deflection generated by the instability wave at the upper and lower boundaries are symmetric and antisymmetric for the symmetric and antisymmetric modes, respectively, as would be expected. \n\nExperiments found that rectangular jets are capable of sustaining both symmetric\nand antisymmetric  \\adddd{oscillation} modes~\\citep{1994Suda,S.kaji}, which can be directly linked to the instability of the jet. In our analysis, both symmetric and antisymmetric instability modes can be considered. \\adddd{But for jet flows from high-aspect-ratio rectangular nozzles, the flapping mode is dominant~\\citep{2019EDINGTON} and the problem can be approximated by a 2D theory}. So in what follows only the antisymmetric mode of instability waves is considered. \n\n\n\n\\subsection{The interaction between shock and instability waves} \n\\label{subsec:sound}\nHaving obtained the shock and instability waves, we are now in a position to\nconsider the $O(\\delta\\epsilon)$ terms in the governing equations. When a\nperfect gas is assumed \\adddd{($\\rho_0 c_0^{2}=\\gamma p_0$)}, the continuity and momentum\nequations can be expressed as\n\\begin{equation}\n        \\frac{\\mathrm{D}_{0}p_{i}}{\\mathrm{D} t^\\prime}+\\rho_{0}c_{0}^{2}\\bnabla\\cdot\\boldsymbol{u}_{i}=-[\\boldsymbol{u}_m\\cdot\\bnabla p_{v}+\\boldsymbol{u}_{v}\\cdot\\bnabla p_{m}+\\gamma p_{m}\\bnabla\\cdot\\boldsymbol{u}_v+\\gamma p_{v}\\bnabla \\cdot \\boldsymbol{u}_{m}],\n        \\label{equ_sound_continuity}\n\\end{equation}\n\\begin{equation}\n        \\rho_{0}\\frac{\\mathrm{D}_{0}\\boldsymbol{u}_{i}}{\\mathrm{D} t^\\prime}+\\bnabla p_{i}=-[\\rho_{0}(\\boldsymbol{u}_m\\cdot\\bnabla \\boldsymbol{u}_{v}+\\boldsymbol{u}_{v}\\cdot\\bnabla \\boldsymbol{u}_{m})+\\rho_{m}\\frac{\\mathrm{D}_{0}\\boldsymbol{u}_v}{D t^\\prime}+\\rho_{v}\\frac{\\mathrm{D}_{0}\\boldsymbol{u}_m}{D t^\\prime}].\n        \\label{equ_sound_momentum}\n\\end{equation}\nSubstituting the momentum equations for the shock wave, i.e.\n(\\ref{equ_shock_momentum}), and the instability wave, i.e. (\\ref{equ_v_momentum}),\ninto~(\\ref{equ_sound_momentum}), we have\n \\begin{equation}\n      \\rho_{0}\\frac{\\mathrm{D}_{0}\\boldsymbol{u}_{i}}{\\mathrm{D} t^\\prime}+\\bnabla p_{i}=-[\\rho_{0}\\bnabla(\\boldsymbol{u}_{m}\\cdot\\boldsymbol{u}_{v})+\\frac{1}{\\rho_{0}c_{0}^{2}}\\bnabla(p_{m}p_{v})].\n      \\label{18}\n  \\end{equation}\n   The interaction field is also irrotational, i.e. $u_{i}=\\bnabla \\phi_{i}$.\n   Then~(\\ref{18}) can be integrated to obtain\n   \\begin{equation}\n      p_{i}= -\\rho_{0}\\frac{\\mathrm{D}_{0}\\phi_{i}}{\\mathrm{D} t^\\prime}-\\rho_{0}\\boldsymbol{u}_{m}\\cdot\\boldsymbol{u}_{v}+\\frac{1}{\\rho_{0}c_{0}^{2}}p_{m}p_{v}.\n      \\label{20}\n  \\end{equation}\n  Combining~(\\ref{equ_sound_continuity}) and (\\ref{20}), and considering the momentum and\n  continuity equations for the $O(\\alpha)$ and $O(\\epsilon)$ terms, we find\n  that $\\phi_{i}$ satisfies the following inhomogeneous wave equation\n  \\begin{equation}\n      \\bnabla^{2}\\phi_{i}-\\frac{1}{c_{0}^{2}}\\frac{\\mathrm{D}_{0}^{2}\\phi_{i}}{\\mathrm{D} t^{\\prime2}}=\\frac{-1}{\\rho_{0}c_{0}^{2}}[2(\\mathbf{u}_m\\cdot\\bnabla p_{v}+\\boldsymbol{u}_v\\cdot\\bnabla p_{m})+(\\gamma-1)(p_{m}\\bnabla\\cdot\\boldsymbol{u}_{v}+p_{v}\\bnabla\\cdot\\boldsymbol{u}_{m})].\n      \\label{equ_sound_inhomogenous}\n        \\end{equation}\n        \n  Next we consider the continuity of the pressure across\n  the vortex sheet. Similar to \\citet{1995Kerchen}, on the two boundaries\n  $y^\\prime=1\/2D$ and $y^\\prime=-1\/2D$, the dynamic boundary\n  condition reduces to\n   \\begin{equation}\n      p_{i+}+h_{m}^\\prime\\frac{\\partial p_{v+}}{\\partial y^\\prime}= p_{i-}+h_{m}^\\prime\\frac{\\partial p_{v-}}{\\partial y^\\prime}+h_{v}^\\prime\\frac{\\partial p_{m-}}{\\partial y^\\prime},\n  \\end{equation}\n  while the kinematic boundary condition requires\n  \\begin{equation}\n         v_{i+}+\\frac{\\partial v_{v+}}{\\partial y^\\prime}h^\\prime_{m}=\\frac{\\partial h_{i}^\\prime}{\\partial t^\\prime}+u_{v+}\\frac{\\partial h_{m}^\\prime}{\\partial x^\\prime},\n  \\end{equation}\n  \\begin{equation}\n      v_{i-}+\\frac{\\partial v_{v-}}{\\partial y^\\prime}h^\\prime_{m}+\\frac{\\partial v_{m-}}{\\partial y^\\prime}h^\\prime_{v}=\\frac{\\partial h_{i}^\\prime}{\\partial t^\\prime}+U\\frac{\\partial h^\\prime_{i}}{\\partial x^\\prime}+u_{v-}\\frac{\\partial h_{m}^\\prime}{\\partial x^\\prime}+u_{m-}\\frac{\\partial h_{v}^\\prime}{\\partial x^\\prime}.\n  \\end{equation}\n  Note that outside the jet flow, there is no\n  perturbation due to shock waves, therefore the right-hand side of\n  (\\ref{equ_sound_inhomogenous}) vanishes. This equation degenerates to a homogeneous\n  wave equation. Using the same $U$ and $D$ to\n  nondimensionalize the velocity potential, we obtain\n  \\begin{equation}\n      \\phi_{i+}=2 U D g_{+}\\mathrm{e}^{-\\mathrm{i}\\omega t},\n      \\label{potential_outside}\n  \\end{equation}\n  \\begin{equation}\n           \\phi_{i-}=2 U D g_{-}\\mathrm{e}^{-\\mathrm{i}\\omega t},\n  \\end{equation}\n  \\add{where $g_\\pm$ are the nondimensionalized potential functions.} \n  Outside the jet flow, the base flow is uniformly 0, and (\\ref{equ_sound_inhomogenous}) reduces to\n    \\begin{equation}\n      \\frac{\\partial^{2}g_{+}}{\\partial x^{2}}+\\frac{\\partial^{2}g_{+}}{\\partial y^{2}}+\\omega^{2}M_{+}^{2}g_{+}=0.\n  \\end{equation}\n \n \n\n \n   \n \n   \n\n  \n \n \n\n  Inside the jet flow, $g_{-}$ satisfies\n    \\begin{eqnarray}\n    \\frac{\\partial^{2}g_{-}}{\\partial x^{2}}+\\frac{\\partial^{2}g_{-}}{\\partial y^{2}}+M_{-}^{2}(\\omega+\\mathrm{i}\\frac{\\partial}{\\partial x})^{2}g_{-} & = &2k_{1}\\mathrm{e}^{\\mathrm{i} \\alpha x}\\frac{A}{U}\\bigg[ \n    \\sinh(\\zeta_{1}y)(B_{1}\\cos(a x)+B_{2}\\sin(a x))\\nonumber\\\\\n    & + &\\sinh(\\zeta_{2}y)(B_{3}\\cos(a x)+B_{4}\\sin(a x)\\bigg],\n    \\label{equ_sound_inhomogenous2}\n  \\end{eqnarray}\nwhere\n\\begin{gather}\n    \\zeta_{1}=m_{-}+\\mathrm{i} a\\beta,\\nonumber\\\\\n    \\zeta_{2}=m_{-}- \\mathrm{i}a\\beta,\n\\end{gather}\n\\begin{gather}\n    B_{1}=\\frac{a\\omega}{2}\\left(\\alpha+\\mathrm{i}\\frac{a\\beta m_{-}}{\\omega-\\alpha}-\\frac{\\gamma-1}{2}(\\omega-\\alpha)M_{-}^{2} \\right),\\nonumber\\\\\n    B_{2}=\\frac{a\\omega}{2}\\left(m_{-}\\beta-\\mathrm{i}\\frac{\\alpha a}{\\omega-\\alpha}+\\frac{\\gamma-1}{2}\\mathrm{i}a M_{-}^{2} \\right),\\nonumber\\\\\n    B_{3}=\\frac{a\\omega}{2}\\left(\\alpha-\\mathrm{i}\\frac{a\\beta m_{-}}{\\omega-\\alpha}-\\frac{\\gamma-1}{2}(\\omega-\\alpha)M_{-}^{2} \\right),\\nonumber\\\\\n    B_{4}=\\frac{a\\omega}{2}\\left(-m_{-}\\beta-\\mathrm{i}\\frac{\\alpha a}{\\omega-\\alpha}+\\frac{\\gamma-1}{2}\\mathrm{i}a M_{-}^{2} \\right).\n\\end{gather}\n  \n  \n \n  \n \n\nBesides, the two boundary conditions can be reorganized as\n\\begin{eqnarray}\n        \\omega g_{+}-\\frac{M_{-}^{2}}{M_{+}^{2}}(\\omega+\\mathrm{i}\\frac{\\partial}{\\partial x})g_{-}&=&\\pm\\frac{A\\beta}{2UM_{+}^{2}}\\sin(\\frac{1}{2}a\\beta )\\cos(a x)\\bigg[\\omega\\big(2k_{1}m_{-}\\cosh(\\frac{1}{2}m_{-})\\nonumber\\\\\n        &\\pm& m_{+}\\mathrm{e}^{\\mp\\frac{1}{2}m_{+}}\\big)\n        -\\frac{\\mathrm{e}^{-\\frac{1}{2}m_{+}}}{\\omega}m_{+}\\frac{M_{-}^{2}}{M_{+}^{2}}a^{2}\\bigg]\\mathrm{e}^{\\mathrm{i} \\alpha x}\n        \\label{equ:boundary condition 1}\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n    (1+\\frac{\\mathrm{i}}{\\omega}\\frac{\\partial}{\\partial x})\\frac{\\partial g_{+}}{\\partial y}-\\frac{\\partial g_{-}}{\\partial y}&=&\\frac{ A }{2 U}\\bigg[{\\mathrm{i}}a\\beta\\sin(a x)\\big[b_{1}\\sin(\\pm\\frac{1}{2}a\\beta )+c\\beta\\cos(\\frac{1}{2}a\\beta ) \\big] \\nonumber\\\\\n    &+&\\cos(a x)\\big[ b_{2}\\sin(\\pm\\frac{1}{2}a\\beta )-c \\alpha\\cos(\\frac{1}{2}a\\beta ) \\big]\\bigg]\\mathrm{e}^{\\mathrm{i} \\alpha x},\n    \\label{equ:boundary condition 2}\n\\end{eqnarray}\nwhere the upper and lower of signs of $\\pm$ and $\\mp$ correspond to the matching conditions on $y=1\/2$ and $y=-1\/2$, respectively, and\n\\begin{gather}\n    b_{1}=\\pm\\frac{2\\alpha k_{1}\\sinh(\\frac{1}{2}m_{-} )}{M_{+}^{2}\\omega}[(\\alpha-\\omega)+\\frac{m_{+}^{2}}{\\alpha}]\\pm\\frac{\\alpha}{M_{-}^{2}}\\frac{\\omega}{\\omega-\\alpha}\\mathrm{e}^{\\mp\\frac{1}{2}m_{+}},\\\\\n        b_{2}=\\pm\\frac{2\\beta k_{1}\\sinh(\\frac{1}{2}m_{-} )}{M_{+}^{2}\\omega}[(\\alpha-\\omega)m_{+}^{2}+\\alpha a^{2}]\\pm m_{-}^{2}\\frac{\\beta}{M_{-}^{2}}\\frac{\\omega}{\\omega-\\alpha}\\mathrm{e}^{\\mp\\frac{1}{2}m_{+}},\\\\\n        c=\\frac{a}{\\omega}\\frac{m_{+}}{M_{+}^{2}}\\mathrm{e}^{-\\frac{1}{2}m_{+}}.\n\\end{gather}\n\nTo obtain the solution to these equations, Fourier \\add{transform} is used. The Fourier transforms $G_{\\pm}(\\lambda,y)$ are defined as\n    \\begin{equation}\n\tG_{\\pm}(\\lambda,y)=\\int_{-\\infty}^{+\\infty} g_{\\pm}(x,y)\\mathrm{e}^{\\mathrm{i} \\lambda x}\\mathrm{d} x,\n\t\\label{equ:fourier}\n    \\end{equation}\n\\add{where $\\lambda$ is the wavenumber in the streamwise direction.} Outside the jet flow, it is easy to find that $G$\nsatisfies\n    \\begin{equation}\n        G_{+}(\\lambda,y)=\n    \\begin{cases}    \n        D_{1}(\\lambda)\\mathrm{e}^{-\\gamma_{+}y}, &\\test{y>1\/2}\\\\[2pt]\n        D_{4}(\\lambda)\\mathrm{e}^{\\gamma_{+}y}, &\\test{y<-1\/2},\n        \\end{cases}\n    \\end{equation}\nwhere\n\\begin{equation}\n    \\gamma_{+}(\\lambda)=\\sqrt{\\lambda^{2}-\\omega^{2}M_{+}^{2}},\n\\end{equation}\nand $D_{1}$ and $D_{4}$ are two undetermined coefficients related to $\\lambda$.\n    \t\\begin{figure}\n\t\t\\centering\n\t\t\\includegraphics[width = 0.4\\textwidth]{figure2.eps}\n\t\t\\caption{The schematic of an effective source term located within one single\n\t\tshock cell. The total sound field is\n\t\tequivalent to a linear superimposition of the results from a number of\n\t\tshock cells.}\\label{fig:schematic}\n\t\\end{figure}\n\nWhen Fourier \\add{transform} is used to the source term of\n(\\ref{equ_sound_inhomogenous2}),\nwe can\nsimplify the problem by noting the periodicity of the function $\\cos(ax)$\nand  $\\sin(a x)$. For example,\n\\begin{IEEEeqnarray}{rCl}\n    &\\int_{0}^{\\infty}&\\sin(a x)\\mathrm{e}^{\\mathrm{i}(\\lambda+\\alpha) x}\\mathrm{d} x=\\sum_{j=0}^\\infty \\int_{2j\\pi\/a}^{2(j+1)\\pi\/a}\n    \\sin(a x)\\mathrm{e}^{\\mathrm{i}(\\lambda+\\alpha) x}\\mathrm{d} x\\nonumber\\\\\n    & = & \n    \\left(\\sum_{j=0}^\\infty \\mathrm{e}^{\\mathrm{i}\\left( \\alpha+\\lambda \\right) 2 j \\pi \/\n    a}\\right)\n    \\frac{1}{2}\\left(\\frac{1}{\\alpha+\\lambda+a}-\\frac{1}{\\alpha+\\lambda-a}\\right)(-\\mathrm{e}^{\\mathrm{i}(\\alpha+\\lambda)\\frac{2\\pi}{a}}+1).\n\\end{IEEEeqnarray}\nClearly, convergence problem arises when $j \\to +\\infty$ since $\\alpha$ has a\nnegative imaginary part. However, this is a difficulty resulting from\nlinearization, not inherent difficulties in the flow physics and its modelling. As\nmentioned earlier, the interaction between shock and instability waves occurs\nwhen the instability waves grow to be of sufficient amplitude, at which place\nthe nonlinear\/linear saturation or even decay begins to take place. Considering\nthat both the instability and shock waves start to decay further\ndownstream of the jet~\\citep{J.COHEN_jfm, 2013_annual_rev},  the effective interaction only takes place within a\nlimited interval \\adddd{spanning} several shock cells~\\citep{1994Suda,Sources, X.D.Li}. In other words, the summation\nonly involves a finite number of terms and therefore the convergence problem\ndoes not occur in realistic jets. In light of this, it is reasonable to only\nfocus on a limited number of shock cells in this paper, e.g. from the third to\nthe fifth \\add{according to previous studies~\\citep{1983Norm,Panda_standingwave,2014TAM,2017_jfm_sources}.} Furthermore, note that the right hand sides of~(\\ref{equ:boundary condition 1}),~(\\ref{equ:boundary condition 2}), and~(\\ref{equ_sound_inhomogenous2}) all have the common factor  $\\left(\\sum\\mathrm{e}^{\\mathrm{i}\\left( \\alpha+\\lambda \\right) 2 j\n\\pi \/ a}\\right)$ after the Fourier \\add{transform}. Hence they can be collected and accounted for later due to\nlinearity of the equation and the two boundary conditions. This means that we\nonly need to consider the interaction within one shock cell, as shown in\nfigure~\\ref{fig:schematic}, and the total interaction field would be a simple\nlinear combination from a number of shock cells. In this way, the effective\nintegration interval when Fourier \\add{transform} is applied to both the governing\nequation and boundary conditions is limited to be within one shock cell.\n\\add{Physically, this corresponds to the effective sound generated by the interaction between the\ninstability waves and one single shock cell, similar to Powell's idea of treating the effective sound as that of monopoles.}\n\n\n\\addd{Note although we limit the integration interval to be within one shock cell, we do not imply that such an effective source is physically localized. The source term on the right side of (\\ref{equ_sound_inhomogenous}) has a periodic nature by construction, but the bounds of integration may be across one or several shock cells due to the linearity of (\\ref{equ_sound_inhomogenous}). This is equivalent to decomposing the problem into several sub-problems, each of which has an effective noise source within one shock cell. The overall sound is a linear combination of the solutions to these sub-problems. In the rest of this paper, we will focus on examining the characteristics of such an effective sound source \\adddd{first and then discuss and compare the total sound from a number of these sources with experiments and numerical simulations.}}\n\nLet us define\n\\begin{equation}\n        \\mathcal{I}_s(\\lambda)=\\int_{0}^{\\frac{2\\pi}{a}}\\sin(a x)\\mathrm{e}^{\\mathrm{i}(\\lambda+\\alpha) x}\\mathrm{d} x=\\frac{1}{2}\\big(\\frac{1}{\\alpha+\\lambda+a}-\\frac{1}{\\alpha+\\lambda-a}\\big)\\big(-\\mathrm{e}^{\\mathrm{i}(\\alpha+\\lambda)\\frac{2\\pi}{a}}+1\\big),\n        \\label{integral_s_s}\n\\end{equation}\n\\begin{equation}\n        \\mathcal{I}_c(\\lambda)=\\int_{0}^{\\frac{2\\pi}{a}}\\cos(a x)\\mathrm{e}^{\\mathrm{i}(\\lambda+\\alpha) x}\\mathrm{d} x=\\frac{\\mathrm{i}}{2}\\big(\\frac{1}{\\alpha+\\lambda+a}+\\frac{1}{\\alpha+\\lambda-a}\\big)\\big(-\\mathrm{e}^{\\mathrm{i}(\\alpha+\\lambda)\\frac{2\\pi}{a}}+1\\big),\n        \\label{integral_s_c}\n\\end{equation}\nwith which the inhomogeneous equation can be written as\n\\begin{equation}\n        \\frac{\\partial^{2}G_{-}}{\\partial y^{2}}-\\gamma_{-}^{2}G_{-}  = \n    2k_{1}\\frac{A}{U}\\bigg[\\sinh(\\zeta_{1}y)(B_{1}\\mathcal{I}_{c}+B_{2}\\mathcal{I}_{s})+ \\sinh(\\zeta_{2}y)(B_{3}\\mathcal{I}_{c}+B_{4}\\mathcal{I}_{s})\\bigg],\n\\end{equation}\nwhere\n    \\begin{equation}\n        \\gamma_{-}=\\sqrt{\\lambda^{2}-M_{-}^{2}(\\omega+\\lambda)^{2}}.\n    \\label{equ:mu-}\n    \\end{equation}\nEquation~(\\ref{equ:mu-}) is equivalent to\n\\begin{equation}\n    \\gamma_{-}=-\\mathrm{i}\\beta\\sqrt{(\\lambda-M_{1})(\\lambda-M_{2})},\n\\end{equation}\nwhere \n\\begin{equation}\n    M_{1}=\\frac{-M_{-}\\omega}{M_{-}+1},\\quad M_{2}=\\frac{-M_{-}\\omega}{M_{-}-1}.\n\\end{equation}\nThe branch cuts \\adddd{passing} $\\lambda=M_{1}$ and $\\lambda=M_{2}$ extend to the\nlower half plane, as illustrated in figure~\\ref{fig:example3}. The function\n$G_{-}(\\lambda,y)$ can be divided into two parts, a particular solution,\n$G^{p}(\\lambda,y)$,\nand a complementary solution, $G^{c}(\\lambda,y)$, i.e.\n    \\begin{equation}\n        G_{-}(\\lambda,y)=G^{p}(\\lambda,y)+G^{c}(\\lambda,y).\n    \\end{equation}\n    The particular solution can be calculated analytically, i.e.\n\\begin{equation}\n     G^{p}(\\lambda,y)  = \n    2k_{1}\\frac{A}{U}\\bigg[\\frac{\\sinh(\\zeta_{1}y)}{\\zeta_{1}^{2}-\\gamma_{-}^2}(B_{1}\\mathcal{I}_{c}+B_{2}\\mathcal{I}_{s})\\nonumber\\\\\n    + \\frac{\\sinh(\\zeta_{2}y)}{\\zeta_{2}^{2}-\\gamma_{-}^2}(B_{3}\\mathcal{I}_{c}+B_{4}\\mathcal{I}_{s})\\bigg ],\n\\end{equation}\nand the complementary solution can be found to be\n    \\begin{equation}\n        G^{c}(\\lambda,y)=D_{2}(\\lambda)\\mathrm{e}^{\\gamma_{-}y}+D_{3}(\\lambda)\\mathrm{e}^{-\\gamma_{-}y},\n    \\end{equation}\n    where $D_{2}$ and $D_{3}$ are two undetermined coefficients. \n    \n    Applying \\add{the}\n    Fourier \\add{transform} to the two boundary conditions, we can solve the\n    undetermined coefficients. For the antisymmetric mode, we obtain\n\\begin{equation}\n\\begin{split}\nD_{1}(\\lambda) &=  \\frac{A\\mathrm{e}^{\\frac{1}{2}\\gamma_{+}}}{2U\\eta M_{+}^{2}}\\bigg[\\gamma_{-}\\coth(\\frac{\\gamma_{-}}{2})\\mathcal{I}_{c}\\sin(\\pm\\frac{1}{2}a\\beta )\\beta\\big[\\omega\\big(2k_{1}m_{-}\\cosh(\\frac{1}{2}m_{-})\\pm m_{+}\\mathrm{e}^{\\mp\\frac{1}{2}m_{+}}\\big)\\\\ \n& - \\frac{1}{\\omega}m_{+}\\frac{M_{-}^{2}}{M_{+}^{2}}a^{2}\\mathrm{e}^{-\\frac{1}{2}m_{+}}\\big]+M_{-}^{2}(\\omega+\\lambda)\\bigg( \\gamma_{-}\\coth(\\frac{\\gamma_{-}}{2})\\frac{2U}{A}G^{p}(\\pm\\frac{1}{2})\\\\\n& - \\frac{2U}{A}{G^{p}}^{\\prime}(\\pm\\frac{1}{2})-i a \\beta \\mathcal{I}_{s}[b_{1}\\sin(\\pm\\frac{1}{2}a\\beta )+c\\beta\\cos(\\frac{1}{2}a\\beta )]\\\\\n&-\\mathcal{I}_{c}[b_{2}\\sin(\\pm\\frac{1}{2}a\\beta )-c\\alpha\\cos(\\frac{1}{2}a\\beta )]\\bigg)\\bigg],\n\\label{equ:resultD1_1_2}\n\\end{split}\n\\end{equation}\n\\begin{equation}\n\\begin{split}\nD_{2}(\\lambda) &=- \\frac{A}{4\\eta U}\\bigg[\\frac{\\omega+\\lambda}{\\omega}\\mathcal{I}_{c}\\gamma_{+}\\beta\\sin(\\pm\\frac{1}{2}a\\beta )\\frac{1}{M_{+}^{2}}\\big[\\omega\\big(2k_{1}m_{-}\\cosh(\\frac{1}{2}m_{-})\\pm m_{+}\\mathrm{e}^{\\mp\\frac{1}{2}m_{+}}\\big)\\\\ \n&-\\frac{1}{\\omega}m_{+}\\frac{M_{-}^{2}}{M_{+}^{2}}a^{2}\\mathrm{e}^{-\\frac{1}{2}m_{+}}\\big]+\\frac{M_{-}^{2}}{M_{+}^{2}}\\frac{(\\omega+\\lambda)^{2}}{\\omega}\\gamma_{+}\\frac{2U}{A}G^{p}(\\pm\\frac{1}{2})\\\\\n&+\\frac{2U}{A}{G^{p}}^{\\prime}(\\pm\\frac{1}{2})+\\mathrm{i}\\omega a \\beta \\mathcal{I}_{s}[b_{1}\\sin(\\pm\\frac{1}{2}a\\beta )+c\\beta\\cos(\\frac{1}{2}a\\beta )]\\\\\n&+\\omega\\mathcal{I}_{c}[b_{2}\\sin(\\pm\\frac{1}{2}a\\beta )-c\\alpha\\cos(\\frac{1}{2}a\\beta )]\\bigg],\n\\end{split}\n\\label{equ:resultD2_1}\n\\end{equation}\nand $D_{4}(\\lambda)=-D_{1}(\\lambda)$, $D_{3}(\\lambda)=-D_{2}(\\lambda)$, where $\\eta(\\lambda)$ is\n\\begin{figure}\n\t\t\\centering\t\n\t\t\\includegraphics[width = 0.8\\textwidth]{figure3.eps}\n\t\t\\caption{The branch points, branch cuts and integral path in complex $\\lambda$ plane. $\\mathrm{P}_1$ and $\\mathrm{P}_2$ are the deformed integral paths when the saddle point approaches to the branch points $-\\omega M_{+}$ and $\\omega M_{+}$, \\add{respectively}.}\\label{fig:example3}\n\\end{figure}\n\\begin{equation}\n    \\eta(\\lambda)=\\omega{\\rm c o t h}(\\frac{1}{2}\\gamma_{-})\\gamma_{-}+\\frac{1}{\\omega}(\\omega+\\lambda)^{2}\\gamma_{+}\\frac{M_{-}^{2}}{M_{+}^{2}}.\n\\end{equation}\n\tIt is straightforward to verify that $\\lambda=-\\alpha$ is a simple zero for $\\eta(\\lambda)$. In fact, $\\eta(-\\alpha)=0$ corresponds to the dispersion relation~(\\ref{dispersion_anti}).\nBesides, $\\lambda=-\\alpha$ is a simple                     \nzero for $\\mathcal{I}_s(\\lambda)$\nand a second-order zero for $\\mathcal{I}_c(\\lambda)$, so $\\lambda=-\\alpha$ is not a pole for $D_{1}(\\lambda)$ and $D_{2}(\\lambda)$. \\adddd{It is found that $D_{1}(\\lambda)$ and $D_{2}(\\lambda)$ have four poles, which are}\n\\begin{gather}\n    Z_{1,2}=\\dfrac{\\omega M_{-}^2\\pm\\sqrt{M_{-}^{2}\\omega^{2}+(1-M_{-}^{2})\\zeta_{1}^{2}}}{1-M_{-}^2},\\\\\n    Z_{3,4}=\\dfrac{\\omega M_{-}^2\\pm\\sqrt{M_{-}^{2}\\omega^{2}+(1-M_{-}^{2})\\zeta_{2}^{2}}}{1-M_{-}^2}.\n\\end{gather}\nThe following inverse Fourier \\add{transform}\n\\begin{equation}\n        g_{+}(x,y)=\\frac{1}{2\\pi}\\int_{-\\infty}^{+\\infty}D_{1}(\\lambda)\\mathrm{e}^{-(\\mathrm{i}\\lambda x+\\gamma_{+}y)}\\mathrm{d}\\lambda\n        \\label{equ:numerical integration}\n\\end{equation}\nyields $g_{+}$. \nThe integration path is near the real axis\nof $\\lambda$, as illustrated in figure~\\ref{fig:example3}. The integration path is \\adddd{indented} to pass above the poles at $\\lambda=Z_{1,4}$, as illustrated in figure~\\ref{fig:example3}, in accordance with the causality argument~\\citep{Briggs}. Because the real part of $\\gamma_{+}$ should be positive when $|\\lambda|\\rightarrow \\infty$ \\adddd{along the integration path}, the branch\ncuts of  $\\gamma_{+}$  \\adddd{passing the} branch points $\\lambda=\\pm \\omega M_{+}$ are chosen to extend to the upper and lower half plane, respectively, as shown in figure~\\ref{fig:example3}. \nThe branch points of $\\gamma_{-}$, i.e. $M_{1}$ and $M_{2}$, are on the \nnegative real $\\lambda$ axis. The branch cuts are chosen to extend down to the lower half plane so as not to  cross the integration path. Using the\nsteepest descent method, and noting that the saddle point is located at\n$\\lambda=-M_{+}\\omega{\\rm cos}\\theta$, where $\\theta={\\rm a r c t a n}(y\/x)$ representing the observer angle,\nwe can express $g_{+}$ as a function of radial distance $r$ and \n$\\theta$ in the far field~($r\\gg1$), i.e.\n    \\begin{equation}\n        g_{+}(r,\\theta)=\\frac{\\sqrt{M_{+}\\omega}}{\\sqrt{2\\pi}}D_{1}(-M_{+}\\omega{\\rm c o s} \\theta){\\rm s i n} \\theta \\frac{\\mathrm{e}^{\\mathrm{i}\\omega(M_{+}r-\\pi\/4)}}{\\sqrt{r}}+O(r^{-3\/2}),\n        \\label{steepest_decent_way}\n    \\end{equation}\n\\adddd{and with (\\ref{20}) and (\\ref{potential_outside}),  the corresponding pressure perturbation (nondimensionalized by $\\sqrt{2\/\\pi}\\rho_{0+}U^2$) can be expressed as}\n   \\begin{equation}\n        p_{+}(r,\\theta)=\\mathrm{i}\\sqrt{M_{+}}\\omega^{\\frac{3}{2}}D_{1}(-M_{+}\\omega{\\rm c o s} \\theta){\\rm s i n} \\theta \\frac{\\mathrm{e}^{\\mathrm{i}\\omega(M_{+}r-t-\\pi\/4)}}{\\sqrt{r}}+O(r^{-3\/2}).\n        \\label{steepest_decent_way_p}\n    \\end{equation}\n\nNote that the saddle\npoint moves between $-\\omega M_{+}$ and $\\omega M_{+}$ as $\\theta$ changes from 0 to $\\pi$, and the integral path is\nforbidden to pass through the branch cut. So when $\\theta=\\pi$ and $0$, the\ncorresponding integral path is deformed along the branch cut and wraps the\nbranch point as shown by $\\mathrm{P}_1$ and $\\mathrm{P}_2$ in\nfigure~\\ref{fig:example3}, respectively. It is similar when the steepest decent path passes through the branch cut at $M_{1}$ and $M_{2}$, in which case the \nintegral path needs to be adjusted to avoid the branch cut. Note when the poles cross the steepest decent path, care must be taken regarding the residue contribution.\n\nAs can be seen from equation~(\\ref{steepest_decent_way}), when the observer angle $\\theta=\\pi$, the potential function $g_{+}(r,\\theta)$ reduces to a high order term $O(r^{-3\/2})$ if $|D_{1}(-\\omega M_{+}\\cos\\theta)|$ is bounded. This implies that sound waves propagating in this direction decays rapidly and nearly vanishes in the far field as $r\\rightarrow \\infty$. \\adddd{This leads to an important feature of the sound directivity that will become clear in the rest of this paper.}\n\n\t\n\n     \n    \n\n \n  \n \n\n\n\n\\section{Results and discussion}\n\\label{section:results and discussion}\nThe sound field due to the interaction between shock and instability waves is shown in this section. In the linear stability analysis, the spatial wavenumber $\\alpha$ is the central parameter determining the characteristics of the instability waves. \nThe dispersion relation calculated from~(\\ref{dispersion_anti}) is shown in figure~\\ref{fig:example4}. The antisymmetric mode is considered. We see that both the wavenumber and growth rates increase as $\\omega$ increases. These are well-established results in the linear stability analysis. Due to a negative \nimaginary part, the instability waves grow exponentially downstream the jet flow, and subsequently interact with shock cell structures. \n    \t\\begin{figure}\n\t\t\\centering\n\t\t\\includegraphics[width = 0.55\\textwidth]{figure4.eps}\n\t\t\\caption{The real and imaginary parts of the solution of the spatial wavenumber $\\alpha$ calculated from the dispersion relation~(\\ref{dispersion_anti}). The antisymmetric mode is considered. Note that the imaginary part of the wavenumber $\\alpha$ is negative, and its absolute value is plotted instead. }\\label{fig:example4}\n\t\\end{figure}\n\n\n\nIn what follows, we first examine the screech frequency prediction using \\adddd{the present} model. Sound propagating at the observer angle $\\theta=150^\\circ$ is used to verify the far-field approximation before the directivity patterns of the fundamental tone and its harmonics are shown. Finally, we examine the near-field pressure fluctuation and discuss the noise generation  mechanism.\n \n\\subsection{The screech frequency}\n\\label{subsection:frequency_prediction}\n\\citet{1953Powell} proposed a model to predict the screech frequency by assuming a constructive interference in the upstream direction $\\theta=180^\\circ$. Following Powell's idea, the screech frequency and its harmonics can be calculated using \\adddd{the present} model. Note similar frequency predictions have been investigated in earlier studies, nevertheless, it is included here as a validation of the model. The shock cell spacing $\\add{s}$  satisfies\n\\begin{equation}\n    \\add{s}\/D=\\frac{2\\pi}{a}=2\\sqrt{M_{-}^{2}-1}.\n    \\label{equ:shock space}\n\\end{equation}\n\\citet{Tam_shock_space} showed that in the jet flow from a rectangular nozzle, the shock cell spacing satisfied\n\\begin{equation}\n    \\add{s}\/D=\\dfrac{2\\sqrt{M_{-}^{2}-1}}{\\sqrt{1+\\bigg(\\dfrac{D}{b}\\bigg)^{2}}},\n    \\label{tam's prediction}\n\\end{equation}\nwhere $b$ is the width of the \\adddd{fully-expanded} jet flow. When the rectangular jet is of high aspect ratio~$(D\/b\\ll1)$, we can see that~(\\ref{tam's prediction}) reduces to~(\\ref{equ:shock space}). \\add{Note that (\\ref{equ:shock space})\nmay not be able to predict the shock spacing accurately when the magnitude of overexpansion\/underexpansion increases~\\citep{X.D.Li_shock}, while a correct representation of shock structures plays a dominant role in predicting the screech frequency, especially in circular jets~\\citep{A1A2modes}. Nevertheless, in this planar model, the jet is assumed to be slightly imperfectly-expanded, and previous studies~\\citep{Tam_shock_space} showed good agreement between the prediction and the experimental data. Therefore, we choose~(\\ref{equ:shock space}) to predict the shock spacing.}\n \n  \n\n \n\nThe convection velocity of instability waves is widely believed to be proportional to the velocity of the fully-expanded jet flow, i.e.\n\\begin{equation}\n    U_{c}=\\kappa U_{-},\n    \\label{equ:convective velocity}\n\\end{equation}\n\\add{where $\\kappa$ is usually taken to be $0.7$.}\nEquations~(\\ref{equ_screech f}),~(\\ref{equ:shock space}), and~(\\ref{equ:convective velocity}) can be combined to yield the nondimensionalized angular frequency of the sound wave\n\\begin{equation}\n    \\label{screech_omega}\n    \\omega=\\frac{a m \\kappa }{M_c +1},\n\\end{equation}\nwhere $m=1, 2, 3, 4...$ \\adddd{correspond to} the fundamental frequency, the first, second and higher harmonics, respectively. \\addd{Considering that $M_c=U_c\/a_\\infty$, where $a_\\infty$ is the speed of sound in the free stream. \\adddd{For a cold jet, $M_c$ can be calculated by} }\n\\addd{\\begin{equation}\n    M_c=\\frac{\\kappa M_{-}}{\\sqrt{1+\\frac{\\gamma-1}{2}M_{-}^2}}.\n    \\label{3.5}\n\\end{equation}\n\\adddd{With~(\\ref{3.5}), $\\omega$ can be readily calculated via~(\\ref{screech_omega}). This formula is consistent with that derived by~\\citet{Tam_shock_space}.}} A comparison between the measured fundamental screech frequency by~\\citet{1953Powell}\nand that predicted by~(\\ref{screech_omega}) is shown in figure~\\ref{frequency_check}. \\adddd{As can be seen, good agreement is achieved. Equations (\\ref{screech_omega}) and (\\ref{3.5}) show that when the jet operating condition is known, the screech frequency can be readily calculated.}\nThe operating conditions and frequencies calculated in this way, as shown in table~\\ref{tab:kd}, will be used in following sections.\n\n\\add{\nNote that this paper does not attempt to model the entire feedback loop, and the reason we include a frequency prediction is mainly to validate the model. \\adddd{Thus, although the feedback theory proposed by Powell is used, it is only used to predict the frequency using (\\ref{screech_omega}). Our focus in this paper is to model the interaction between the shock and instablity waves.} It is worth noting that a new  feedback mechanism for circular jets has been proposed in a number of recent papers~\\citep{Gojon_modeStaging,edgington_modestaging,Xiang-Ru_2020,NewC_1}. The present paper, however, focuses on a 2D jet, and it is not yet clear what role the guided jet modes play in this case. Besides, even for circular jets, it is known that the convection velocity of the upstream-travelling jet guided mode is very close to the speed of sound. Therefore, it can be expected little change in the frequency prediction would occur even if the guided jet mode is taken as the closure mechanism.}\n\n\t\\begin{figure}\n\t\\centering\n\t\\label{1frequency spectrum}\n\t\\includegraphics[width = 0.55\\textwidth]{figure5.eps}\n\t\\caption{\\addd{Comparison of the fundamental screech frequency between  Powell's experiment~\\citep{1953Powell} and the model's prediction.}}\n\t\\label{frequency_check}\n\t\\end{figure}\n\n\\begin{table}\n  \\begin{center}\n\\def~{\\hphantom{0}}\n  \\begin{tabular}{lccc}\n      $M_{-}$  & $\\omega_{1}$   &   $\\omega_{2}$ & $\\omega_{3}$ \\\\[3pt]\n       1.5   &\\addd{1.05} & \\addd{2.10} & \\addd{3.15}\\\\\n       1.3   & \\addd{1.48} &\\addd{2.86} &\\addd{4.45}\\\\\n       1.2  & \\addd{1.91} & \\addd{3.81} &\\addd{5.73}\\\\\n  \\end{tabular}\n  \\caption{The operating conditions and calculated frequencies of the screech tones, where $\\omega_{1}$ denotes the angular frequency of the fundamental tone, $\\omega_{2}$ its first harmonic, and $\\omega_{3}$ the second harmonic.}\n  \\label{tab:kd}\n  \\end{center}\n\\end{table}\n\n\\subsection{Directivity of the sound field }\n\\label{subsection_directivity}\n\n\\add{The distinct directivity pattern of jet screech is perhaps one of its most important features and has been well-reported in various experiments. In this section, we aim to predict the noise directivity using the model developed in section 2. As mentioned in section 1, this paper concerns the acoustic emission due to the interaction between shock and instability waves, and therefore does not consider the entire feedback process of screech. However, the existence of the feedback would not \\adddd{alter} the fact that the noise is generated due to the shock-instability interaction. Therefore, \\adddd{provided the screech frequency is specified, the present model can be used to compare with the screech directivity.}}\n\nBefore used to study the directivity of the resulting sound, equation~(\\ref{steepest_decent_way_p}) is verified by numerically integrating~(\\ref{equ:numerical integration}) at $\\theta=150^\\circ$. The comparison between the prediction using~(\\ref{steepest_decent_way_p}) and~(\\ref{equ:numerical integration}) is shown in figure~\\ref{sound_outside}. The SPL is defined to be\n\\begin{equation}\n    \\mathrm{SPL}=10\\log_{10}\\frac{|p_{+}|^2}{|p_{\\mathrm{ref}}|^2},\n    \\label{equ:SPL}\n\\end{equation}\n\\adddd{where the reference pressure $p_\\mathrm{ref}=2\\times 10^{-5}$.} We see that good agreement is achieved when $r$ is beyond $5$, where the difference between two methods is within 1 dB. When $r\\geq 20 $, the difference reduces to 0.2 dB. \\adddd{Consequently $r=5$, in which case $kr\\approx7.9$, may be used to approximately separate the acoustic near and far field~\\citep{krgg1}.}\n\n\t\\begin{figure}\n\t\\centering\n\t\\includegraphics[width = 0.55\\textwidth]{figure6.eps}\n\t\\caption{\\addd{Comparison of the SPL from the far-field approximation and that from the numerical integration~($\\theta=150^\\circ$). The Mach number of the fully-expanded jet is 1.5. The origin $r=0$ represents the beginning of the first shock cell, and $r$ is the nondimensionalized radial distance.}}\n\t\t\\label{sound_outside}\n\t\\end{figure}\n\n Whether the sound source of jet screech is spatially localized or distributed along the jet flow is still open to debate. As mentioned above, Powell proposed the monopole array theory to predict the directivity patterns. Many other researchers~\\citep{S.kaji,1997JFM_Raman,Sources} also found that the fundamental screech tone emitted from several shock cells downstream the jet flow. However, other researchers~\\citep{1997_POF_SHWalker,2017_jfm_sources} observed that the screech was produced from a particular shock cell, e.g. the third or fourth one. In addition, \"shock-clapping\"~\\citep{1994Suda} and \"shock leakage\"~\\citep{2003Suzeki} were observed at the third and fourth shock cells downstream the jet, which were suggested to be the source of screech, particularly for higher harmonics~\\citep{2020_sound_sources}. \\add{This suggests that the number of shock cells that need to be included is not clear. \\adddd{However}, it would be interesting to compare and contrast the noise directivity patterns due to the interaction between the instability waves and various numbers of shock cells.} Therefore, in what follows we will examine the directivity patterns due to the interaction between the instability wave and a single shock cell, \\adddd{and then discuss that from several shock cells.}\n \n\n\nIt is known that in realistic jets the instability waves exhibit a characteristic structure of wave packets~\\citep{2005Lele,2013_annual_rev,2019Wong}. The amplitude of the instability waves varies slowly within one wavelength~\\citep{2006_jfm_wavepackets}, while the whole wave packet shows a Gaussian envelope~\\citep{2011_jsv_wavepackets}, \\add{or more precisely an exponentially-modified Gaussian envelope as demonstrated by a recent work~\\citep{exponetially_Gaussian}}. We see that the local growth rate within a wavelength is varying, but the effects of the local growth rate on the sound characteristics are not clear \\adddd{across the wave packet}. In this paper, we will also examine the effects of the local growth rate by showing results \\add{with various values of $\\alpha_{\\mathrm{i}}$}.\n\n\n\\subsubsection{Directivity pattern of \\add{sound due to one-cell interaction}}\n\nWhen the \\adddd{fully-expanded Mach number} is given, the directivity patterns of the fundamental tone and its harmonics can be calculated from~(\\ref{steepest_decent_way_p}). The operating condition is shown in table~\\ref{tab:kd}. Note that  the wavenumber is obtained from~(\\ref{dispersion_anti}), as we  consider the antisymmetric mode.\n \t\\begin{figure}\n\t\\centering\n\t\\includegraphics[width = 1.0\\textwidth]{figure7.eps}\n\t\\caption{\\addd{The directivity of sound in the far field obtained by~(\\ref{steepest_decent_way_p}). $r $ is fixed to be 1. Labels (1), (2), (3) represent the results for a fully-expanded jet Mach number of 1.5, 1.3, and 1.2, respectively. Columns (a), (b), (c) are the results of the fundamental tone, the first and second harmonics, respectively. The antisymmetric mode of instability waves is taken, and the imaginary part of wavenumber $\\alpha_{\\mathrm{i}}\\neq 0$. }\\adddd{In addition, $\\mathcal{U}$ in~(\\ref{equ:A}) is taken to be 1.}}\n\t\t\\label{n=1_consider}\n\t\\end{figure}\nThe directivity patterns of the fundamental tone and its first two harmonics under three different operating conditions are shown in figure~\\ref{n=1_consider}. The SPL is defined by~(\\ref{equ:SPL}). Labels (1), (2), (3) represent the results for the fully-expanded jet Mach number of 1.5, 1.3, and 1.2, respectively. Columns (a), (b), (c) are the results of the fundamental tone, the first and second harmonics, respectively. \\addd{From figure~\\ref{n=1_consider}, it is clear that the \\addd{effective} directivity of the fundamental tone due to a single shock cell is not that of a monopole. Instead, it consists of two\nlobes. One primary lobe radiates upstream, while the other radiates downstream with a weaker intensity. Although this represents the effective directivity due to one shock-cell interaction, we see that it possesses some inherent directivity that resembles the total sound field measured in experiments. For example, the effective directivity of the fundamental tones accords with} both numerical~\\citep{numerical_directivity_of_rectangular} and experimental~\\citep{1997_POF_SHWalker,Sources} results for rectangular jets of high aspect ratios, \\add{where both an upstream lobe and a downstream lobe of similar intensities appear. However, a precise match of every lobe position may not be achieved; this is expected because the prediction is only from one shock-cell interaction, whereas the numerical and experimental results are from the entire screeching jet.} Moreover, in all three cases, the fundamental tone is reinforced at the upstream direction, but drops quickly as $\\theta$ approaches $180^\\circ$. In Kerschen's original paper~\\citep{1995Kerchen}, the directivity pattern reaches its maximum at $48^\\circ$ with only one lobe in the downstream direction. Little sound radiates in the upstream direction. Our present model shows that when a two-dimensional vortex sheet and a more realistic shock structure are considered, the screech\ndirectivity from a single shock cell has a major radiation lobe in the upstream direction. This result shows a better \\adddd{qualitative} agreement with experiments. It is also interesting to note that the maximal radiation angles shown in figure 7 appear to depend on the fully-expanded jet Mach number. A similar tendency was also reported in earlier experiments for round jets~\\citep{1992Powell}. \n\nAnother important result is that there is a large lobe perpendicular to the jet flow for the first harmonic in all three conditions, \\adddd{which again resembles the total noise directivity measured in experiments}~\\citep{1953Powell, 1997_POF_SHWalker, 2020_sound_sources}. Note that the peak radiation angle was reported to be not exactly at $90^\\circ$, but slightly towards the downstream direction~\\citep{numerical_directivity_of_rectangular, M.Kandula, 2020_sound_sources}, \\adddd{which is similar to the prediction by the present model}. For the second harmonic,~\\add{it was experimentally observed that the \\add{effective} directivity pattern showed two lobes, one directed slightly upstream, and one downstream~\\citep{Sources}; this feature is \\adddd{consistent} with the prediction.  In addition, we find that in a recent experiment conducted by~\\citet{2020_sound_sources}, the main radiation angle for the second harmonic is between $40^\\circ$ and $110^\\circ$, while little radiation appeared around $90^\\circ$. We see from figure~\\ref{n=1_consider} that this \\adddd{is also reflected} in the prediction.} \n\n\\addd{From figure~\\ref{n=1_consider}, It is straightforward to see that the effective noise directivity due to the interaction between the instability waves and one shock cell is not of the monopole type, but shows an intrinsic shape that is close to that of the overall screech directivity. This shows that the unique directivity of jet screech is not caused by pure interference between an array of monopoles as assumed by Powell. Therefore, to properly model and understand the directivity of screech, one has to use quantitative models such as the one developed in this paper. However, it is worth noting that this model does not imply that the screech source is localized as a \"single\" source, what we show is just the effective noise directivity due to interactions between the instability waves and a single shock cell.}\n\n\\add{To examine the effects of the local growth rate of the instability waves on sound generation, the result obtained using~(\\ref{steepest_decent_way_p}) with only the real part of $\\alpha$ considered is shown in figure~\\ref{n=1_noconsider}.} The SPL is similarly defined by~(\\ref{equ:SPL}). Only the fundamental tone and its first harmonic are presented to compare with those reported by Powell~\\citep{19533Powell}. \\addd{Considering that the original directivity results reported by~\\citet{19533Powell} were presented in the form of schlieren photographs and were therefore not suitable for a direct comparison, only a qualitative comparison is presented.}\nAs shown in figure~\\ref{n=1_noconsider}, the fundamental tones in all three cases are only reinforced in the upstream direction, while in Powell's experiment, sound waves at the fundamental frequency can only be observed propagating upstream, as shown in figure 4 of the original paper~\\citep{19533Powell}. \\addd{The model prediction is in agreement with the experimental data}. In addition, a quick decay also occurs when the observer angle approaches $180^\\circ$, which is similar to the case when the imaginary part of $\\alpha$ is not zero. While for the first harmonic, a large  lobe perpendicular to the jet flow is predicted by our model. In Powell's experiment, when a reflector was placed, a downstream propagating sound wave~(as shown in figure~5 of the original paper~\\citep{19533Powell}) twice of the fundamental frequency emerged. This implied that there was a strong beaming to the side of the jet flow, which is in good \\adddd{qualitative} agreement with the prediction. \n\n\\add{To further investigate the effects of the local growth rate of the instability waves on directivity patterns, we change the imaginary part of $\\alpha$  to $2\/3$ of its original value. The resulting directivity patterns are shown in figure~\\ref{n=2\/3}. As can be seen, the fundamental tones in all three cases radiate primarily to the upstream direction, while small lobes appear downstream of the jet flow. Compared with the results shown in figure~\\ref{n=1_consider} these lobes are both thinner and weaker, whereas in the case of $\\alpha_{\\rm{i}}=0$ there are no observable lobes downstream of the jet flow, as illustrated in figure~\\ref{n=1_noconsider}(a). For the first harmonic, a large lobe appears perpendicular to the jet flow. Compared with the results shown in figure~\\ref{n=1_consider}, the directivity patterns seem to shrink and move closer to $90^\\circ$ to the jet. In particular, two small lobes appearing in figure~\\ref{n=1_consider}(3b) appear to collapse to a single wide lobe, as shown in figure~\\ref{n=2\/3}(3b).}\n\nThe directivity pattern when a \\add{positive} imaginary part of $\\alpha$ is used can be similarly studied. It can be shown that little change  occurs when $\\alpha$ is replaced by $\\alpha^{*}$~($\\alpha^{*}_{i}=-\\alpha_{i}$). So we will omit a \\adddd{repetitive} discussion for brevity. Figures~\\ref{n=1_consider},~\\ref{n=1_noconsider}, and~\\ref{n=2\/3} show that the directivity pattern depends on the local growth rate of the instability wave, and a change in the imaginary part of $\\alpha$ would lead to a corresponding  change in the resulting directivity pattern.\nAs mentioned at the beginning of  section~\\ref{subsection_directivity}, it is not clear exactly where the interaction between instability and shock waves occurs. The amplitude of instability waves may have experienced a growth, a saturation, or even decay before reaching the point of interaction. Comparing figures~\\ref{n=1_consider},~\\ref{n=1_noconsider}, and~\\ref{n=2\/3}, we see that the noise directivity is very sensitive to the local growth rate, and this may be used to explain the discrepancies observed across different experiments. For example, Powell's original results showed a clearly dominant radiation only in the upstream direction and a strong $90^\\circ$ radiation at the first harmonic. This could be explained well if the local instability waves experience a saturation. On the other hand,~\\citet{1997_POF_SHWalker},~\\citet{numerical_directivity_of_rectangular_2}, and~\\citet{GJWU}  showed that two lobes could be observed at the fundamental frequency, and a relatively weak radiation at $90^\\circ$ for the \\addd{first} harmonic. This may be explained if the instability waves are in a growth or decay stage at the effective point of interaction.\n\n\n \t\\begin{figure}\n\t\\centering\n\t\\includegraphics[width = 0.75\\textwidth]{figure8.eps}\n\t\\caption{\\addd{The directivity of sound in the far field obtained by~(\\ref{steepest_decent_way_p}). $r $ is fixed to be 1. Labels (1), (2), (3) represent the results for a fully-expanded jet Mach number of 1.5, 1.3 and 1.2, respectively. Columns (a), (b) are the results of the fundamental tone and the first harmonic, respectively. The antisymmetric mode of instability waves is taken, and the imaginary part of wavenumber $\\alpha_{\\mathrm{i}}= 0$. }\\adddd{In addition, $\\mathcal{U}$ in~(\\ref{equ:A}) is taken to be 1.}}\n\t\t\\label{n=1_noconsider}\n\t\\end{figure}\n\n\t\t\\begin{figure}\n\t\\centering\n\t\\includegraphics[width = 0.75\\textwidth]{figure9.eps}\n\t\\caption{\\addd{The directivity of sound in the far field obtained by~(\\ref{steepest_decent_way_p}). $r $ is fixed to be 1. Labels (1), (2), (3) represent the results for a fully-expanded jet Mach number of 1.5, 1.3 and 1.2, respectively. Columns (a), (b) are the results of the fundamental tone and the first harmonic, respectively. The antisymmetric mode of instability waves is taken, and the imaginary part of wavenumber $\\alpha_{\\mathrm{i}}$ changes to $2\/3$ of the original value.} \\adddd{In addition, $\\mathcal{U}$ in~(\\ref{equ:A}) is taken to be 1.}}\n\t\t\\label{n=2\/3}\n\t\\end{figure}\n\n\\subsubsection{Directivity patterns from several shock cells}\nAs mentioned at the beginning of section~\\ref{subsection_directivity}, some researchers report that screech appears to originate from several shock cells downstream the jet flow, for example, from the second  to the fourth shock cell~\\citep{1994Suda,Sources}. \\addd{Since our model \\adddd{can include multiple shock structures, we can study and compare the sound field produced by the interaction between instability waves and several shock cells with simulations and experiments}}. \\addd{Note that to precisely predict the screech amplitude, it is likely that every stage of the feedback loop needs to be considered~\\citep{absolute_instability} and the nonlinearity that is inevitable within the loop needs to be included. However, in this paper, we only study the shock-instability interaction and use a linear model ($\\mathcal{U}$=1 in~(\\ref{equ:A})). Therefore, the prediction would not be able to match the data in terms of the absolute amplitude. However, we can still plot the predictions and the numerical or experimental data in one figure and focus instead on comparing the shapes of the directivity pattern.  The SPL of the model prediction is again defined by~(\\ref{equ:SPL}), but rescaled according to the experimental or numerical data.} \\addd{The predictions of the monopole array theory are also included \\adddd{for} comparison.} \n\n\\addd{The results are first compared with the study by~\\citet{GJWU}, where LES simulations were conducted and well validated against the experimental data reported by~\\citet{Wu_experiment_1} and~\\citet{Wu_experiment_2}. In both the experiment and the numerical simulation, a rectangular nozzle with an aspect ratio of 4:1 was used. The designed Mach number of the nozzle was 1.44, while the Mach number of the fully-expanded jet flow was 1.69.} It was stated by~\\citet{GJWU} that the measured directivity patterns resulted from the interference among spatially distributed sources. Therefore, multiple shock cells are included in our model to facilitate a comparison. Considering that the amplitude of instability waves shows a Gaussian~\\citep{2011_jsv_wavepackets}, \\add{or more precisely exponentially-modified Gaussian~\\citep{exponetially_Gaussian}} intensity distribution downstream of the jet flow, in this paper \\add{we use three shock cells, the interaction between which and the instability waves leads to different effective source strengths.} The middle cell is chosen to have the maximal strength. In front of the middle cell, the instability waves still grow and have not reached the maximum intensity, while after that the instability waves begin to decay but are still of sufficient intensity to generate sound. \\addd{Thus, the relative strengths of the three interactions are assumed to be 0.45, 1, 0.7, respectively.} Similar assumptions have been also made by~\\citet{1983Norm} and \\citet{ numerical_directivity_of_rectangular}. \\addd{It was known that the effect of varying source strengths on the directivity of the fundamental and the first harmonic was unimportant with regard to the principal lobe, and was appreciable only in the secondary or minor lobes~\\citep{M.Kandula}. }In light of linearity, in what follows we first calculate the sound by one shock cell using our model, and then combine the other two with a spatial phase difference $ \\mathrm{e}^{\\mathrm{i}\\left(\\lambda_{0}+\\mathrm{real} (\\alpha)\\right)2 \\pi\/a}$, where $\\lambda_{0}=-\\omega M_{+}\\cos\\theta$.\n \t\\begin{figure}\n\t\\centering\n\t\\includegraphics[width = 0.75\\textwidth]{figure10.eps}\n\t\\caption{\\addd{The comparison between the numerical results~\\citep{GJWU}, the present model, and the monopole array theory~\\citep{19533Powell}. The Mach number of the fully-expanded jet flow is 1.69. The red solid line denotes the model prediction, the red line with markers the numerical data, and the black dashed line represents the prediction of the monopole array theory. Columns (a) and (b) represent the fundamental tone and its first harmonic, respectively.}}\n\t\t\\label{Norum_1.49}\n\t\\end{figure}\n\t\n\n\t\n    \\begin{figure}\n\t\\centering\n\t\\includegraphics[width = 0.75\\textwidth]{figure11.eps}\n\t\\caption{\\addd{The comparison between the experimental data~\\citep{farfield_rectangular}, the present model, and the monopole array theory~\\citep{19533Powell}. The fully-expanded  Mach number for (a) and (b) is 1.5 and 1.6, respectively, while the designed Mach number of the nozzle is 1.35. The red solid line denotes the model prediction, the orange line with markers the experimental results, and the black dashed line represents the prediction of the monopole array theory. }}\n\t\t\\label{Norum_1.19}\n\t\\end{figure}\n\nThe result is shown in figure~\\ref{Norum_1.49}, where \ncolumns (a) and (b) represent directivity patterns of the fundamental tone and the first harmonic, respectively. \\addd{As can be seen, for the directivity pattern of the fundamental tone, the numerical data shows two lobes. The major lobe points to the upstream direction, while the other peaks \\adddd{at} around $30^\\circ$ with a slightly weaker intensity. Note that the radiation intensity seems to decrease as the observer angle approaches $0^\\circ$. The results from the monopole array theory also show two lobes, which are of equal intensity and peak at $180^\\circ$ and $0^\\circ$, respectively. The weaker intensity of the downstream lobe appears not captured in the model. In addition, as $\\theta$ approaches $0^\\circ$, the monopole array theory predicts an increasingly large noise radiation, which contradicts the numerical data. On the other hand, we see that the present model predicts a similar major lobe in the upstream direction, and a weaker lobe in the downstream direction. The relative intensity and positions of the two lobes agree better with the numerical data than the monopole array theory. Moreover, the predicted acoustic radiation decreases quickly as the observer angle approaches $0^\\circ$, which is in good agreement with the numerical data. Note however the maximum radiation angle of the downstream lobe appears at around $40^\\circ$, slightly different from the numerical data. But considering the many assumptions made in the model, such deviation may be \\adddd{deemed} acceptable.}\n\n\\addd{For the first harmonic, the numerical results exhibit three lobes. Two dominant lobes peak at $\\theta=30^\\circ$ and $\\theta=83^\\circ$, respectively, and one secondary lobe points to $155^\\circ$. It is evident that a quick decrease occurs as the observer angle approaches $180^\\circ$ and $0^\\circ$, respectively. The results obtained by the monopole array theory also show three lobes, which are of the same intensity and peak at $180^\\circ$, $88^\\circ$, and $0^\\circ$, respectively. The corresponding \nerrors compared with the numerical data are $+25^\\circ$, $+5^\\circ$, and $-30^\\circ$, respectively. Moreover, the monopole array theory predicts monotonously increasing acoustic radiations as the observer angle approaches $180^\\circ$ or $0^\\circ$, which is not able to match the numerical data. For the present model prediction, a narrow lobe peaks \\adddd{at} around $84^{\\circ}$ with a dominant intensity, and two weaker lobes appear \\adddd{at} \\addd{$\\theta=31^\\circ$} and \\addd{$\\theta=135^\\circ$}, respectively. The corresponding differences compared with the numerical data are $+1^\\circ$, $+1^\\circ$, and $-20^\\circ$, respectively, which is in more satisfactory agreement with the numerical results than those predicted by the monopole array theory. In addition, the rapid decay as the observer angle approaches $180^\\circ$ and $0^\\circ$ can be predicted well by this model. \\adddd{Note however the present model cannot correctly predict the relative amplitude of the upstream and downstream lobes. The reason is not yet clear.} In summary, although \\adddd{both} models are not capable of predicting the screech amplitude, the present model  shows a more satisfactory agreement with the numerical data in terms of the peak angle. In addition, unlike \\adddd{the monopole array theory}, it can capture the rapid decay of the noise intensity as observer angles approach $0^\\circ$ and $180^\\circ$.}\n\n\n\\addd{The results obtained by the present model are subsequently compared with~\\citet{farfield_rectangular}, where a series of experiments were conducted \\adddd{at} the NASA Langley Research Center. Several microphones were positioned on a circular arc from $\\theta=30^\\circ$ to $135^\\circ$ at an increment of $15^\\circ$. The aspect ratio and the designed Mach number of the rectangular nozzle \\adddd{were} 3.7 and 1.35, respectively. Two sets of experimental data are chosen for comparison, of which the corresponding fully-expanded Mach numbers are 1.5 and 1.6, respectively. \\adddd{Note that the available data only spans the observer angle between $30^\\circ$ and $135^\\circ$ at an increment of $15^\\circ$. It is known that \nvery weak noise is radiated in this range, and the peak radiation angles are likely to fall outside this range at the fundamental frequency. Therefore, for a robust comparison we only compare the first harmonic results, where it is known to radiate primarily at side angles.} The results predicted by the monopole array theory are also included for comparison. The number and relative strengths of the effective sources are kept the same as those used in figure 10. }\n\n\\addd{As can be seen in figure~\\ref{Norum_1.19}, in the case of $M_{-}=1.5$, the experimental data shows a major lobe to the side of the jet, and another lobe  in the upstream direction with a slightly weaker intensity. In addition, note that in the downstream direction, the acoustic radiation becomes much weaker, as can be seen \\adddd{at} the observer angle $\\theta=30^\\circ$. The results from the monopole array theory exhibit three lobes of nearly the same intensity, one in the upstream direction, one in the downstream direction, and another to the side of the jet. The agreement with the experimental data is good \\adddd{when} $60^\\circ<\\theta<135^\\circ$, but much less so in the downstream direction. The present model prediction also exhibits three lobes, one dominant lobe to the side of the jet, one weaker lobe \\adddd{in} the upstream direction, and another lobe peaks at $25^\\circ$ with a much weaker intensity. Both the position and the relative intensity of the three lobes agree well with the experimental data. }\n\n\\addd{In the case of $M_{-}=1.6$,  as can be seen in figure~\\ref{Norum_1.19}(b), the experimental data shows a dominant lobe to the side of the jet, while another lobe appears slightly downstream with a weaker intensity. Note that a much weaker acoustic wave radiates to the downstream direction. The prediction obtained by the monopole array theory shows three lobes of the same intensity. \\adddd{Again, the agreement with the experimental data is good} when $60^\\circ < \\theta < 120^\\circ$, but much less satisfactory in the downstream direction. The present model prediction also exhibits three lobes, one dominant lobe to the side of the jet, one weaker lobe in the upstream direction, and another peaks at $30^\\circ$ with a much weaker intensity. Both the position and the relative intensity of the latter two lobes agree well with the experimental data. In both the monopole array theory and the present model, it is difficult to drama a conclusion about the agreement when $\\theta$ approaches $135^\\circ$ due to the very sparse data points. Because the experimental results only cover an observer angle of $30^\\circ$ to $135^\\circ$, the directivity patterns as the observer angle approaches $180^\\circ$ and $0^\\circ$ \\adddd{cannot} be examined. \\adddd{However, the numerical results in figure 10 show a quick decay as the observer angle approaches $180^\\circ$ and $0^\\circ$. Similar phenomenon has been widely reported in numerous experiments for circular nozzles, such as those by~\\citet{1983Norm} and \\citet{1992Powell}. Such an important feature can be well captured by the present model, compared to earlier models.}}\n\n\\addd{Note that the present model makes use of a number of linear approximations, for example, both the shock and instability waves are of small magnitude. \\adddd{It is mentioned that supersonic jets may be regarded as weakly imperfectly-expanded when $|M_{-}^2-M_{1}^2|\\le1$~\\citep{tam_machwave,1985Tam}. It can be seen that both the LES simulation and the experiments satisfy this condition. Therefore, the present model may be used to compare with the two cases.} However, deviation may occur if intense shocks are involved. In such cases, we would not expect accurate predictions, but it may be possible that some important features of the \\adddd{nonlinear screech} may still be captured by the linear model.}\n\n\\acc{In summary, in this section we show that the effective noise directivity due to the interaction between the instability waves and one shock cell has an intrinsic shape that is close to that of the overall screech directivity. In particular, figure 7(a) shows that the directivity for the fundamental tone has a major lobe in the upstream direction and  a minor lobe in the downstream direction. Comparing with figure 11(a), we see that  incorporating multiple shock-cell interactions results in two lobes that are both thinner and closer to $180^\\circ$ and $0^\\circ$, respectively. This can be expected, because multiple acoustic sources satisfying (3.4) would lead to constructive intereference near $180^\\circ$ (and may or may not be so near $0^\\circ$ depending on operating conditions). Therefore, the two radiation lobes after incorporating multiple effective sources would become thinner and closer to the jet centerline. Similar trends can be observed for the first harmonic by comparing figures 7 and 11 }\n\n\n\n\n\n\t\n\\subsection{Near-field pressure and noise generation mechanism}\n\\begin{figure}\n    \\centering\n    \\includegraphics[width = 0.6\\textwidth]{figure12.eps}\n        \\caption{\\addd{The \\addd{normalized} near-field pressure immediately outside the jet~($x_{0}$ is the starting position of the effective source) due to the  one-cell interaction~($\\alpha_{\\mathrm{i}}\\neq 0$). The Mach number of the fully-expanded jet flow is 1.5. The shock spacing is 2.236, therefore the effective sound source is located between 0 and 2.236. The labels (a), (b), and (c) represent the results at the fundamental frequency, its first and second harmonics, respectively.}}\n    \\label{fig:spl_1.5}\n\\end{figure}\n\t\\begin{figure}\n    \\centering\n    \\includegraphics[width = 0.6\\textwidth]{figure13.eps}\n    \\caption{\\addd{The \\addd{normalized} near-field pressure immediately outside the jet~($x_{0}$ is the starting position of the effective source) due to the  one-cell interaction~($\\alpha_{\\mathrm{i}}\\neq 0$). The Mach number of the fully-expanded jet flow is 1.3. The shock spacing is 1.661, therefore the effective sound source is located between 0 and 1.661. The labels (a), (b), and (c) represent the results at the fundamental frequency, its first  and second harmonics, respectively.}}\n    \\label{fig:spl_1.3}\n\\end{figure}\n\t\\begin{figure}\n    \\centering\n    \\includegraphics[width = 0.6\\textwidth]{figure14.eps}\n    \\caption{\\addd{The \\addd{normalized} near-field pressure immediately outside the jet~($x_{0}$ is the starting position of the effective source) due to the  one-cell interaction~($\\alpha_{\\mathrm{i}}\\neq 0$). The Mach number of the fully-expanded jet flow is 1.2. The shock spacing is 1.327, therefore the effective sound source is located between 0 and 1.327. The labels (a), (b), and (c) represent the results at the fundamental frequency, its first harmonic and second harmonics, respectively.}}\n    \\label{fig:spl_1.2}\n\\end{figure}\n To further examine the noise generation due to the interaction between shock and instability waves, we can calculate the near-field pressure perturbation \\adddd{$p_{+}$} by numerically integrating~(\\ref{equ:numerical integration}). In the results shown below, all lengths are nondimensionalized by the height of the jet $D$. The shock spacing can be obtained from~(\\ref{equ:shock space}). Figure~\\ref{fig:spl_1.5} shows the pressure perturbation immediately outside the jet due to the interaction between instability waves and one shock cell. Labels (a), (b), and (c) represent the results at the fundamental frequency, its first and second harmonics, respectively.\nIt can be seen that at the fundamental frequency, the near-field pressure has a dominated distribution in the upstream direction, while at its first harmonic it shows a strong distribution perpendicular to the jet flow. This is in good agreement with the far-field directivity pattern shown in figure~\\ref{n=1_consider}. For the second harmonic, two major distribution lobes are visible around $80^\\circ$ and $110^\\circ$, whereas the radiation at $\\theta=90^\\circ$ is relatively weak. These results are in good agreement with the directivity patterns in the far field, as shown in figure~\\ref{n=1_consider}(1). When the Mach number of the fully-expanded jet changes to 1.3 or 1.2, as shown in figures~\\ref{fig:spl_1.3} and~\\ref{fig:spl_1.2}, respectively, similar agreement between the near- and far-field is achieved, and we omit a repetitive description for brevity.\n\nIt is interesting to note that for the fundamental frequency the near-field pressure perturbations span the entire shock cell, while for the harmonics they appear to be somewhat localized near the end of the shock, as illustrated in figures~\\ref{fig:spl_1.5}(b-c),~\\ref{fig:spl_1.3}(b-c), and~\\ref{fig:spl_1.2}(b-c). This phenomenon was also  reported in a recent experiment conducted by~\\citet{2020_sound_sources}. \\adddd{However, since the present model has a periodic nature by construction, we cannot determine whether the sources are  physically distributed or localized. But it is interesting to note these near-field behaviours.}\n\\begin{figure}\n    \\centering\n    \\includegraphics[width = 0.55\\textwidth]{figure15.eps}\n    \\caption{Schematic of the Mach wave radiation in supersonic jets. The phase velocity of the near-field pressure fluctuations along the jet flow is $u_{x}$, while the phase velocity of the radiated sound is $c_{0}$.}\n    \\label{fig:machwave}\n\\end{figure}\n\n \n  \n   \n\nUsing the model and results of the near-field pressure fluctuations, we now are in a position to examine the noise generation mechanism due to the interaction between shock and instability waves. As illustrated in section~\\ref{subsec:sound}, the velocity potential takes the form\n\\begin{equation}\n    \\phi_{i+}=  \\frac{U D}{\\pi}\\int_{-\\infty}^{+\\infty}D_{1}(\\lambda)\\mathrm{e}^{-(\\mathrm{i}\\lambda x+\\mathrm{i}\\omega t+\\gamma_{+}y)}\\mathrm{d}\\lambda\\mathrm{e}^{-\\mathrm{i}\\omega t}.\n    \\label{equ:3.8}\n\\end{equation}\nIn the far field,  $\\phi_{i+}$ can be estimated by the saddle point method. The saddle point $\\lambda_{0}$ is $-\\omega M_{+}\\cos\\theta$, where $\\theta$ represents the observer angle to the downstream direction. It can be shown that the $D_{1}(\\lambda)$ is connected with the Fourier \\add{transform} of the near-field pressure fluctuations~(along a fixed $y_{0}$). It is known that in supersonic jets, the phase velocity of the near-field pressure fluctuations along the jet flow can be supersonic relative to the ambient speed of sound, which leads to the Mach\nwave radiation~\\citep{tam_machwave}. As shown in figure~\\ref{fig:machwave}, the Mach angle satisfies\n\\begin{equation}\n    \\theta^{*}=\\arccos(\\frac{c_{0}}{u_{x}}),\n    \\label{equ:mach wave}\n\\end{equation}  \nwhere $\\theta^{*}$ represents the direction of Mach wave radiation, and $u_{x}$, $c_{0}$ denote the phase velocities along the jet flow and the radiation direction, respectively. In our case, the phase velocity of the near-field pressure fluctuations in the $+x$ direction is equal to $-\\omega\/\\lambda$, while the Mach wave has the phase velocity of $1\/M_{+}$ in the radiation direction. So from~(\\ref{equ:mach wave}), we can obtain \n\\begin{equation}\n    \\theta^{*}=\\arccos (-\\frac{\\lambda}{M_{+}\\omega}).\n    \\label{3.10:equ}\n\\end{equation}\nIt is straightforward to find that $\\lambda=-\\omega M_{+}\\cos\\theta^{*}$, which is exactly the same as the saddle point $\\lambda_{0}$. In fact, the saddle point precisely matches the $x$ component of the wavenumber of the Mach wave propagating to the $\\theta$ direction. As the observer angle~(or the Mach wave radiation angle) changes from $0$ to $\\pi$, the saddle point $\\lambda_{0}$ changes from $-\\omega M_{+}$ to $\\omega M_{+}$ correspondingly. Therefore, the noise radiated at angle $\\theta$ is directly related to $D_{1}(\\lambda_{0})$ through the Mach wave mechanism, as shown in~(\\ref{equ:3.8}). We may therefore examine the directivities of the sound generation by examining $|D_{1}(\\lambda)|$ between $-\\omega M_{+}$ and $\\omega M_{+}$. Figure~\\ref{fig:mechanism_3} shows $|D_{1}(\\lambda)|$ as a function of $\\lambda$. We see that sound radiates primarily to the upstream direction~($\\lambda>0$), which is in good agreement with the experimental data.\n\n However, compared with the result in figure~\\ref{n=1_consider}, there is no quick decay of $D_{1}(\\lambda)$ as $\\lambda\\rightarrow\\omega M_{+}$, so the sound at $180^\\circ$ appears to be the strongest. This appears to contradict figure~\\ref{n=1_consider}. This is because $|D_{1}(\\lambda_{0})|$ can only show the overall shape of the directivity pattern, but the directivity is in fact determined by $|D_{1}(\\lambda_{0})\\sin\\theta|$ instead. We can show this by rewriting~(\\ref{equ:3.8}) as\n\\begin{equation}\n    \\phi_{i+}=  \\frac{U D}{\\pi}\\int_{-\\infty}^{+\\infty}D_{1}(\\lambda)\\mathrm{e}^{-r(\\mathrm{i}\\lambda \\cos\\theta+\\gamma_{+}\\sin\\theta)}\\mathrm{d}\\lambda\\mathrm{e}^{-\\mathrm{i}\\omega t},\n    \\label{equ:whatever}\n\\end{equation}\nwhere $r$ again denotes the radial distance, and $\\theta$ represents the observer angle. We consider the substitution\n\\begin{equation}\n    \\lambda=-\\omega M_{+}\\cos(\\theta+\\beta),\n\\end{equation}\nwith which~(\\ref{equ:whatever}) can be rewritten as\n\\begin{equation}\n      \\phi_{i+}=  \\frac{U D}{\\pi}\\int_{P^{*}}D_{1}(-\\omega M_{+}\\cos(\\theta+\\beta))\\sin(\\theta+\\beta)\\mathrm{e}^{\\mathrm{i}\\omega M_{+} r \\cos\\beta}\\mathrm{d}\\beta\\mathrm{e}^{-\\mathrm{i}\\omega t},\n      \\label{equ:3.13}\n\\end{equation}\nwhere $P^{*}$ denotes the new integration contour in the complex $\\beta$ plane. We see that the exponent term in~(\\ref{equ:3.13}) is independent of $\\theta$, and now if we use the saddle point method~(the saddle point is $\\beta_{0}=0$), we obtain\n\\begin{equation}\n      \\phi_{i+}=  \\frac{U D}{\\pi}D_{1}(\\lambda_{0})\\sin\\theta F(r,\\beta),\n      \\label{equ:3.14}\n\\end{equation}\nwhere $F(r,\\beta)$ is independent of $\\theta$.\nSo the result which determines the far-field directivity pattern is indeed $|D_{1}(\\lambda_{0})\\sin\\theta|$.\n\\begin{figure}\n    \\centering\n   \\includegraphics[width = 0.55\\textwidth]{figure16.eps}\n   \\caption{\\addd{$|D(\\lambda)|$ at the frequency of the fundamental tone. The Mach number of the fully-expanded jet flow is 1.3.}}\n    \\label{fig:mechanism_3}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width = 0.3\\textwidth]{figure17.eps}\n    \\caption{The schematic of a monopole located at $(0,0)$ and its radiated sound.}\n    \\label{fig:monopole}\n\\end{figure}\n\nTo better understand this, we take the classical sound field of a monopole as an example. Consider a monopole located at $(0,0)$ in the $x-y$ plane, as shown in figure~\\ref{fig:monopole}. Its velocity potential $\\phi^{*}$ satisfies the homogeneous Helmholtz equation, i.e.\n\\begin{equation}\n    \\bnabla^{2}\\phi^{*}+\\omega^{2}M_{+}^{2}\\phi^{*}=0.\n    \\label{equ:helmholze}\n\\end{equation}\nIts axisymmetric solution in the infinite space takes the form $\\phi^{*}=H_{0}^{1}(\\omega M_{+}r)$, where $H^{1}_{0}(\\omega M_{+}r)$ is the $0$th-order Hankel function of the first kind and $r$ is the radial distance to the sound source. Note such a solution has a uniform directivity. Although we have obtained the exact solution, we still repeat the calculation process to demonstrate how the factor $\\sin\\theta$ emerges. \n\n Similar to~(\\ref{equ:numerical integration}), Fourier \\add{transform} can be used to solve~(\\ref{equ:helmholze}), and the solution is\n \\begin{equation}\n        \\phi^{*}(x,y)=\\frac{1}{2\\pi}\\int_{-\\infty}^{+\\infty}D^{*}(\\lambda)\\mathrm{e}^{-(\\mathrm{i}\\lambda x+\\gamma_{+}y)}\\mathrm{d}\\lambda.\n        \\label{equ:numerical integration_2}\n\\end{equation}\nSince $\\phi^{*}$ is already known, $D^{*}(\\lambda)$ can be obtained by taking the Fourier \\add{transform} of $H_{0}^{1}(\\omega M_{+} r)$ along $y=0$. It is straightforward to find that\n\\begin{equation}\n    D^{*}(\\lambda)=\\dfrac{2}{\\sqrt{\\omega^{2}M_{+}^{2}-\\lambda^{2}}},\n    \\label{equ:d_lambda}\n\\end{equation}\nwhere suitable branch of $\\sqrt{\\omega^{2}M_{+}^{2}-\\lambda^{2}}$ is chosen. In the far field, the saddle point method can be used to estimate~(\\ref{equ:numerical integration_2}). The saddle point is $\\lambda_{0}$, and the final result reduces to\n    \\begin{equation}\n        \\phi^{*}(r,\\theta)=\\frac{\\sqrt{M_{+}\\omega}}{\\sqrt{2\\pi}}D^{*}(\\lambda_{0}){\\rm s i n} \\theta \\frac{\\mathrm{e}^{\\mathrm{i}\\omega(M_{+}r-\\pi\/4)}}{\\sqrt{r}}+O(r^{-3\/2}).\n        \\label{steepest_decent_way_D_star}\n    \\end{equation}\nHere $D^{*}(\\lambda_{0})=2\/\\omega M_{+}\\sin\\theta$, and $\\phi^{*}(r,\\theta)=\\sqrt{2\/\\pi\\omega M_{+}r}\\mathrm{e}^{\\mathrm{i}\\omega(M_{+}r-\\pi\/4)}$, which is exactly the far-field approximation of $H_{0}^{1}(\\omega M_{+} r)$. We can see that the directivity is indeed determined by $|D^{*}(\\lambda_{0})\\sin\\theta|$ rather than $|D^{*}(\\lambda_{0})|$. In fact, the coefficient $D^{*}({\\lambda}_{0})=2\/\\omega M_{+}\\sin\\theta\\rightarrow \\infty$  as the observer angle $\\theta\\rightarrow 180^\\circ$. It is $|D^{*}(\\lambda_{0})\\sin\\theta|$ that remains bounded and is independent of $\\theta$ as expected from the solution $H^{1}_{0}(\\omega M_{+} r).$\n\n Figure~\\ref{fig:mechanism_3} only shows results for the fundamental frequency at $M_{-}=1.3$. Similar results can be obtained for higher harmonics at other Mach numbers. We see that noise is primarily generated through the Mach wave mechanism.\n  \\add{Note that Mach wave radiation can also occur in perfectly-expanded supersonic jets via jet instability waves. However, in this paper, we focus on the interaction between shock and instability waves, where the near-field fluctuations with supersonic phase speed lead to sound generation. This is not to be confused with the convectional Mach wave radiation due to the jet instability waves in  perfectly-expanded supersonic jets.}\n \n\n\n\n \n \n \n\n\n\n\n \n\n\\section{Conclusion}\n\\label{section:conclusion}\nAn analytical model is developed in this paper to predict the sound arising from the interaction between shock and instability waves in imperfectly-expanded 2D jets. Both shock and instability waves are assumed to be of small amplitudes so that linear theories may be used. A vortex-sheet model is used to describe the base jet flow, and 2D Euler equations are subsequently linearised around this base flow to determine the governing equations for shock, instability waves and their interaction, respectively. The interaction between shock and instability waves is determined by solving an inhomogeneous wave equation while simultaneously matching kinematic and dynamic conditions on the vortex sheets. The generated sound in the far field is obtained in a closed form after Fourier \\add{transform} is used in conjunction with the saddle point method. \n\nThe screech frequencies are determined by using the constructive\ninterference assumption proposed by Powell and show good agreement with experimental results. The model can be used to predict the sound due to the instability waves interacting with one shock cell, as well as that with a number of shock cells. \\adddd{The directivity of the sound due to the one-cell interaction is shown to resemble that of the total sound field. It is interesting to note that the noise directivity is sensitive to the local growth rate of the instability waves interacting with the shock cells to generate sound and may be used to explain the discrepancies observed across different experiments. When multiple shock cells are included, the present model shows better agreement with experiments and simulations than the monopole array theory. In particular, the present model corrextly captures the rapid decay of the acoustic radiation when $\\theta$ approaches $180^\\circ$ and $0^\\circ$, respectively.} In particular, noise radiation primarily occurs in the upstream direction but becomes weaker as the observer angle gradually approaches 180 degrees, which is in better agreement with experimental results compared with earlier models. \n\nThe near-field pressure fluctuation due to the shock-instability interaction is subsequently studied. It is shown that the near-field pressure fluctuation has a distribution that is consistent with the far-field directivity patterns. By examining the wavenumber matching of the near-field pressure, we find that noise is generated primarily through the Mach wave mechanism. It is shown that the model developed in this paper can correctly capture the essential physics and may be used to further study the screech in imperfectly-expanded supersonic jets.\n\\section*{Acknowledgments}\nThe authors gratefully acknowledge the funding under Marine S\\&T Fund of Shandong Province for Pilot National Laboratory for Marine Science and Technology (Qiangdao) (No. 2022QNLM010201). The second author (B.L.) thanks Professor A. P. Dowling for an earlier stimulating discussion on jet instability waves.\n\nDeclaration of Interests. The authors report no conflict of interest.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\chapter{Introduction}\n\n\\section{Periodic twisted cohomology}\n\n\\subsection{}\n\nThe twisted de Rham cohomology $H_{dR}(M,\\omega)$ of a manifold $M$ equipped with a closed   three form $\\omega\\in \\Omega^3(M)$\nis the two-periodic cohomology of the complex\n\\begin{equation}\\label{system1002}\n\\Omega(M,\\omega)_{per}\\colon \\dots\\to \\Omega^{ev}(M)\\stackrel{d_\\omega}{\\to} \\Omega^{odd}(M)\\stackrel{d_\\omega}{\\to}\\Omega^{ev}(M)\\to \\dots\\ ,\n\\end{equation}\nwhere $d_\\omega:=d_{dR}+\\omega$ is the sum of the de Rham differential and the operation\nof taking the wedge product with the form $\\omega$. The two-periodic twisted de Rham cohomology is interesting\nas the target of the Chern character from twisted $K$-theory \\cite{MR2172633}, \\cite{MR1977885}, \\cite{MR1911247}, or as a cohomology theory which admits a $T$-duality isomorphism  \\cite{MR2080959}, \\cite{MR2130624}.\n\n\n\n\n\\subsection{}\n\nIn \\cite{bss} we developed a sheaf theory for smooth stacks. Let $f\\colon G\\to X$ be\na gerbe with band $U(1)$ over a smooth stack $X$, and consider a closed three-form   $\\omega\\in \\Omega_X^3(X)$ which\nrepresents the image of the Dixmier-Douady class of the gerbe $G\\to X$ in de\nRham cohomology. The main result of \\cite{bss} states\nthat there exists an isomorphism \n\\begin{equation}\\label{system1000}Rf_*f^*\\underline{\\mathbb{R}}_\\mathbf{X}\\xleftarrow{\\sim} \\Omega_X[[z]]_\\omega\\end{equation}\nin the bounded below derived category $D^+({\\tt Sh}_{\\tt Ab}\\mathbf{X})$ of sheaves of abelian groups on $X$. Here\n$\\underline{\\mathbb{R}}_\\mathbf{X}$ denotes the constant sheaf with value $\\mathbb{R}$ on $X$. Furthermore, \n$\\Omega_X[[z]]_\\omega$ is the sheaf of formal power series of smooth forms on $X$,\nwhere $\\deg(z)=2$, and its differential is given by\n$d_\\omega:=d_{dR}+\\omega\\frac{d}{dz}$. The isomorphism is not canonical, but depends on the\nchoice of a connection on the gerbe $G$ with characteristic  form $\\omega$.\n\n\\subsection{}\n\n\nThe complex (\\ref{system1002}) can be defined for a smooth stack $X$ equipped with a three-form $\\omega\\in \\Omega^3_X(X)$. It is the complex of global sections of a sheaf of two-periodic complexes $\\Omega_{X,\\omega, per}$ on $X$.\nThe complex of sheaves $\\Omega_X[[z]]_\\omega$ is not two-periodic. The relation between $\\Omega_X[[z]]_\\omega$\nand $\\Omega_{X,\\omega, per}$ has been discussed in \\cite[1.3.23]{bss}. Consider the diagram\n\\begin{equation}\\label{uifhuwefewf111}\\mathcal{D}\\colon \\Omega(X)[[z]]_\\omega\\stackrel{\\frac{d}{dz}}{\\leftarrow}\\Omega(X)[[z]]_\\omega\\stackrel{\\frac{d}{dz}}{\\leftarrow}\\Omega(X)[[z]]_\\omega\\stackrel{\\frac{d}{dz}}{\\leftarrow}\\dots\\ .\\end{equation}\nThen there exists an isomorphism\n\\begin{equation}\\label{uifhuwefewf}\n\\Omega_{X,\\omega, per} \\cong {\\tt holim} \\mathcal{D}\\ .\n\\end{equation}\n\n\n\\subsection{}\n\nAs mentioned above, the isomorphism (\\ref{system1000}) depends on the choice\nof a connection on the gerbe $G$. Moreover, the diagram\n$\\mathcal{D}$ depends on these choices via $\\omega$.\nIn order to construct a natural two-periodic cohomology one must find a\nnatural replacement \nof the operation $\\frac{d}{dz}$ which acts on the left-hand side\n$Rf_*f^*\\underline{\\mathbb{R}}_\\mathbf{X}$ of (\\ref{system1000}). It is the first goal of this paper to\ncarry this out properly.\n\n\\subsection{}\n\nOne can do this construction in the framework of smooth stacks developed in \\cite{bss}.\nBut for the present paper we choose the setting of topological stacks.  Only in Subsection \\ref{ufefiwufwefe}\nwe work in smooth stacks and discuss the connection with \\cite{bss}.\nIn Section \\ref{system3003} we develop some aspects of the theory of locally compact stacks and the sheaf theory in this context. For the purpose of this introduction we freely use notions and constructions from this theory.\nWe hope that the ideas are  understandable by analogy with the usual case of sheaf theory on locally compact spaces.\n\n\n\n\n\\subsection{}\n\nLet $G\\to X$ be a $U(1)$-banded gerbe over a locally compact stack.\nThe main object of the present paper is a periodization functor\n $$P_G:D^+({\\tt Sh}_{\\tt Ab}\\mathbf{X})\\to D({\\tt Sh}_{\\tt Ab}\\mathbf{X})$$\nwhich is functorial in $G\\to X$, and where $D^+({\\tt Sh}_{\\tt Ab}\\mathbf{X})$ and\n$D({\\tt Sh}_{\\tt Ab}\\mathbf{X})$ denote the bounded below and unbounded derived categories of sheaves of abelian groups on the site $\\mathbf{X}$ of the stack $X$.\nA simple construction of the isomorphism class of $P_G(F)$ is given in Definition \\ref{system188}.\nThe functorial version is much more complicated. Its construction is completed in Definition \\ref{system18}.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{}\n\nLet us sketch the construction of $P_G$. Recall that gerbes with band $U(1)$ over a locally compact stack $Y$ are\nclassified by $H^3(Y;\\mathbb{Z})$, and automorphisms of a given $U(1)$-gerbe are\nclassified by $H^2(Y;\\mathbb{Z})$ \\cite{heinloth}.\nWe\nconsider the diagram\n$$\\xymatrix{T^2\\times G\\ar[d]_p\\ar[dr]\\ar[rr]^u&&T^2\\times G\\ar[d]^p\\ar[dl]\\\\\nG\\ar[dr]_f&T^2\\times X\\ar[d]&G\\ar[dl]^f\\\\&X&}\\ ,$$\nwhere the  automorphism $u$ of gerbes over $T^2\\times X$  is classified by\n$\\ori_{T^2}\\times 1\\in H^2(T^2\\times X;\\mathbb{Z})$, and where   $\\ori_{T^2}$ denotes the orientation class of the  two-torus. We define a natural transformation\n$$D\\colon Rf_*f^*\\to Rf_* f^*\\colon D^+({\\tt Sh}_{\\tt Ab}\\mathbf{X})\\to D^+({\\tt Sh}_{\\tt Ab}\\mathbf{X})$$\nof degree $-2$ as the composition \n$$D\\colon Rf_* f^*\\stackrel{units}{\\to}Rf^*Rp_*Ru_*u^*p^*f^*\\stackrel{fpu=fp}{\\to}Rf_*Rp_*p^*f^*\\stackrel{\\int_p}{\\to} Rf_*f^*\\ ,$$\nwhere $\\int_p\\colon Rp_*p^*\\to {\\tt id}$ is the integration map of the oriented $T^2$-bundle $T^2\\times G\\to G$.\n\n\n\n\n\n\n\n\n\n\n\n\n\nFor $F\\in  D^+({\\tt Sh}_{\\tt Ab}\\mathbf{X})$ we form the diagram\n$$\\mathcal{S}_G(F)\\colon Rf_* f^*(F)\\stackrel{D}{\\leftarrow} Rf_*f^*(F)[2] \\stackrel{D}{\\leftarrow}Rf_* f^*(F)[4]\\stackrel{D}{\\leftarrow} \\dots$$\nin $D({\\tt Sh}_{\\tt Ab}\\mathbf{X})$.\n\\begin{ddd}\nWe define the periodization\n$P_G(F)\\in D({\\tt Sh}_{\\tt Ab}\\mathbf{X})$\nof $F$ by \n$$P_G(F):={\\tt holim} \\mathcal{S}_G(F)\\in D({\\tt Sh}_{\\tt Ab}\\mathbf{X})\\ .$$\n\\end{ddd}\n\nNote that this introduction is meant as a sketch. In particular, one has to be\naware of the fact that the notion of ${\\tt holim}$ in a triagulated category is\nambiguous and has to be used with great care, as will be explained below and\nin the body of the paper. At present, the above definition only fixes the\nisomorphism class of $P_G(F)$.\n\\subsection{}\n\n\nThe same construction can be applied in  the case of smooth stacks $X$.\nIt is an immediate consequence of Theorem \\ref{theo:D_is_derivative} that there exists an isomorphism \nof the diagrams $S_G(\\underline{\\mathbb{R}}_\\mathbf{X})$ and $\\mathcal{D}$ (see (\\ref{uifhuwefewf111})). Equation  (\\ref{uifhuwefewf})\nimplies the following result.\n\\begin{kor}\nIf $X$ is a smooth manifold, then there exists an isomorphism\n$$P_G(\\underline{\\mathbb{R}}_\\mathbf{X})\\cong \\Omega_{X,\\omega,per} $$\nin $D({\\tt Sh}_{\\tt Ab}\\mathbf{X})$.\nIn particular we have an isomorphism of two-periodic cohomology groups\n$H^*_{dR}(X,\\omega)\\cong H^*(X;P_G(\\underline{\\mathbb{R}}_\\mathbf{X}))$.\n\\end{kor}\n\nThe existence of this isomorphism played the role of a design criterion for the construction of the periodization functor $P_G$. \n\n\n\\subsection{}\n\nThe operation $D\\colon Rf_*f^*(F)\\to Rf_*f^*(F)$ is a well-defined morphism in the derived category.\nIn particular, we get a well-defined diagram $\\mathcal{S}_G(F)\\in D({\\tt Sh}_{\\tt Ab}\\mathbf{X})^{\\mathbb{N}^{op}}$,\nwhere we consider the ordered set $\\mathbb{N}$ as a category. This determines the isomorphism class of the object $P_G(F)\\in D({\\tt Sh}_{\\tt Ab}\\mathbf{X})$.\nWe actually want to define a periodization \\emph{functor}\n$$P_G\\colon D^+({\\tt Sh}_{\\tt Ab}\\mathbf{X})\\to D({\\tt Sh}_{\\tt Ab}\\mathbf{X})\\ ,$$\nwhich also depends functorially on the gerbe $G\\to X$. These functorial properties are\nrequired in our applications to $T$-duality, or if one wants to formulate a statement about the naturality of a Chern character from $G$-twisted $K$-theory with values in the periodic twisted  cohomology $H^*(X;P_G(\\underline{\\mathbb{R}}_\\mathbf{X}))$.\n\n \n\nIn order to define $P_G(F)$ in a functorial way we must refine the diagram\n$\\mathcal{S}_G(F)\\in D({\\tt Sh}_{\\tt Ab}\\mathbf{X})^{\\mathbb{N}^{op}}$ to a diagram in\n$D(({\\tt Sh}_{\\tt Ab}\\mathbf{X})^{\\mathbb{N}^{op}})$. This is the technical heart of the present\npaper. The details of this construction are contained in Section\n\\ref{system7} and will be completed in Definition \\ref{system18}. Along the\nway, we have to use the enhancement of the category of sheaves to bounded\nbelow complexes of flasque sheaves. \n\n\\subsection{}\n\nThe periodization functor\n$P_G$ can be applied to arbitrary objects in $D^+({\\tt Sh}_{\\tt Ab}\\mathbf{X})$. In Proposition \\ref{system30}\nwe calculate examples which indicate some interesting arithmetic features of this functor.\n\n\\section{$T$-duality}\\label{sec:subs-topol-t}\n\n\\subsection{}\n\nTopological $T$-duality is a concept which models the underlying topology of mirror symmetry in algebraic geometry or $T$-duality in string theory. We refer to \\cite{math.GT\/0501487} for a more detailed discussion of the literature. In the present paper we introduce the concept of $T$-duality for pairs $(E,G)$ of a $U(1)$-principal bundle $E\\to B$  over a topological stack $B$ together with a topological gerbe $G\\to E$ with band $U(1)$ using the notion of a $T$-duality diagram.\n\n\\subsection{}\n\nConsider a diagram\n\\begin{equation}\\label{system1004}\n\\xymatrix{&p^*G\\ar[dl]^q\\ar[dr]\\ar[rr]^u&&\\hat p^* \\hat G\\ar[dl]\\ar[dr]^{\\hat q}&\\\\G\\ar[dr]^f&&E\\times_B\\hat E\\ar[dl]^p\\ar[dr]^{\\hat p}&&\\hat G\\ar[dl]^{\\hat f}\\\\&E\\ar[dr]^\\pi&&\\hat E\\ar[dl]^{\\hat \\pi}&\\\\&&B&&}\\ ,\n\\end{equation}\nwhere $\\pi,\\hat \\pi$ are $U(1)$-principal bundles, and $f,\\hat f$ are gerbes with band $U(1)$.\n In \\ref{system1003} \nwe describe the isomorphism class of the universal $T$-duality diagram over the classifying stack $\\mathcal{B} U(1)$. \n\\begin{ddd}[Definition \\ref{eruihfrvc}]\nThe diagram (\\ref{system1004}) is a $T$-duality diagram, if it is locally   isomorphic to the universal $T$-duality diagram.\n\\end{ddd}\nThe pair $(\\hat G,\\hat E)$ is then called a $T$-dual of $(E,G)$.\n\n\\subsection{}\n\nIn Lemma \\ref{lem:identify_T_duality} we will check that this generalizes the\nconcept of T-duality (for $U(1)$-bundles) from\nthe classical situation of\nprincipal bundles in the category of spaces\n\\cite{MR2246781,math.GT\/0501487} and the\nslightly more general situation of such bundles in orbispaces \\cite{MR2246781}\nto arbitrary $U(1)$-actions. The situation of semi-free actions is discussed\n(in a completely different way) in \\cite{pande-2006}. It is an\ninteresting open problem to relate his approach to the approach used here.\n\n\\subsection{}\n\nOne of the main themes of topological $T$-duality is the $T$-duality transformation in twisted cohomology theories. \nIn \\cite{MR2246781} we observed that if the $T$-duality transformation is an isomorphism, then\nthe corresponding twisted cohomology theory must be two-periodic.\n\nThis applies e.g. to twisted $K$-theory. In fact, one can argue that twisted $K$-theory is the universal twisted cohomology theory for which the $T$-duality transformation is an isomorphism\\footnote{We thank M. Hopkins for pointing out a proof of this fact.}.\n\n\n\n\\subsection{}\\label{system1005}\n\nOur construction of $P_G$ is designed such that the corresponding $T$-duality transformation is an isomorphism. To this end we define the periodic $G$-twisted cohomology of $E$\nwith coefficients in $\\pi^*F$, $F\\in D^+({\\tt Sh}_{\\tt Ab}\\mathbf{B})$, by\n$$H^*_{per}(E,G;\\pi^*F):=H^*(E;P_G(\\pi^*F))\\ .$$\nIn this case the $T$-duality transformation\n$$T\\colon H^*_{per}(E,G;\\pi^*F)\\to H^*_{per}(\\hat E,\\hat G;\\hat\\pi^*F)$$\nis induced by the composition\n\\begin{eqnarray*}\nR\\pi_*P_G(\\pi^*F)&\\stackrel{unit}{\\to} &R\\pi_*Rp_*p^*P_G(\\pi^*F)\\\\&\\cong& R\\pi_*Rp_*P_{p^*G}(p^*\\pi^*F)\\\\&\\stackrel{u^*}{\\cong}& R\\pi_*Rp_*P_{\\hat p^*\\hat G}(p^*\\pi^*F)\\\\&\\stackrel{\\pi p=\\hat \\pi \\hat p}{\\to}&R\\hat \\pi_*R\\hat p_*P_{\\hat p^*\\hat G}(\\hat p^*\\hat \\pi^*F)\\\\&\\stackrel{\\cong}{\\to}&R\\hat \\pi_*R\\hat p_*\\hat p^* P_{\\hat G}(\\hat \\pi^*F)\\\\&\\stackrel{\\int_{\\hat p}}{\\to}&\nR\\hat \\pi_* P_{\\hat G}(\\hat \\pi^*(F))\\ .\n\\end{eqnarray*}\nNote that here we use the functoriality of the periodization in an essential way.\n \n\\begin{theorem}[Theorem \\ref{system66}]\nThe $T$-duality transformation in twisted periodic cohomology is an isomorphism.\n\\end{theorem}\n\n\\subsection{}\n\nIf $G\\to X$ is a gerbe over a nice non-singular space $X$, then\n$H^*_{per}(X,G;\\underline{\\mathbb{R}}_{\\mathbf{X}})$ is the correct target of a Chern character from\ntwisted $K$-theory. \nIf $X$ is a topological stack with non-trivial automorphisms of points, then\nthis no longer correct. At the moment we do understand the special case of\norbispaces. In \\cite[Sec.~1.3]{math.KT\/0609576} we give a detailed motivation\nfor the introduction of the twisted delocalized cohomology.\n\nLet $G\\to X$ be a topological gerbe with band $U(1)$ over an orbispace $X$.\nIn \\cite[Definition 3.4]{math.KT\/0609576} we show that it gives rise to a\nsheaf $\\mathcal{L}\\in {\\tt Sh}_{\\tt Ab} \\mathbf{LX}$, where $LX$ is the loop orbispace of  $X$.\n\nThe $G$-twisted delocalized periodic cohomology of $X$ (with complex\ncoefficients) is defined as (see \\cite[Definition 3.5]{math.KT\/0609576}) \n$$H^*_{deloc,per}(X,G):=H^*(LX;P_{G_L}(\\mathcal{L}))\\ ,$$where\n$G_L\\to LX$ is defined by the pull-back\n$$\\xymatrix{G_L\\ar[d]\\ar[r]&G\\ar[d]\\\\LX\\ar[r]&X}\\ .$$\n\nLet us now consider a $T$-duality diagram (\\ref{system1004}) over an orbispace $B$.\nThen we define a $T$-duality transformation\n$$T\\colon H^*_{deloc,per}(E,G)\\to H^*_{deloc,per}(\\hat E,\\hat G)$$\nby a modification of the construction \\ref{system1005}.\n\n\\begin{theorem}[Theorem \\ref{system1006}]\nThe $T$-duality transformation in twisted delocalized periodic cohomology is an isomorphism.\n\\end{theorem}\n\nSo the situation with twisted delocalized periodic cohomology is better  than with orbispace $K$-theory.\nAt the moment we do not know a proof that the $T$-duality transformation in twisted orbifold $K$-theory\nis an isomorphism (see the corresponding comments in \\cite{MR2246781}).\nUsing the fact that the Chern character is an isomorphism, our result implies that\nthe $T$-duality transformation in  twisted orbifold and orbispace $K$-theory\nis an isomorphism after complexification.\n\n\n\n\n\\section{Duality for sheaves on locally compact stacks}\n\n\\subsection{}\n\nIn Section \\ref{system3003} of the present paper we develop some features of a  sheaf theory for locally compact stacks.\nOur main results are the construction of the basic setup, of the functor $f^!$,  and the integration $\\int_f$ for oriented fiber bundles. Section \\ref{system3003}  not only  provides the technical  background for the applications of sheaf theory in the previous sections, but also contains some additional material of independent interest (in particular the results connected with  $f^!$).\n\n\\subsection{}\n\nA presheaf $F$ of sets on a topological space $X$ associates to each open subset $U\\subseteq X$ a set of sections $F(U)$, and to every inclusion $V\\to U$ of open subsets a functorial restriction map\n$F(U)\\to F(V)$, $s\\mapsto s_{|V}$. In short, a presheaf  it is contravariant a functor from the category $(X)$ open subsets of $X$ to sets. A presheaf is a sheaf of it has the following two properties:\n\\begin{enumerate}\n\\item If $s,t\\in F(U)$ are two sections and there exists an open covering $(U_i)$ of $U$ such that $s_{|U_i}=t_{|U_i}$ for all $i$, then $s=t$.\n\\item If $(U_i)$ is an open covering of $U$ and $(s_i)$ is a collection of sections $s_i\\in F(U_i)$\nsuch that $s_{i|U_i\\cap U_j}=s_{j|U_i\\cap U_j}$ for all pairs $i,j$, then there exists a section $s\\in F(U)$ such that\n$s_{|U_i}=s_i$ for all $i$.\n\\end{enumerate}\nThe notion of a sheaf is thus determined by the Grothendieck topology on $(X)$ given by the collections of open coverings of open subsets. We will call $(X)$ the small site associated to $X$.\n\n \nIf $X$ is a topological stack, then  the open substacks form a two-category which  does not give the appropriate setting for sheaf theory on $X$. For example, if $G$ is a finite group, then the quotient stack $[*\/G]$\nis quite non-trivial but does not have  proper open substacks.\nOn the other hand its identity one-morphism has the two-automorphism group $G$, and in a non-trivial theory sheaves should reflect the two-automorphisms.\n\n\\subsection{}\n\nFor applications to twisted cohomology a setting for sheaf theory on smooth stacks has been introduced in \\cite{bss}. In the present paper we develop a similar theory for topological stacks. There are various choices to be made in order to define the site of a stack in topological spaces. The   sheaf theories associated to these choices will have many features in common, but will differ in others. The main goal of the present paper is the construction of the periodization functor $P_G$ associated to a $U(1)$-banded gerbe $G\\to X$. One of the main ingredients of the construction is an integration $\\int_f$ for oriented fiber   bundles $f$ with a closed topological manifold as fiber.\nIn order to define the integration map we need a projection formula which\nexpresses a compatibility of the pull-back and push-forward operations  with\ntensor products, see Lemma \\ref{system81}. Already  for the projection formula\nin ordinary sheaf theory one needs local compactness assumptions. For this\nreason we decided to work generally with locally compact stacks and spaces\nthough much of the theory would go through under more general or different\nassumptions.\n\n\n\n\n\\subsection{}\n\nA stack in topological spaces is topological if it admits an atlas $A\\to X$. From the atlas we can derive a groupoid $A\\times_XA\\rightrightarrows A$ which represents $X$ in an appropriate sense.\nThe stack is called locally compact if one can find an atlas $A\\to X$ such that the resulting groupoid\nis locally compact (i.e. $A$ and $A\\times_XA$ are locally compact spaces).\n\nThe site $\\mathbf{X}$ associated to a locally compact stack is the category of locally compact spaces $(U\\to X)$ over $X$ such that the  morphisms are  morphisms of spaces over $X$ (i.e. pairs of a morphism between the spaces and a two-morphism filling the obvious triangle.) We require that the structure morphism $U\\to X$ has local sections.\n The topology on $\\mathbf{X}$\nis again given by the collections of coverings by open subsets of the objects $(U\\to X)$.\nFor many constructions and calculations the restriction functors from sheaves on $\\mathbf{X}$ to sheaves on $(U)$ play a distinguished role. They are used to build the connection between operations with sheaves on the stack $X$ and corresponding classical operations in sheaf theory on the spaces $U$.\n\n\n\\subsection{}\n\nFor the theory of stacks in topological spaces in general we refer to \\cite{heinloth}, \\cite{math.KT\/0609576}, \\cite{math.AG\/0503247}.\nSome special aspects of locally compact stacks are discussed in Subsection \\ref{wuiefwefwefqwd} of the present paper.\n\nIn our treatment of sheaf theory on the site $\\mathbf{X}$ \nwe  give a description of the closed monoidal structure on the categories of sheaves  and presheaves of abelian groups ${\\tt Sh}_{\\tt Ab}\\mathbf{X}$ and $\\Pr_{\\tt Ab}\\mathbf{X}$ on $\\mathbf{X}$.   The interplay between sheaves and presheaves will be important when we study the compatibility of the monoidal structures with the functors  $$f^*:{\\tt Sh}_{\\tt Ab}\\mathbf{Y} \\leftrightarrows {\\tt Sh}_{\\tt Ab}\\mathbf{X}: f_*$$\nassociated to a morphism of locally compact stacks $f:X\\to Y$. \nIn general these functors do not come from a morphisms of sites but are constructed in an ad-hoc manner. Because of this we must check under which conditions properties expected from the classical theory  carry over to the present case.\n\n\n\n\n\n\n\nThe  derived versions of these functors on the bounded below and unbounded derived categories $D^+({\\tt Sh}_{\\tt Ab}\\mathbf{X})$ and $D({\\tt Sh}_{\\tt Ab}\\mathbf{X})$ will play an important role in the present paper. In order to deal with the unbounded derived category we use an approach via model categories.\n\n\\subsection{}\n\nBesides the development of the basic set up which we will not discuss further in the introduction\nlet us now explain the two main results which may be of independent interest. \n\\begin{theorem}[Theorem \\ref{main123}]\\label{zueefewf}\nIf $f:X\\to Y$ is a proper representable map between locally compact stacks such that $f_*$ has finite cohomological dimension, then the functor $Rf_*:D^+({\\tt Sh}_{\\tt Ab}\\mathbf{X})\\to D^+({\\tt Sh}_{\\tt Ab}\\mathbf{Y})$ has a right-adjoint,\ni.e. we have an adjoint pair\n\\begin{equation}\\label{hiduwqdwqdwqdq}\nRf_*:D^+({\\tt Sh}_{\\tt Ab}\\mathbf{X})\\leftrightarrows D^+({\\tt Sh}_{\\tt Ab}\\mathbf{Y}):f^!\\ .\n\\end{equation}\n\n\\end{theorem}\nWe  think that one could prove a more general theorem stating  the existence of a right adjoint\nof a functor $Rf_!$ where $f_!$ is the push-forward with proper support along an arbitrary map between\nlocally compact stacks such that $f_!$ has finite cohomological dimension, though we have not checked all details.  \n\nThis theorem generalizes a well-known result  (\\cite{MR1610971}, \\cite[Ch.~3]{MR1299726} in ordinary sheaf theory. Its importance is due to the classical calculation \n\\begin{equation}\\label{wuiefwefwef}\nf^!(F)\\cong f^*(F)[n]\n\\end{equation} (compare \\cite[Prop.3.3.2]{MR1299726}) for $F\\in D^+({\\tt Sh}_{\\tt Ab}(Y))$, if $f:X\\to Y$ is an oriented  locally trivial bundle of closed connected topological $n$-dimensional manifolds on a locally compact space $Y$. If we would know such an  isomorphism in the present case (for sheaves on the sites $\\mathbf{X}, \\mathbf{Y}$ and stacks $X,Y$), then  we could define the integration map as the composition\n$$\\int_f:Rf_*f^*(F)\\stackrel{\\sim}{\\to} Rf_*f^!(F)[-n]\\stackrel{counit}{\\to} F[-n]\\ ,$$\nwhere the last map is the co-unit of the adjunction (\\ref{hiduwqdwqdwqdq}).\n\nUnfortunately, at the moment we are not able to calculate $f^!(F)$ in any interesting example.\nHowever, we can construct the integration map in a direct manner avoiding the knowledge of (\\ref{wuiefwefwef}).\n\nSome elements of the theory developed here are formally similar to the work \\cite{MR2312554}\non sheaves on the lisse \\'etale site of an Artin stack. In this framework in \\cite{laszlo-2005} a functor\n$f^!$ was introduced  between derived categories of constructible sheaves.  On the one hand the methods seem to be completely different. On the other hand this functor has the expected behavior for\nsmooth maps, i.e. it satisfies a relation like (\\ref{wuiefwefwef}).  At the moment we do not see  even a formal relation between  the construction of \\cite{laszlo-2005} with the construction in the present paper which could be exploited for a calculation of $f^!(F)$. \n\n\\subsection{}\n\n\n\n\n\n\n\n\nThe following Theorem is the result of Subsection \\ref{system103}.\n\\begin{theorem}\nIf the map $f:X\\to Y$ of locally compact stacks  is an oriented locally trivial fiber bundle with a closed connected topological $n$-dimensional manifold as fiber,\nthen there exists an integration map, a natural transformation of functors\n$$\\int_f:Rf_*f^*\\to {\\tt id}[-n]:D^+({\\tt Sh}_{\\tt Ab} \\mathbf{X})\\to D^+({\\tt Sh}_{\\tt Ab}\\mathbf{X})$$\nwhich has the expected compatibility with pull-back and compositions.\n\\end{theorem}\nIn Subsection \\ref{system4001} we extend the push-forward and pull-back operations to the unbounded derived categories and construct the integration map in this setting.\n\n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\chapter{Gerbes and periodization}\\label{system3001}\n\n\\section{Sheaves on the locally compact site of a stack}\n\n\\subsection{}\\label{system1007}\n\nLet ${\\tt Top}$ denote the site of topological spaces. The topology is generated by  covering families\n${\\tt cov}_{\\tt Top}(A)$ of the objects $A\\in {\\tt Top}$, where ${\\tt cov}_{\\tt Top}(A)$ is the set  of\ncoverings by collections of open subsets.\n\nA stack will be a stack on the site ${\\tt Top}$. Spaces are considered as stacks\nthrough the Yoneda embedding.\n\n\nA map  $A\\to X$ from a space $A$ to a stack $X$ which is surjective,\nrepresentable, and has local sections is called an atlas. \nWe refer to \\ref{staqwndwqodwqd} for definitions and more details about stacks in topological spaces.\n\\begin{ddd}\nA topological stack\nis a stack which admits an atlas. \n\\end{ddd}\n\n\n\n\\begin{ddd}\\label{qwuidiuwqdwqdwqd1fwefw}\nA topological space is locally compact if it is Hausdorff and every point admits a compact neighborhood.\n A stack is called locally compact if it admits an atlas $A\\to X$ such that $A$ \nand  $A\\times_XA$ are locally compact.\n\\end{ddd}\n\nIf $X$ is a locally compact stack, then the site of $X$ is the subcategory ${\\tt Top}_{lc}\/X$ of locally compact spaces over $X$ such that the structure map $A\\to X$ has local sections. The  topology is induced from ${\\tt Top}$. We denote this site by $\\mathbf{X}$ or ${\\tt Site}(X)$. See \\ref{hshsqiwhswqs} for more details.\n\n\\subsection{}\n\nAs will be explained in \\ref{sec:morph_of_sheaves}, a morphism of locally compact\nstacks $f\\colon X\\to Y$ gives rise to an adjoint pair of functors\n$$f^*\\colon {\\tt Sh}\\mathbf{Y}\\leftrightarrows{\\tt Sh}\\mathbf{X}:f_*\\ .$$\nThe functor\n$f_*$ is left-exact on the categories of sheaves of abelian groups and admits\na right-derived\n$$Rf_*\\colon  D^+({\\tt Sh}_{\\tt Ab}\\mathbf{X})\\to D^+({\\tt Sh}_{\\tt Ab}\\mathbf{Y})$$\nbetween the bounded below derived categories, compare  \\ref{lem:lrexact}.\n\n\n\n \n\n\\subsection{}\\label{system102}\n\n\n\nLet $M$ be some space.\n\\begin{ddd}\nA map between topological stacks $f:X\\to Y$ is a locally trivial fiber bundle with fiber $M$ if\nfor every space $U\\to X$ the pull-back $U\\times_YX\\to U$ is a locally trivial\nfiber bundle of spaces with fiber $M$.\n \\end{ddd}\n\n\n\nAssume that $M$ is a closed connected and  orientable $n$-dimensional topological manifold.\n\\begin{ddd}\\label{iquhuiqwdddqwfefefefefeffefeffefe}\nLet $f\\colon X\\to Y$ be a map of locally compact stacks which\nis a locally trivial fiber bundle  with fiber $M$. It is called orientable if\nthere exists an isomorphism $R^nf_*(\\underline{\\Z}_\\mathbf{X})\\cong \\underline{\\Z}_\\mathbf{Y}$. An orientation of $f$ is a choice of such an isomorphism.\n\\end{ddd}\n\n\n\n\n\n\\subsection{}\n\nLet $f\\colon X\\to Y$ be a locally trivial oriented fiber bundle with $n$-dimensional fiber $M$\nover a locally compact stack $Y$. Under these assumption we can generalize the integration map\n(see \\cite[Sec.~3.3]{MR1299726}) \n\\begin{theorem}[Definition \\ref{rdphi}]\\label{system2}\nIf $f\\colon X\\to Y$ be a locally trivial oriented fiber bundle over a locally compact stack with fiber a closed topological manifold of dimension $n$, then we have an integration map, i.e. a natural transformation of functors\n$$\\int_f\\colon Rf_*\\circ f^*\\to {\\tt id}\\colon D^+({\\tt Sh}_{\\tt Ab}\\mathbf{Y})\\to D^+({\\tt Sh}_{\\tt Ab}\\mathbf{Y})$$\nof degree $-n$.\n\\end{theorem}\n\n\\subsection{}\n\n\nWe consider a map of locally compact stacks $f\\colon X\\to Y$ which is a locally trivial oriented fiber bundle  with fiber a closed topological manifold of dimension $n$. \nFurthermore let $U\\to X$ be a morphisms of locally compact stacks which has local sections. Then we form the Cartesian\\footnote{{In the present paper by a Cartesian diagram in the two-category of stacks we mean a $2$-Cartesian diagram. In particular, the square commutes up to a $2$-isomorphism which we often omit to write in order to simplify the notation. More generally, when we talk about a commutative diagram in stacks, then we mean a diagram of $1$-morphisms together with a collection of $2$-isomorphism filling all faces in a compatible way, and again we will usually not write the $2$-isomorphisms explicitly.}} diagram\n$$\\xymatrix{V\\ar[r]^v\\ar[d]^g&X\\ar[d]^f\\\\U\\ar[r]^u&Y}\\ .$$\nNote that $g:V\\to U$ is again a locally trivial oriented fiber bundle  with fiber a closed topological manifold of dimension $n$. The orientation of $f$ (which gives the marked isomorphism below) induces an orientation of $g$ by\n$$R^ng_*(\\underline{\\Z}_\\mathbf{V})\\cong R^ng_*v^*(\\underline{\\Z}_\\mathbf{X})\\stackrel{(\\ref{wqdwqdqw})}{\\cong}u^*R^nf_*(\\underline{\\Z}_\\mathbf{X})\\stackrel{!}{\\cong} u^*(\\underline{\\Z}_\\mathbf{Y})\\cong \\underline{\\Z}_{\\mathbf{U}}\\ .$$\n\n\n\n\n \n\\begin{lem}\\label{ghtq}\nThe following diagrams commute\n\\begin{equation}\\label{fde4}\\xymatrix{u^*\\circ Rf_*\\circ f^*\\ar[r]^\\cong\\ar[d]^{u^*\\int_f}&Rg_*\\circ v^*\\circ f^*\\ar[d]^\\cong\\\\\nu^*&Rg_*\\circ g^*\\circ u^* \\ar[l]^{\\int_g}}\n\\quad \n \\xymatrix{Ru_*\\circ Rg_*\\circ g^*\\ar[r]^\\cong\\ar[d]^{Ru_*\\int_g}&Rf_*\\circ Rv_*\\circ g^*\\ar[d]^\\cong\\\\\n   Ru_*&Rf_*\\circ f^*\\circ Ru_* \\ar[l]^{\\int_f Ru_*}}\n\\ .\\end{equation}\n\\end{lem}\n\\begin{proof}\nCommutativity of the first\ndiagram follows immediately from the stronger (because valid in the derived\ncategory of unbounded complexes)  Lemma\n\\ref{system300}. Commutativity of the second diagram is proved in Lemma\n\\ref{lem:funct_int1}, but only for the bounded below derived category. \n\\end{proof}\n\n\n \n\\section{Algebraic structures on the cohomology of a gerbe}\n \n\n\\subsection{}\\label{ddef2}\n\nLet $X$ be a locally compact  stack and\n$f\\colon G\\to X$ be a topological gerbe with band $U(1)$.  \nThen $G$ is a locally compact stack.\nIndeed, we can choose an atlas $A\\to X$ such that $A$ and $A\\times_XA$ are locally compact, and there exists a section \n$$\\xymatrix{&G\\ar[d]\\\\A\\ar@{.>}[ur]\\ar[r]&X}\\ .$$\nThen $A\\to G$ is an atlas and $A\\times_GA\\to A\\times_XA$ is a locally trivial $U(1)$-bundle. In  particular, $A\\times_GA$ is a locally compact space. \n\n \n\n\n\\subsection{}\\label{uidqwdqwdqdqwdd}\n\n\n\n\nBy $T^2$ we denote the two-dimensional torus. We fix an orientation of $T^2$.\nWe consider the pull-back ${\\tt pr}_2^*G\\cong T^2\\times G\\to T^2\\times X$. The isomorphism classes of automorphisms of this gerbe are classified by $H^2(T^2\\times X;\\mathbb{Z})$. Let\n$$\\xymatrix{{\\tt pr}^*_2G\\ar[rr]^\\phi\\ar[dr]& &{\\tt pr}_2^*G\\ar[dl]\\\\&T^2\\times X&}$$ \nbe an automorphism classified by\n$\\ori_{T^2}\\times 1_X\\in H^2(T^2\\times X;\\mathbb{Z})$.\nWe consider the diagram\n\\begin{equation} \n  \\xymatrix{{\\tt pr}_2^* G\\ar[rd]\\ar[rr]^\\phi\\ar[d]^{p}&&{\\tt pr}_2^*G\\ar[ld]\\ar[d]^{p}\\\\\nG\\ar[dr]^f&T^2\\times X\\ar[d]&G\\ar[dl]^f \\\\&X&}\\ .\n\\label{eq:dtdul}\n\\end{equation}\n\n{\nNotice that $\\phi$ is unique up to a non-canonical $2$-isomorphism. In the present paper we\nprefer a more canonical choice. We will fix the morphism $\\phi$ once and for all\nin the special case that $X$ is a point and $G=\\mathcal{B} U(1)$, i.e. we fix a diagram\n$$\\xymatrix{T^2\\times \\mathcal{B} U(1)\\ar[d]\\ar[rr]^{\\phi_{univ}}\\ar[dr]&&T^2\\times \\mathcal{B} U(1)\\ar[d]\\ar[dl]\\\\\\mathcal{B} U(1)\\ar[dr]&T^2\\ar[d]&\\mathcal{B} U(1)\\ar[dl]\\\\&{*}&}\\ .$$\nIf $G\\to X$ is a topological gerbe with band $U(1)$, then we obtain the induced diagram by taking products\n$$\\xymatrix{G\\times T^2\\times  \\mathcal{B} U(1)\\ar[d]\\ar[rr]^{{\\tt id}_G\\times \\phi_{univ}}\\ar[dr]&&G\\times T^2\\times \\mathcal{B} U(1)\\ar[d]\\ar[dl]\\\\G\\times \\mathcal{B} U(1)\\ar[dr]&X\\times T^2\\ar[d]&G\\times \\mathcal{B} U(1)\\ar[dl]\\\\&X&}\\ .$$\nWe now replace the products $\\mathcal{B} U(1)\\times G$ by the tensor product of gerbes\nas explained in \\cite[6.1.9]{math.AT\/0701428} and identify $\\mathcal{B} U(1)\\otimes G$ with $G$ using the canonical isomorphism in order to get\n$$\\xymatrix{{\\tt pr}_2^*G\\ar[d]^p\\ar[rr]^{\\phi}\\ar[dr]&&{\\tt pr}_2^* G \\ar[d]^p\\ar[dl]\\\\G \\ar[dr]^f& T^2\\times X\\ar[d]&G\\ar[dl]_f\\\\&X&}\\ .$$\nIn this way we have constructed a $2$-functor from the $2$-category of $U(1)$-banded gerbes over $X$ to the $2$-category of diagrams of the form (\\ref{eq:dtdul}).\nBy taking prefered models for the products we can, if we want, assume a strict equality\n$f\\circ p\\circ \\phi_G=f\\circ p$.\n}\n\n\n\n\\subsection{}\n\nObserve that the map of locally compact stacks $p\\colon {\\tt pr}_2^*G\\to G$ is a locally trivial oriented fiber bundle with fiber $T^2$.\nTherefore we have the integration map (see \\ref{system2})\n$$\\int_p\\colon Rp_*\\circ p^*\\to{\\tt id}\\ .$$\n\\begin{ddd}\\label{defofd}\nWe define a natural endo-transformation $D_G$ of the functor \n$$Rf_*\\circ f^*\\colon D^+({\\tt Sh}_{\\tt Ab}\\mathbf{X})\\to\nD^+({\\tt Sh}_{\\tt Ab}\\mathbf{X})$$ of degree $-2$ which associates to $F\\in D^+({\\tt Sh}_{{\\tt Ab}}\\mathbf{X})$ the morphism  \n\n\\begin{multline*}\n  Rf_*\\circ f^*(F)\\stackrel{units}{\\longrightarrow}Rf_*\\circ Rp_*\\circ\n  R\\phi_*\\circ \\phi^* \\circ p^*\\circ f^*(F)\\\\\n\\xrightarrow{f\\circ p\\circ\n    \\phi= f\\circ p} Rf_*\\circ Rp_*\\circ p^*\\circ f^*(F) \\stackrel{\\int_p}{\\to}\n  Rf_*\\circ f^*(F)\\ .\n\\end{multline*}\n \\end{ddd}\n\n\n\\subsection{}\nIt follows from Lemma \\ref{ghtq} that $D_G$ is compatible with pull-back diagrams.\nIn fact, consider a Cartesian diagram \n$$\\xymatrix{ G^\\prime\\ar[d]^{f^\\prime}\\ar[r]&G\\ar[d]^f\\\\\nX^\\prime\\ar[r]^g&X}\\ .$$\n{Using the canonical construction explained in \\ref{uidqwdqwdqdqwdd} we  extend this to a morphism between diagrams of the form (\\ref{eq:dtdul}).}\n Then we have  the  commutative diagram\n$$\\xymatrix{g^*\\circ Rf_*\\circ f^*\\ar[r]^\\sim\\ar[d]^{g^*D_G}&Rf^\\prime_*\\circ (f^\\prime)^*\\circ g^*\\ar[d]^{D_{G^\\prime}\\circ g^*}\\\\\ng^*\\circ Rf_*\\circ f^*\\ar[r]^\\sim&Rf^\\prime_*\\circ (f^\\prime)^*\\circ g^*}\\ .$$\n \n\n\\subsection{}\n\nWe compute the action of $D_G$ in the case of the trivial gerbe $f:G\\to *$ and the sheaf $\\underline{F}\\in {\\tt Sh}_{\\tt Ab}{\\tt Site}(*)$ represented by a discrete abelian group $F$. Note that\n$Rf_*\\circ f^*(\\underline{F})$ is an object of $D^+({\\tt Sh}_{\\tt Ab} {\\tt Site}(*))$. We get an object \n$Rf_*\\circ f^*(\\underline{F})(*)\\in D^+({\\tt Ab})$ by evaluation at the object $(*\\to *)\\in {\\tt Site}(*)$.\n\\begin{lem}\\label{system5}\nThere exists an isomorphism\n$$H^*(Rf_*\\circ f^*(\\underline{F})(*))\\cong F\\otimes \\mathbb{Z}[[z]]\\ ,$$\nwhere $\\deg(z)=2$. On cohomology the transformation $D_G$ is given by  $D_G={\\tt id}\\otimes \\frac{d}{dz}$.\n\\end{lem}\n\\begin{proof}\n We choose a lift\n$*\\to G$. Forming iterated fiber products we get a simplicial space\n$$\\dots *\\times_G*\\times_G*\\times_G*\\to *\\times_G*\\times_G*\\to *\\times_G*\\to *\\ .$$\nNote that $*\\times_G*\\cong U(1)$.\nOne checks that the simplicial space is equivalent to the simplicial space\n$BU(1)^\\cdot$, the classifying space of the group $U(1)$,\n$$U(1)\\times U(1)\\times U(1)\\to U(1)\\times U(1)\\to U(1)\\to *\\ .$$\nLet $(U\\to *)\\in {\\tt Site}(*)$. \nIf $H\\in {\\tt Sh}_{\\tt Ab}\\mathbf{G}$, then  we consider an injective resolution $0\\to H\\to I^\\cdot$.\nThe evaluation $I^\\cdot(U\\times BU(1)^\\cdot)$ gives a cosimplicial complex, and after normalization, a double complex. Its total complex represents $Rf_*(H)(U\\to *)$ (see \\cite[Lemma 2.41]{bss} for a proof of the corresponding statement in the smooth context).\nWe calculate the cohomology of  $Rf_*(H)(U\\to *)$ using the associated spectral sequence.\nIts second page has the form $$E_2^{p,q}\\cong H^p(U\\times BU(1)^q;H)\\ .$$\n\nWe now specialize to the sheaf $H=f^*(\\underline{F})\\cong \\underline{F}_G$, where $F$ is a discrete abelian group, and $U=*$. In this case the spectral sequence is the usual spectral sequence which calculates the cohomology of the realization of the simplicial space $BU(1)^\\cdot$ with coefficients in $F$.\nNote that $H^*(B U(1);\\mathbb{Z})\\cong  \\mathbb{Z}[[z]]$ as rings with $\\deg(z)=2$. Since it is torsion free as an abelian group\nwe get\n$$H^*(R^*f_*\\circ f^*(\\underline{F})(*))\\cong  F\\otimes  H^*(B U(1);\\underline{\\Z})\\cong F\\otimes  \\mathbb{Z}[[z]]\\ .$$\nIn a similar manner we calculate $Rf_*\\circ Rp_*\\circ p^*\\circ f^*(\\underline{F})(*)$. Its cohomology is\n$H^*(T^2\\times B U(1);F)$, hence we have\n$$H^*(Rf_*\\circ Rp_*\\circ p^*\\circ f^*(\\underline{F}) (*))\\cong F\\otimes H^*(T^2\\times B U(1);\\mathbb{Z})\\cong F\\otimes \\Lambda(u,v)\\otimes  \\mathbb{Z}[[z]]\n\\ ,$$\nwhere $u,v\\in H^1(T^2,\\mathbb{Z})$ are the canonical generators.\n\n\n\nFor every topological group $\\Gamma$ we have a natural map\n$\\Gamma \\to \\Omega(B\\Gamma)$. By adjointness we get a map\n$c:U(1)\\times \\Gamma\\to U(1)\\wedge \\Gamma\\to B\\Gamma$.\nWe will need a simplicial model $c^\\cdot$ of this map.\nWe consider the standard simplicial model $\\SSSS^\\cdot$ of $U(1)$ with two non-degenerate simplices, one in degree $0$, and one in degree $1$. Then $\\SSSS^\\cdot\\times \\Gamma$ is a simplicial model\nof $U(1)\\times \\Gamma$. It suffices to describe the map $c^\\cdot$ on the non-degenerate part of $\\SSSS^\\cdot\\times \\Gamma$. The component $c^0$ maps\n$\\SSSS^0\\times \\Gamma$ to the base point $*$ of $B\\Gamma^\\cdot$.\n The component $c^1$ is the natural identification of the non-degenerate copy of $\\Gamma\\subset\n\\SSSS^1\\times \\Gamma$ with $\\Gamma\\cong B\\Gamma^1$.\n\nWe now specialize to the case $\\Gamma=U(1)$.\nWe get a map\n$c:T^2\\cong U(1)\\times U(1)\\to B U(1)$, or on the simplicial level, a map\n$c^\\cdot:\\SSSS^\\cdot\\times U(1)\\to B U(1)^\\cdot$.\nWe have $H^*(BU(1);\\mathbb{Z})\\cong \\mathbb{Z}[[z]]$ with $z$ odd degree $2$, and one checks that $uv=c^*(z)\\in H^2(T^2;\\mathbb{Z})$\n(after choosing an appropriate basis $u,v\\in H^1(T^2;\\mathbb{Z})$).\n\nNote that $BU(1)^\\cdot$ is a simplicial abelian group. \nThe discussion above shows that the automorphism $\\phi\\colon G\\to G$ in (\\ref{eq:dtdul}) with $X=*$ and classified by $uv\\in H^2(T^2;\\mathbb{Z})$ can be arranged so that it induces\nan automorphism of bundles of $BU(1)^\\cdot$-torsors\n\\begin{equation}\\label{uqfbqwdqwqwd}\n\\xymatrix{\\SSSS^\\cdot\\times U(1)\\times BU(1)^\\cdot\\ar[dr]\\ar[rr]_{\\phi^{\\cdot}}^{(t,x)\\mapsto (t,c^\\cdot(t)x)}&&\\SSSS^\\cdot\\times U(1)\\times BU(1)^\\cdot\\ar[dl]\\\\&\\SSSS^\\cdot\\times U(1)&}\\ .\n\\end{equation}\n\nUnder this isomorphism \nthe action of\n\\begin{equation}\n\\phi^*\\colon H^*(Rf_*\\circ Rp_*\\circ p^*\\circ f^*(\\underline{F})(*)) \\to H^*(Rf_*\\circ Rp_*\\circ\np^*\\circ f^*(\\underline{F})(*))\n\\label{eq:action_of_phi}\n\\end{equation}\n is induced by  $z\\mapsto z+u v$, $u\\mapsto u$, $v\\mapsto v$. \nIn order to see this note that\n$m^*(z)=z_1+z_2$, where  $m:BU(1)\\times BU(1)\\to BU(1)$ is the multiplication,\nand $H^*(BU(1)\\times BU(1);\\mathbb{Z})\\cong \\mathbb{Z}[[z_1,z_2]]$.\nAfter realization the  map $\\phi^\\cdot$ leads to  the composition\n$$T^2\\times BU(1)\\stackrel{({\\tt id}_{T^2}, c)\\times {\\tt id}}{\\to}T^2\\times  BU(1)\\times BU(1)\\stackrel{{\\tt id}_{T^2}\\times m}{\\to} T^2\\times BU(1)$$\nwhich maps $$z\\stackrel{({\\tt id}_{T^2}\\times m)^*}{\\mapsto} z_1+z_2\\stackrel{(({\\tt id}_{T^2},c)\\times {\\tt id})^*}{\\mapsto} uv+z\\ .$$\nIn cohomology of the evaluations at the point the integration\nmap\n$$\\int_p\\colon \nRf_*\\circ Rp_*\\circ p^*\\circ f^*(\\underline{F})\\to Rf_*\\circ f^*(\\underline{F})$$\ninduces the map \n$F\\otimes \\Lambda(u,v)\\otimes  \\mathbb{Z}[[z]] \\to F\\otimes  \\mathbb{Z}[[z]]$ which takes\nthe coefficient at $uv$. This implies the assertions of Lemma \\ref{system5}. \n\\end{proof}\n\n\n\\section{Identification of the transformation $D_G$ in the smooth case}\\label{ufefiwufwefe}\n\n\n\\subsection{}\nIn this subsection we work in the context of \\cite{bss} of manifolds and smooth stacks.\nIt can be considered as a supplement to \\cite{bss} concerning the  transformation $D_G$ introduced in Definition \\ref{defofd} which can be defined in the smooth context in  a parallel manner.\n\nIf $X$ is a smooth stack, then $\\Omega_X$ denotes the sheaf of de Rham complexes on $X$.\nIt associates to $(U\\to X)\\in \\mathbf{X}$ the de Rham complex $\\Omega_X(U\\to X):=\\Omega(U)$ of the manifold $U$. Note that in this subsection $\\mathbf{X}$ denotes the site of a smooth stack introduced in \\cite{bss}.\n\nIf $\\omega\\in \\Omega_X^3(X)$ is a closed $3$-form, then we form the sheaf of twisted de Rham complexes $ \\Omega_X[[z]]_\\omega$. Its evaluation at\n$(U\\to X)\\in \\mathbf{X}$ is the complex $\\Omega_X[[z]]_\\omega(U\\to X):=\\Omega(U)[[z]]\\cong \\Omega(U)\\otimes_\\mathbb{Z}\\Z[[z]]$ with differential\n$d_{dR}+\\omega \\frac{d}{dz}$.\nIn this formula the form $\\omega$ acts by wedge multiplication with the pull-back of $\\omega$ to $U$.\n \nLet  $f\\colon G\\to X$ be a gerbe with band $U(1)$ over a \\textit{smooth manifold} $X$. \nThe choice of a gerbe connection determines a closed $3$-form $\\omega\\in \\Omega_X^3(X)$ which  represents the\nDixmier-Douady class of the gerbe. By\n\\cite[Theorem 1.1]{bss} we have an\nisomorphism\n\\begin{equation}\\label{wukhwefwefw}\n  Rf_* f^* \\underline{\\mathbb{R}}_\\mathbf{X}\\stackrel{\\sim}{\\to} \\Omega_X[[z]]_\\omega \n\\end{equation}\nin the derived category $D^+({\\tt Sh}_{\\tt Ab}\\mathbf{X})$.\n \n \n \n\n \n\n\\subsection{}\n\n\\begin{theorem}\\label{theo:D_is_derivative}\nWe have a commutative diagram\n  \\begin{equation*}\n    \\begin{CD}\n      Rf_* f^*\\underline{\\mathbb{R}}_\\mathbf{X} @<{\\cong}<(\\ref{wukhwefwefw})< \\Omega_X[[z]]_\\omega\\\\\n      @VV{D_G}V   @VV{\\frac{d}{dz}}V\\\\\n      Rf_*f^*\\underline{\\mathbb{R}}_\\mathbf{X} @<{\\cong}<(\\ref{wukhwefwefw})< \\Omega_X[[z]]_\\omega.\n    \\end{CD}\n  \\end{equation*}\n\\end{theorem}\n{\\it Proof.$\\:\\:\\:\\:$}\nThe isomorphism (\\ref{wukhwefwefw})  was constructed in \\cite[Section 3]{bss} using a particular model of\n$Rf_*f^*(\\underline{\\mathbb{R}}_\\mathbf{X})$.  We first recall its construction.\nLet $A\\to G$ be  an atlas.\nFor $(U\\to X)\\in \\mathbf{X}$ \nwe form the simplicial object\n$(A_U^\\cdot\\to G)\\in \\mathbf{G}^{\\Delta^{op}}$ with $n$th piece $$\nA_U^n:=\\underbrace{A\\times_G \\dots \\times_G A}_{n+1\\text{ factors}}\\times_X U\\to G\\ .$$\nThe boundaries and degenerations are given by the projections and diagonals as usual.\n\nIf $F\\in C^+(\\Pr_{\\tt Ab}\\mathbf{G})$ is a bounded below complex of  presheaves, then\nwe form the simplicial complex of presheaves\n$(U\\to X)\\mapsto F(A_U^\\cdot\\to G)$.\nWe let $C_A(F)\\in C^+(\\Pr_{\\tt Ab}\\mathbf{X})$ denote the presheaf of associated total complexes.\nSometimes we will write $C_A^{m,n}(F) $ for the summand of bidegree $(m,n)$, where the first entry\n$m$ denotes the cosimplicial degree.\n\nIf $F$ is a complex of flabby sheaves, then by \\cite[Lemma 2.41]{bss}\nwe have a natural isomorphism $Rf_*(F)\\cong C_A(F)$. Here we use in particular that the functor $C_A$ preserves sheaves.\n\nNote that the resolution $\\underline{\\mathbb{R}}_G\\to \\Omega_G$ of the constant sheaf with value $\\mathbb{R}$ by the sheaf of de Rham complexes is a flabby resolution (see \\cite[Subsection 3.1]{bss}). Therefore we have a natural isomorphism $Rf_*(\\underline{\\mathbb{R}}_G)\\cong C_A(\\Omega_G)$.\n\n\nWe choose an atlas $A\\to X$ given by the disjoint union of a collection of open subsets of $X$  such that there exists a lift in\n$$\\xymatrix{&G\\ar[d]^f\\\\A\\ar[r]\\ar@{.>}[ur]&X}\\ .$$\nThis lift is an atlas $A\\to G$ of $G$.\n We furthermore\n choose a connection datum $(\\alpha,\\beta)\\in \\Omega^1(A\\times_G A)\\times\n\\Omega^2(A)$. The one-form  $\\alpha$ is a connection of\nthe $U(1)$-principal bundle $A\\times_G A\\to A\\times_X A$. It is related with the two-form $\\beta$ by \n$d_{dR}\\alpha=\\delta \\beta$. This equation implies that $\\delta d_{dR}\\beta=0$ so that \n$d_{dR}\\beta$\nassembles to a uniquely determined closed form $\\omega\\in \\Omega^3_X(X)$ (compare \\cite[Section 3.2]{bss}).\nThe  $3$-form $\\omega$ represents the Dixmier-Douady class of the gerbe $G\\to X$ and will be used for twisting the de Rham complex. \n\n\n\n\n The isomorphism (\\ref{wukhwefwefw}) is given by an explicit quasi-isomorphism \\begin{equation}\\label{dgqwudwqdwq}\n\\Omega_X[[z]]_\\omega\\to C_A(\\Omega_G)\\ .\n\\end{equation}\\\nNote that $\\Omega_X[[z]]_\\omega$ and $C_A(\\Omega_G)$ are sheaves of\nassociative $DG$-algebras central over the sheaf of $DG$-algebras $\\Omega_X$, and that $z$ generates\n $\\Omega_X[[z]]_\\omega$.\nThe quasi-isomorphism (\\ref{dgqwudwqdwq}) is the unique morphism of sheaves of associative $DG$-algebras, central over $\\Omega_X$, with\n\\begin{equation*}\nz\\mapsto (\\alpha,\\beta)\\in\nC_A^{1,1}(\\Omega_G)(X)\\oplus C_A^{0,2}(\\Omega_G)(X) \\ .\n\\end{equation*}\nFor more details we refer to \\cite[Subsection 3.2]{bss} \n\n \n\n\n\n\n\n\n\n\n\n\\subsection{}\n\nFor $i=1,\\dots,n$ there are $U(1)$-principal bundle  structures  \n$$p_i:\\underbrace{A\\times_G\\cdots \\times_G A}_{n+1\\text{ factors}}\\to \\underbrace{A\\times_G\\cdots \\times_G A}_{i \\text{ factors}}\\times_X\\underbrace{\n  A\\times_G\\cdots \\times_G A}_{n-i+1 \\text{ factors}}\\ .$$\nFurthermore, we have embeddings\n$$j_i:\\underbrace{A\\times_G\\cdots \\times_G A}_{n\\text{ factors}}\n\\to \\underbrace{A\\times_G\\cdots \\times_G A}_{i \\text{ factors}}\\times_X\\underbrace{\n  A\\times_G\\cdots \\times_G A}_{n-i+1 \\text{ factors}}$$\ngiven by \n$$j_i:= \\underbrace{{\\tt id}_A\\cdots \\times {\\tt id}_A}_{i-1 \\text{ factors}}\\times \\Delta_A\\times \\underbrace{\n  {\\tt id}_A\\cdots \\times {\\tt id}_A}_{n-i \\text{ factors}}\\ ,$$\nwhere $\\Delta:A\\to A\\times_XA$ is the diagonal.\n\nIf $(U\\to X)\\in \\mathbf{X}$, then \nthe maps $p_i$ and $j_i$ induce similar maps on the product $\\dots\\times_XU$ of these manifolds over $X$ with $U$\nwhich we denote by the same symbols.\nFor $i=1,\\dots,n$ we define the map of degree $-1$\n$$v_i\\colon \\Omega(A_U^n)\\to \\Omega(A_U^{n-1})$$\nas the composition of the integration over the fiber of $p_i$ with the pull-back along $j_i$, i.e.\n$v_i:=j_i^*\\circ \\int_{p_i}$.\nSince the construction of $v_i$ is natural with respect to $U$\nwe can view $v_i$ as a morphism of sheaves $C_A^{n,m}(\\Omega_G)\\to C_A^{n-1,m-1}(\\Omega_G)$.  \nWe define the family of morphisms \n$$D_n:=\\sum_{i=1}^n (-1)^i v_i:\nC_A^{n,*}(\\Omega_G)\\to C_A^{n-1,*-1}(\\Omega_G)\n$$\nand let $D:C_A(\\Omega_G)\\to C_A(\\Omega_G)$ be the endomorphism of sheaves of degree $-2$\ngiven by $D_n$ in bidegree $(n,*)$.\n\n\n\n\n\n\n\n\\subsection{}\n\n\\begin{lem}\nThe map $D:C_A(\\Omega_G)\\to C_A(\\Omega_G)$ is a derivation\nof  $\\Omega_X$-modules.\n\\end{lem}\n\\begin{proof}\nNote that $v_j$ commutes with the de Rham differential. Moreover, if\n$$q_k\\colon \\underbrace{A\\times_G\\dots\\times_GA}_{n+1 \\text{ factors}}\\to \\underbrace{A\\times_G\\dots\\times_GA}_{n \\text{ factors}}$$ is the projection which leaves out the $k$-th\nfactor $(k=0,\\dots,n$), then we have the relations\n\\begin{equation*}\n  \\begin{split}\n   v_jq_k^* &= q_{k-1}^* v_j, \\qquad  j<k\\\\\n    v_jq_k^* &= q_k^* v_{j-1}, \\qquad j>k+1\\\\\n    v_jq_k^* &= 0, \\qquad j=k,k+1.\n  \\end{split}\n\\end{equation*}\nObserve that in the last case $q_k$ factors over the bundle \n which is used for the integration in the definition\nof $v_k$ or $v_{k+1}$, and the composition of a pullback along a bundle projection followed by an \nintegration along the same bundle projection vanishes.\nThese relations imply by a direct calculation that $D$ is a chain map \n  for the \\v{C}ech-de Rham differential of\n$C_A(\\Omega_G)$. \n\nMoreover, it follows immediately from the definition of $D$  that it is\n$\\Omega_X$-linear (even $\\Omega_A$-linear).\n\nIt is again a straightforward calculation to verify that $D$ is a derivation for that\nassociative product on $C_A(\\Omega_G)$ (compare \\cite[2.4.9]{bss} for the\nproduct structure).\n\\end{proof}\n\n\n\n\\subsection{}\n\n\\begin{lemma}\\label{dz_D}\n  We have a commutative diagram\n\\begin{equation*}\n  \\begin{CD}\n    \\Omega_X[[z]]_\\omega @>{(\\ref{dgqwudwqdwq})}>> C_A(\\Omega_G) \\\\\n    @VV{\\frac{d}{dz}}V @VV{D}V\\\\\n    \\Omega_X[[z]]_\\omega @>{(\\ref{dgqwudwqdwq})}>> C_A(\\Omega_G) .\n  \\end{CD}\n\\end{equation*}\n\\end{lemma}\n\\begin{proof}\nSince $\\alpha$ is the connection one-form of  a $U(1)$-connection on the total space   of the $U(1)$-principal bundle $p_1:A\\times_G A\\to A\\times_X A$  we have\n$\\int_{p_1}\\alpha=1$. Consequently,\n$D(\\alpha,\\beta)=1$.\nThis implies the assertion,  since $D$  and $\\frac{d}{dz}$  are $\\Omega_X$-linear\nderivation, and $z$ generates $ \\Omega_X[[z]]_\\omega$.\n\\end{proof}\n In view of Lemma \\ref{dz_D}, in order to finish the proof of Theorem \\ref{theo:D_is_derivative} is suffices to show that the operation $D$ coincides with\nthe operation of \n$\\int_p\\circ \\phi^*\\circ p^*$ on $C_A(\\Omega_G)$.\n\n\\subsection{}\n\n\n\nLet $M^\\cdot$ be a simplicial manifold and consider the bundle \n$U(1)\\times M^\\cdot\\to M^\\cdot$. \nWe describe the integration map\n$$\\int:\\Omega(U(1)\\times M^\\cdot)\\to  \\Omega(M^\\cdot)$$ in the\nsimplicial picture, i.e. as a map\n$$\\int:\\Omega(\\SSSS^\\cdot\\times M^\\cdot)\\to  \\Omega(M^\\cdot)\\ .$$\nFor $n\\ge 1$ the manifolds\n$\\SSSS^{n}\\times M^n$ consists of\n$n$ copies $\\sigma_1(M^n),\\dots,\\sigma_n(M^n)$ of $M^n$ which correspond to the points of $\\SSSS^n$ which are degenerations of the non-degenerated point of $\\SSSS^1$  (where the index measures which $1$-simplex in the boundary\n  is non-degenerate), and an additional copy of $M^n$ corresponding the point of $\\SSSS^n$ which is the  degeneration of the point in $\\SSSS^0$.  For $k=1,\\dots,n+1$ let  $j_k: M^n\\to \\SSSS^{n+1}\\times M^{n+1}$ be the map $M^n\\to \\sigma_k(M^{n+1})\\subset \\SSSS^{n+1}\\times M^{n+1}$, which corresponds the $k$th degeneration\n$[n+1]\\to [n]$.   We now define a chain map of total complexes\n$$\\int:\\Omega(\\SSSS^\\cdot\\times M^\\cdot)\\to \\Omega(M^\\cdot)$$\nof degree $-1$ which is given by \n\\begin{equation}\\label{uefqwefqewfqfwqw}\n\\sum_{k=1}^{n+1} (-1)^kj_k^*:\\Omega(\\SSSS^{n+1}\\times M^{n+1})\\to \\Omega(M^n)\\ ,\n\\end{equation}\nand is zero on $\\Omega(\\SSSS^0\\times M^0)$.\nThis map realizes the integration in the simplicial picture.\n\\subsection{}\n\nFor $(U\\to X)\\in \\mathbf{X}$\nthe automorphism of gerbes $\\phi:T^2\\times G\\to T^2\\times G$ induces an automorphism\nof simplicial sets\n$$\\phi^\\cdot:\\SSSS^\\cdot\\times U(1)\\times A_U^\\cdot\\to \\SSSS^\\cdot\\times U(1)\\times A_U^\\cdot$$\nwhich we now describe explicitly by an extension of the special case (\\ref{uqfbqwdqwqwd}) \nto general base spaces.\n\nIf  $t\\in \\SSSS^n\\times U(1)$ belongs to $U(1)\\cong\\sigma_k(U(1))\\subset \\SSSS^n\\times U(1)$, $k=1,\\dots,n$, then $\\phi^\\cdot(t,a)=(t, m_k(t,a))$, where $m_k:U(1)\\times A_U^n\\to A_U^n$ is the action of $U(1)$ on the principal fibration $p_k$. \nIf $t\\in  \\SSSS^n\\times U(1)$ belongs to the degeneration of $\\SSSS^0\\times U(1)$, then $\\phi^\\cdot(t,a)=(t, a)$.\nThis formula provides a simplicial description of the action of\n$$\\phi^*:C_{\\SSSS^\\cdot\\times U(1)\\times A}(\\Omega_{G})\\to C_{\\SSSS^\\cdot\\times U(1)\\times A}(\\Omega_{G})\\ .$$\n\n\n\nCombining the description of the integration map (\\ref{uefqwefqewfqfwqw}) with this formula for the action of $\\phi^*$\nit is now straightforward to show the equality of maps\n$$D=\\int_p\\circ \\phi^*\\circ p^*:C_A(\\Omega_G)\\to C_A(\\Omega_G)\\ .$$\n\\hspace*{\\fill}$\\Box$ \\\\[0cm]\\noindent\n\n\n\n\n \n\n\n\n\n\n\n\n\n\n\n\n\n\n \\section{Two-periodization --- up to isomorphism}\\label{system12}\n\n\\subsection{}\n\nLet $f\\colon G\\to X$   be a topological gerbe with band $U(1)$ over a locally compact stack $X$. In Definition \\ref{defofd} we have constructed a natural endomorphism\n$D_G\\in {\\tt End}(Rf_*\\circ f^*)$ of degree $-2$.\nTo any object\n$F\\in D^+({\\tt Sh}_{\\tt Ab} \\mathbf{X})$ we  associate the inductive system\n\\begin{equation}\\label{system10}\n\\mathcal{S}_G(F)\\colon Rf_*\\circ f^*(F)\\stackrel{D_G}{\\leftarrow} Rf_*\\circ f^*(F)[2]\\stackrel{D_G}{\\leftarrow}Rf_*\\circ f^*(F)[4]\\stackrel{D_G}{\\leftarrow}\\dots\n\\end{equation} indexed by $\\{0,1,2\\dots\\}$.\n\n \nUsing the inclusion $D^+({\\tt Sh}_{\\tt Ab}\\mathbf{X})\\to D({\\tt Sh}_{\\tt Ab}\\mathbf{X})$ of the bounded below into the unbounded derived category of sheaves of abelian groups on $X$ we can consider\n$\\mathcal{S}_G(F)\\in D({\\tt Sh}_{\\tt Ab} \\mathbf{X})^{\\mathbb{N}^{op}}$, where the ordered set of integers  $\\mathbb{N}$\nis considered as a category.\n\n\\subsection{}\n\nUsing the triangulated structure of $D({\\tt Sh}_{\\tt Ab} \\mathbf{X})$ one can define for each object\n$\\mathcal{S}\\in D({\\tt Sh}_{\\tt Ab} \\mathbf{X})^{\\mathbb{N}^{op}}$   an object\n${\\tt holim}\\mathcal{S}\\in D({\\tt Sh}_{\\tt Ab} \\mathbf{X})$ which is unique up to non-canonical isomorphism\n(see \\cite{MR1812507}).\nAn explicit construction of this homotopy limit uses the extension of maps in \n$D({\\tt Sh}_{\\tt Ab} \\mathbf{X})$ to exact triangles by a mapping cylinder construction.\nIn particular, we obtain ${\\tt holim} \\mathcal{S}_G(F)$  by the  extension to a triangle of the map $1-\\hat D$ in\n$${\\tt holim} \\mathcal{S}_G(F) \\to \\prod_{i\\ge 0}Rf_*\\circ f^*(F)[2i]\\stackrel{1-\\hat D}{\\longrightarrow} \\prod_{i\\ge 0}Rf_*\\circ f^*(F)[2i]\\to {\\tt holim} \\mathcal{S}_G(F)[1] \\ ,$$\nwhere\n$$\\hat D\\colon \\prod_{i\\ge 0}Rf_*\\circ f^*(F)[2i]\\to \\prod_{i\\ge 0}Rf_*\\circ f^*(F)[2i]$$\nmaps the sequence $(x_i)_{i\\ge 0}$ to the sequence\n$(D_Gx_{i+1})_{i\\ge 0}$. \n\n\\subsection{}\n\n \nWe can now define the periodization $P_G(F)\\in D({\\tt Sh}_{\\tt Ab} \\mathbf{X})$ of an object\n$F\\in D^+({\\tt Sh}_{\\tt Ab} \\mathbf{X})$.\n\n\\begin{ddd}\\label{system188}\n For $F\\in D^+({\\tt Sh}_{\\tt Ab} \\mathbf{X})$ we define $P_G(F)\\in D({\\tt Sh}_{\\tt Ab} \\mathbf{X})$  by\n$$P_G(F):={\\tt holim} \\mathcal{S}_G(F)\\ .$$\nNote that $P_G(F)$ is well-defined up to non-canonical isomorphism.\n\\end{ddd}\n\n\n\n\n\n\n\n\n \n\n \n\n\\subsection{}\\label{choicesiasg}\n\nThe operator \n$$\\prod_{i\\ge 0} D_G\\colon \\prod_{i\\ge 0}Rf_*\\circ f^*(F)[2i]\\to (\\prod_{i\\ge 0}Rf_*\\circ f^*(F)[2i])[-2]$$ commutes with $\\hat D$ and  therefore induces a map $W\\colon P_G(F)\\to P_G(F)[-2]$ via an extension in the diagram\n\\begin{equation*}\n  \\begin{CD}\n    P_G(F) @>W>> P_G(F)[-2]\\\\\n    @VVV @VVV\\\\\n   \\prod_{i\\ge 0}Rf_*\\circ f^*(F)[2i] @>{\\prod_{i\\ge 0} D_G}>> \\prod_{i\\ge\n     0}Rf_*\\circ f^*(F)[2i])[-2]\\\\\n   @VV{1-\\hat D}V @VV{1-\\hat D}V\\\\\n    \\prod_{i\\ge 0}Rf_*\\circ f^*(F)[2i] @>{\\prod_{i\\ge 0} D_G}>> \\prod_{i\\ge\n      0}Rf_*\\circ f^*(F)[2i])[-2]\\\\\n    @VVV @VVV\\\\\n    P_G(F)[1] @>{W}>> P_G(F)[1][-2]\\ .\n  \\end{CD}\n\\end{equation*}\nNote that such an extension exists by the axioms of a triangulated category, but it might not\nbe unique.  \n\nThe following proposition asserts that $P_G(F)$ is two-periodic.\n\\begin{prop}\\label{system3}\nThe map\n$W\\colon P_G(F)\\to P_G(F)[-2]$ is an isomorphism.\n\\end{prop}\n{\\it Proof.$\\:\\:\\:\\:$}\nFor notational convenience, we consider the following general situation.\nLet $D(A)$ be the unbounded derived category of a Grothendieck abelian\ncategory. Note that ${\\tt Sh}_{\\tt Ab}(\\mathbf{X})$ is such a category (see Section \\ref{system11}).\nWe consider an object $X\\in D(A)$ together with a morphism\n$D\\colon X\\to X[-2]$. We can assume that $D$ is represented\nby a map of complexes $D\\colon X\\to X[-2]$. We obtain the extension $1-\\hat D$ to a triangle\n\\begin{equation}\\label{system4}\nY\\to \\prod_{i\\ge 0} X[2i]\\stackrel{1-\\hat D}{\\to} \\prod_{i\\ge 0} X[2i]\\to Y[1]\n\\end{equation}\nwhere $Y:= \\prod_{i\\ge 0} X[2i]\\oplus ( \\prod_{i\\ge 0} X[2i])[1]$ with the differential \n$$\\delta:=\\left(\\begin{array}{cc}d&1-\\hat D\\\\0&-d\\end{array}\\right)\\ ,$$\nwhere $d$ is the differential of $X$. The induced map $W\\colon Y\\to Y[-2]$ is given by\n$$W:=\\left(\\begin{array}{cc}\\prod_{i\\ge 0}D&0\\\\0&\\prod_{i\\ge 0}D\\end{array}\\right)\\ .$$\nLet $$E\\colon \\prod_{i\\ge 0}X[2i]\\to (\\prod_{i\\ge 0}X[2i])[2]$$\nbe the shift $E(x_i)_{i\\ge 0}:=(x_{i+1})_{i\\ge 0}$.\nNote that $E$ commutes with $1-\\hat D$, too. Therefore\nwe obtain the extension\n $S\\colon Y\\to  Y[2]$ in the  diagram\n$$\\xymatrix{ Y\\ar[r]\\ar[d]^S& \\prod_{i\\ge 0}X[2i]\\ar[d]^{E}\\ar[r]^{1-\\hat D}&\\prod_{i\\ge 0}X[2i]\\ar[d]^{E}\\ar[r]&Y[1]\\ar[d]^S\\\\\nY[2] \\ar[r]&( \\prod_{i\\ge 0}X[2i])[2]\\ar[r]^{1-\\hat D}&(\\prod_{i\\ge 0}X[2i])[2]\\ar[r]& Y[1][2]}\\ .$$\nby the matrix \n $$S:=\\left(\\begin{array}{cc}E&0\\\\0&E\\end{array}\\right)\\ .$$\nProposition \\ref{system3} is a consequence of the  following Lemma.\n\\begin{lem}\\label{uwzgfwehjcfbsdac}\nWe have the equalities  $W\\circ S={\\tt id}=S\\circ W$.\n\\end{lem}\n\\begin{proof}\nFirst observe that $\\prod_{i\\ge 0} D\\circ E=\\hat D=E\\circ \\prod_{i\\ge 0} D$. Therefore\n$W\\circ S=S\\circ W= \\left(\n  \\begin{smallmatrix}\n    \\hat D & 0\\\\ 0 & \\hat D\n  \\end{smallmatrix}\\right)$.\nIn order to show that $W\\circ S={\\tt id}$\nwe  show that the map\n$$I:=\\left(\\begin{array}{cc}\\hat D&0\\\\0&\\hat D\\end{array}\\right)\\ .$$\non $Y$ is homotopic to the identity and therefore is equal to the identity in\nthe derived category. This follows from   \n$$1-I=\\delta\\circ J+J\\circ \\delta$$\nwith\n$$J:=\\left(\\begin{array}{cc}0&0\\\\mathbf{1}}  % if this does not work: \\boldsymbol{1&0\\end{array}\\right)\n\\ \n.$$ \n\\end{proof}\n\n\\subsection{}\n\nWe continue with the notation introduced in the proof of Proposition \\ref{system3}.\nApplying a homological functor to the triangle (\\ref{system4})  we get the long exact sequence\n$$\\dots\\to H^*(Y)\\to\\prod_{i\\ge 0}H^*(X[2i])\\to \\prod_{i\\ge 0}H^*(X[2i])\\to H^{*}(Y[1])\\to \\ .$$\nIf we analyze the middle map and compare it with the ordinary definition of limits in abelian categories we get the following result.\n\\begin{kor} \\label{lim1seq}\nWe have an exact sequence:\n$$0\\to {\\lim_i}^1 H^*(X[2i])[-1] \\to H^*(Y)\\to \\lim_i H^*(X[2i]) \\to 0\\ .\n$$\n \\end{kor}\n\n\\subsection{}\n\nNote that the construction\n$${\\tt holim}\\colon D(A)^{\\mathbb{N}^{op}}\\to D(A)$$\nis not a functor. The construction of the homotopy limit ${\\tt holim}(S)$ for $S\\in D(A)^{\\mathbb{N}^{op}}$  via mapping cylinders uses explicit representatives of the maps of the system $S$ and depends non-trivially on this choice.\n\nA homotopy limit functor ${\\tt holim}\\colon D(A^{\\mathbb{N}^{op}} )\\to D(A)$ can be defined as the right-derived functor \nof $\\lim\\colon A^{\\mathbb{N}^{op}} \\to A$. Note that in the domain we take the derived category of the abelian category of $\\mathbb{N}^{op}$-diagrams in $A$ as opposed to $\\mathbb{N}^{op}$-diagrams in the derived category of $A$. In Section \\ref{system7} we will use this idea and refine $P_G$ to a periodization functor\n$$P_G\\colon D^+({\\tt Sh}_{\\tt Ab}\\mathbf{X})\\to D({\\tt Sh}_{\\tt Ab}\\mathbf{X})$$\nwhich is a triangulated functor and natural in $G\\to X$.\nThe main idea is the construction of a refinement of the diagram\n(\\ref{system10}) to a diagram in $D(({\\tt Sh}_{\\tt Ab}\\mathbf{X})^{\\mathbb{N}^{op}})$, see \\ref{system16} (the details are in fact more complicated).\n\n\n\n\n\n \n\n\n\n\n\n\n\n\n\n\n\n\\section{Calculations}\n\n\\subsection{}\n\nIn this subsection we calculate $P_G(\\underline{F})$ in the special case, where\n$G\\to *$ is the (trivial) $U(1)$-gerbe over the point, and $\\underline{F}\\in {\\tt Sh}_{\\tt Ab}{\\tt Site}(*)$ is the sheaf\nrepresented by a discrete abelian group $F$. We will calculate the abelian group $H^*(*;P_G(\\underline{F}))$.\nThis cohomology is two-periodic so that we only have to distinguish the even and the odd-degree case.\nIn the table below $\\mathbb{A}^\\mathbb{Q}_f$ denotes the group of finite adeles of $\\mathbb{Q}$, which contains $\\mathbb{Q}$ via the diagonal embedding.\n\\begin{prop}\\label{system30}\nWe have the following table for the cohomology $H^*(*;P_G(\\underline{F}))$.\n$$\\begin{array}{|c|c|c|}\\hline F&H^{ev}(*;P_G(\\underline{F}))&H^{odd}(*;P_G(\\underline{F}))\\\\\\hline\\hline\n\\mathbb{Z}&0&\\mathbb{A}^\\mathbb{Q}_f\/\\mathbb{Q}\\\\\\hline\n\\mathbb{Q}&\\mathbb{Q}&0\\\\\\hline\n\\mathbb{Z}\/n\\mathbb{Z}&0&0\\\\\\hline\n\\mathbb{Q}\/\\mathbb{Z}&\\mathbb{A}_f^\\mathbb{Q}&0\\\\\\hline\n  \\end{array}\\ .$$\n\\end{prop}\n\n\\subsection{}\nTo prove Proposition \\ref{system30}, we use the exact sequence \\ref{lim1seq} where\n$$H^*(X)=H^*(*;Rf_*\\circ f^*(\\underline{F}))\\cong F\\otimes \\mathbb{Z}[ [z]]\\cong F[[z]]$$\nby Lemma \\ref{system5} with $z$ of degree $2$. We must \n discuss the cohomology of the complex\n$$0\\to \\prod_{i\\ge 0} F[[z]][2i]\\stackrel{1-\\hat D}{\\to}  \\prod_{i\\ge 0}F[[z]][2i]\\to 0\\ ,$$\nwhere $\\hat D(x_i)_{i\\ge 0}=(D_Gx_{i+1})_{i\\ge 0}$ with $D_G=\\frac{d}{dz}$.\nThis means that we have to study the solution theory for the system\n\\begin{equation}\\label{system6}\nx_i-\\frac{d}{dz}x_{i+1}=a_i\\ ,\\quad  i\\ge 0\\ ,\\quad x_i\\in F[[z]]\\ .\n\\end{equation}\n \n\\subsection{}\\label{sec:calculate_period_cohom}\nLet us start with the case $F=\\mathbb{Q}$. Since we can divide by arbitrary integers\nthe operator $D_G$ is surjective and we can for any $(a_i)_{i\\in\\mathbb{N}}$ solve this system inductively. Therefore the cokernel $\\lim_i^1 \\mathbb{Q}[u]$\nof $1-\\hat D$ is trivial. A solution of the homogeneous system is uniquely determined by the choice of $x_0$ and the constant terms of the $x_i$, $i\\ge 1$. Note that the constant term of $x_i$ is in degree $-2i$. It follows that\n$$H^{ev}(*;P_G(\\underline{\\mathbb{Q}}))\\cong   \\mathbb{Q}\\ ,\\quad  H^{odd}(*;P_G(\\underline{\\mathbb{Q}}))\\cong 0\\ .$$\n \\subsection{}\nWe now discuss torsion coefficients $F=\\mathbb{Z}\/n\\mathbb{Z}$. Write $x_i=\\sum x_{i,k} z^k$,\n$a_i=\\sum a_{i,k}z^k$ with $x_{i,k}, a_{i,k}\\in \\mathbb{Z}\/n\\mathbb{Z}$. Then we have to solve \n\\begin{equation*}\n  \\sum_{k=0}^\\infty x_{i,k}z^k -(k+1) x_{i+1,k+1} z^k=\\sum_{k=0}^\\infty\n  a_{i,k}z^k\\qquad\\forall i\\ge 0.\n\\end{equation*}\nEquating coefficients  this system\ndecouples into finite systems \n\\begin{eqnarray*}\nx_{i,kn}-(kn+1)x_{i+1,kn+1}&=&a_{i,kn}\\\\\nx_{i,kn+1}-(kn+2)x_{i+1,kn+2}&=&a_{i,kn+1}\\\\\n&\\vdots&\\\\\nx_{i,kn+n-2}-(kn+n-1)x_{i+1,kn+n-1}&=&a_{i,kn+n+2}\\\\\nx_{i,kn+n-1-r}+\\underbrace{(kn+n)x_{i+1,kn+n}}_{=0}&=&a_{ikn+n-1} \\  ,\\end{eqnarray*} \nwhere $i,k\\ge 0$.\nWe see that we can always solve this system uniquely by backwards induction.\nWe get $$H^{ev}(*;P_G(\\underline{\\mathbb{Z}\/n\\mathbb{Z}}))\\cong 0\\ ,\\quad H^{odd}(*;P_G(\\underline{\\mathbb{Z}\/n\\mathbb{Z}}))\\cong 0\\ .$$\n \n \\subsection{}\nLet us now assume that $F=\\mathbb{Q}\/\\mathbb{Z}$. Since this group is divisible we can solve\nthe system (\\ref{system6}) for every $(a_i)_{i\\in\\mathbb{N}}$. It follows that\n$$H^{odd}(*;P_G(\\underline{\\mathbb{Q}\/\\mathbb{Z}}))\\cong 0\\ .$$\nWe now discuss the solution of the homogeneous system in degree $0$.\nWe can choose $x_0$ arbitrary. If we have found $x_i$ for $i=0,\\dots,n-1$, then we must solve\n$x_{n-1}=nx_n$ in the next step. We see that $x_n$ is well-defined up to the image of $\\mathbb{Z}\/n\\mathbb{Z}\\cong n^{-1}\\mathbb{Z}\/\\mathbb{Z}\\subset \\mathbb{Q}\/\\mathbb{Z}$.\nWe see that \n$H^{ev}(*;P_G(\\underline{\\mathbb{Q}\/\\mathbb{Z}}))$ admits a sequence of quotients\n$$H^{ev}(*;P_G(\\underline{\\mathbb{Q}\/\\mathbb{Z}}))\\to \\dots \\to Q^n\\to Q^{n-1}\\to \\dots \\to Q^0$$\nwhere $Q^{n}\\cong \\mathbb{Q}\/\\mathbb{Z}$ and $Q^n\\to Q^{n-1}$ is given by multiplication\nwith $n$ for all $n\\in\\mathbb{N}$.\nThe limit \n\\begin{equation*}\n\\mathbb{A}^\\mathbb{Q}_f\\cong \\lim_{\\stackrel{\\longleftarrow}{n\\in \\mathbb{N}}}\n(\\mathbb{Q}\/n!\\mathbb{Z})\n\\end{equation*}\nis the ring $\\mathbb{A}_f^\\mathbb{Q}$ of finite adeles of $\\mathbb{Q}$, and $\\mathbb{Q}\\subset \\mathbb{A}^\\mathbb{Q}_f$ is a subgroup.\nWe thus get\n$$H^{ev}(*;P_G(\\underline{\\mathbb{Q}\/\\mathbb{Z}}))\\cong \\mathbb{A}^\\mathbb{Q}_f\\  .$$\n\n \\subsection{}\nFinally assume that $F=\\mathbb{Z}$. We must again consider the system (\\ref{system6}) of equations above. \nLet us discuss this system in degree $2r$. Then the relevant coefficients of\n$x_i$ and $a_i$ are sequences of integers, and (writing out only these) \n$dx_{i+1}=(r+i+1)x_{i+1}$.\nWe see that the homogeneous equation has only the trivial solution since otherwise the integer\n$x_0$ must be divisible by $n+i+1$ for all $i\\ge 0$. Hence\n$$H^{ev}(*;P_G(\\underline{\\Z}))\\cong 0\\ .$$ \nIn order to calculate  $H^{odd}(*;P_G(\\underline{\\Z}))$ we consider the  exact sequence\n$$0\\rightarrow \\mathbb{Z}\\to \\mathbb{Q}\\to \\mathbb{Q}\/\\mathbb{Z}\\to 0\\ .$$\nIt gives rise to an exact sequence of sheaves \n$$0\\rightarrow \\underline{\\Z}\\to \\underline{\\mathbb{Q}}\\to \\underline{\\mathbb{Q}\/\\mathbb{Z}}\\to 0\\ .$$ and a long exact cohomology sequence.\nIn Section \\ref{funct_per} we will construct a functorial version of $P_G$\nwhich is a triangulated functor, and which coincides with the isomorphism class\nconstructed above. Using this functor, we get a triangle\n$$P_G(\\underline{\\Z})\\to P_G(\\underline{\\mathbb{Q}})\\to P_G(\\underline{\\mathbb{Q}\/\\mathbb{Z}})\\to P_G(\\underline{\\Z})[1]$$ \nand therefore a long exact cohomology sequence \n$$H^{*}(*;P_G(\\underline{\\Z}))\\to H^*(*;P_G(\\underline{\\mathbb{Q}}))\\to H^*(*;P_G(\\underline{\\mathbb{Q}\/\\mathbb{Z}}))\\to H^{*}(*;P_G(\\underline{\\Z}))[1]\\ .$$\nBy the calculations for $\\mathbb{Q}$ and $\\mathbb{Q}\/\\mathbb{Z}$ we get exact sequences\n$$0\\to \\mathbb{Q}\\stackrel{c}{\\to} \\mathbb{A}_f^\\mathbb{Q} \\to H^{odd}(*;P_G(\\underline{\\Z}))\\to 0\\ ,$$\nwhere $c$ is the canonical embedding. Therefore\n$$  H^{odd}(*;P_G(\\underline{F}))\\cong  \\mathbb{A}^\\mathbb{Q}_f\/\\mathbb{Q} \\ .$$\n\\hspace*{\\fill}$\\Box$ \\\\[0cm]\\noindent\n \n\\chapter{Functorial periodization}\\label{system7}\n\n\\section{Flabby resolutions}\n\n\\subsection{}\\label{nr1}\n\nLet $\\mathbf{X}$ be a  site, e.g. the site of a locally compact stack. For $U\\in \\mathbf{X}$\nlet $\\tau:=(U_i\\to U)_{i\\in I}\\in {\\tt cov}_\\mathbf{X}(U)$ be a covering family.\nThen we consider $V:=\\bigsqcup_{i\\in I} U_i\\to U$. Forming iterated fiber products  we obtain a  simplicial object\n$V^\\cdot$ in $\\mathbf{X}$ with $$V^n=\\underbrace{V\\times_U\\dots\\times_UV}_{n+1 \\:\\:\\mbox{\\scriptsize factors}}\\ .$$\nIf $F\\in \\Pr\\mathbf{X}$ is a presheaf on $X$, then we form the cosimplicial set\n$C^\\cdot(\\tau,F):=F(V^\\cdot)$.\n \n\\subsection{}\n\nIf $F$ is a presheaf of abelian groups, then we form the \\v{C}ech complex\n$\\check C(\\tau,F)$ which is the  chain complex associated to the cosimplicial abelian group  $C^\\cdot(\\tau,F)$.\n\nIf $F$ is a sheaf, then $H^0\\check C(\\tau,F)\\cong F(U)$. We recall the following definition (see \\cite[Definition 3.5.1]{MR1317816}).\n\n\n \n\n \n\\begin{ddd}[see  3.5.1, \\cite{MR1317816}]\\label{def:flabby}\nA sheaf $F\\in {\\tt Sh}_{\\tt Ab}\\mathbf{X}$ is called flabby if for all $U\\in \\mathbf{X}$ and $\\tau\\in {\\tt cov}_\\mathbf{X}(U)$ we have\n$H^i\\check C(\\tau,F)\\cong 0$ for all $i\\ge 1$.\n\\end{ddd}\nBy \\cite[Cor. 3.5.3]{MR1317816} a sheaf $F\\in {\\tt Sh}_{\\tt Ab}\\mathbf{X}$ is flabby if and only if $R^ki(F)=0$ for all $k\\ge 1$, where $i:{\\tt Sh}_{\\tt Ab}\\mathbf{X}\\to \\Pr_{\\tt Ab}\\mathbf{X}$ is the inclusion of sheaves into presheaves.\n\nAs an immediate consequence of the definition a sheaf $F\\in {\\tt Sh}_{\\tt Ab}\\mathbf{X}$ is flabby if and only if the restriction $F_U$  of $F$ to the site $(U)$ is flabby for all $(U\\to X)\\in \\mathbf{X}$ (see \\ref{obstr1a} for the notation).\n\n\\subsection{}\n\nLet now $X$ be a locally compact stack and $\\mathbf{X}$ be the site of $X$.\nOccasionally, in the present paper we need the stronger notion of a flasque sheaf.\n\\begin{ddd}\\label{flasquedef}\nA sheaf $F\\in {\\tt Sh}_{\\tt Ab}\\mathbf{X}$ is called flasque if for every $(U\\to X)\\in \\mathbf{X}$ and\nopen subset $V\\subseteq U$ the restriction $F(U\\to X)\\to F(V\\to X)$ is surjective.\n\\end{ddd}\nIn the literature, e.g. in \\cite{MR1299726} or \\cite{MR1481706}, \nthis is used as the  definition of flabbiness. \n\\begin{lem}\\label{wgszguwqsws}\nA flasque sheaf is flabby.\n\\end{lem}\n\\begin{proof}\nFor $U\\in \\mathbf{X}$ let $\\Gamma_U:{\\tt Sh}_{\\tt Ab}\\mathbf{X}\\to {\\tt Ab}$ be the section functor $F\\mapsto \\Gamma_U(F):=F(U)$.\nFor $V\\subseteq U$ we have\n$\\Gamma_V(F_U)=\\Gamma_V(F)$.\nA sheaf  $F\\in {\\tt Sh}_{\\tt Ab}\\mathbf{X}$ is flasque by definition if and only  if $F_U$ is flasque for all $U\\in \\mathbf{X}$. But a flasque sheaf is $\\Gamma_V$-acyclic\nfor every $V\\subseteq U$ by \\cite[Ch. 2, Thm. 5.4]{MR1481706} (note that in this reference our flasque is called flabby).  By \\cite[Cor. 3.5.3]{MR1317816} it is flabby in the sense of \\ref{def:flabby}.\n\nThis argument shows that $F_U$ is flabby for all $(U\\to X)\\in \\mathbf{X}$ and   implies that $F$ itself is flabby.\n\\end{proof}\n\nWe do not know if the converse of Lemma \\ref{wgszguwqsws} is true.\nTherefore we must be careful when using results from the literature.\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{}\n\n \n\\begin{lem}\\label{hdqoiwdwqw}\nIf $f:X\\to Y$ is a representable map of locally compact stacks, then\na flabby sheaf is $f_*$-acyclic.\n\\end{lem}\n\\begin{proof}\nLet $F\\in {\\tt Sh}_{\\tt Ab}\\mathbf{X}$ be a flabby sheaf. We must show that $R^kf_*(F)=0$ for all $k\\ge 1$.\nWe have a morphism of sites $f^\\sharp:\\mathbf{Y}\\to \\mathbf{X}$, see \\ref{obensharp}.\nThe functor ${}^pf_*:\\Pr\\mathbf{X}\\to \\Pr\\mathbf{Y}$ is given by ${}^pf_*F:=F\\circ f^\\sharp$.\nIt is in particular exact. Therefore we have $Rf_*\\cong i^\\sharp\\circ {}^pf_*\\circ Ri$.\nSince a flabby sheaf is $i$-acyclic we conclude that $R^ki(F)=0$ for $k\\ge 1$.\nThis implies $R^kf_*(F)=0$ for $k\\ge 1$. \n\\end{proof}\n\n\n\n\n\n\n\n\n\n\\subsection{}\n\n \n  \\begin{lem}\\label{flabbypres}\nIf a morphism $f\\colon X\\to Y$ of locally compact stacks has local sections, then the functor $f^*\\colon {\\tt Sh}_{\\tt Ab}\\mathbf{Y}\\to\n{\\tt Sh}_{\\tt Ab}\\mathbf{X}$ preserves flabby sheaves.\n\\end{lem}\n\\begin{proof}\nLet $F\\in {\\tt Sh}_{\\tt Ab}\\mathbf{Y}$ be flabby.\nWe consider an object  $(U\\to X)\\in \\mathbf{X}$ and a covering family $\\tau \\in {\\tt cov}_\\mathbf{X}(U)$.\nThen we must show that the higher cohomology groups of $\\check{C}(\\tau,f^*F)$ vanish.\n\nWe obtain  a covering family $f_\\sharp\\tau \\in {\\tt cov}_\\mathbf{Y}(f_\\sharp U)$, see \\ref{sharp}.\nLet $V^\\cdot$ be the simplicial object associated to $\\tau$ as in \\ref{nr1}. Since\n$f_\\sharp$ preserves fiber products in the sense of \\cite[1.2.2(ii)]{MR1317816} we see that\n$f_\\sharp V^\\cdot$ is the simplicial object in $\\mathbf{Y}$ associated to $f_\\sharp \\tau$.\nThe rule  $f^*F(U)\\cong F(f_\\sharp U)$ (see again \\ref{sharp}) gives the isomorphism of\ncosimplicial sets\n$f^*F(V^\\cdot)\\cong F(f_\\sharp V^\\cdot)$ and hence\nan isomorphism of complexes\n$$\\check C(\\tau,f^*F)\\cong \\check C(f_\\sharp\\tau,F)\\ .$$ \nSince $F$ is flabby the higher cohomology groups of the right-hand side vanish.\n\\end{proof}\n\n\n \n\n\n\\subsection{}\n \nWe now construct a canonical flabby resolution functor $${\\mathcal Fl}\\colon {\\tt Sh}_{\\tt Ab}\\mathbf{X}\\to C^+({\\tt Sh}_{\\tt Ab}\\mathbf{X})\\ ,\\quad  {\\tt id}\\to {\\mathcal Fl}\\ .$$ \nIt associates to a $F$ a sort of Godement resolution which consists in fact of flasque sheaves.\n\n For a space $U$ let $(U)$ denote the site of open subsets of $U$\nwith the topology of open coverings. We will first construct flabby resolution\nfunctors  \n$${\\mathcal Fl}_U\\colon {\\tt Sh}_{\\tt Ab}(U)\\to C^+({\\tt Sh}_{\\tt Ab} (U))\\ ,\\quad {\\tt id}\\to {\\mathcal Fl}_U$$ for all  $(U\\to X)\\in \\mathbf{X}$\nwhich are compatible with the morphisms $V\\to U$ in $\\mathbf{X}$.\nFor $F\\in {\\tt Sh}_{\\tt Ab}\\mathbf{X}$ we obtain a collection\nof flabby resolutions $(F_U\\to {\\mathcal Fl}_U(F_U))_{U\\in \\mathbf{X}}$, which by \n\\ref{desc_sheaves_on_U} \ngive rise to a resolution\n$F\\to {\\mathcal Fl}(F)$.\nIn the following we discuss these steps in detail. \n \n\n\\subsection{}\n\n\n\n\nLet $p_U\\colon \\hat U\\to U$ be the identity map, where $\\hat U$ is the set $U$ with the discrete topology. Let $F\\in {\\tt Sh}_{\\tt Ab}(U)$. We set\n${\\mathcal Fl}_U^0(F):=(p_U)_*\\circ p_U^*(F)$ and let $F\\to {\\mathcal Fl}_U^0(F)$ be given by the unit\n${\\tt id}\\to (p_U)_*\\circ p_U^*$.\n\n\\begin{lem}\\label{lem:flab_exact}\n  The sequence $0\\to F\\to (p_U)_*\\circ p_U^*F$ is exact.\n\\end{lem}\n\\begin{proof}\nLet $w\\in U$. We must show that the induced map on stalks\n$F_w\\to ((p_U)_*\\circ p_U^*F)_w$ is injective. This immediately follows from the description\n$$((p_U)_*\\circ p_U^*F)_w=\\colim_{w\\in W\\subseteq U}\n\\prod_{v\\in W} F_v\\ .$$  \n\\end{proof} \n\n\n \n \n\n\\hspace*{\\fill}$\\Box$ \\\\[0cm]\\noindent\n\n\\subsection{}\n\nWe now  construct ${\\mathcal Fl}_U(F)$ inductively. Assume that we have\nalready  constructed ${\\mathcal Fl}_U^0(F)\\to \\dots \\to {\\mathcal Fl}_U^k(F)$. Then we let\n$${\\mathcal Fl}_U^{k+1}(F):=(p_U)_*\\circ p_U^*({\\tt coker}({\\mathcal Fl}_U^{k-1}(F)\\to {\\mathcal Fl}_U^{k}(F))$$ and\n${\\mathcal Fl}_U^k(F)\\to {\\mathcal Fl}_U^{k+1}(F)$ be again given by\n$${\\mathcal Fl}_U^k(F)\\to {\\tt coker}({\\mathcal Fl}_U^{k-1}(F)\\to {\\mathcal Fl}_U^{k}))\\stackrel{unit}{\\to} {\\mathcal Fl}^{k+1}_U(F)\\ .$$\nIn this way we construct an exact complex\n$$0\\to F\\to{\\mathcal Fl}^0_U(F)\\to {\\mathcal Fl}^1_U(F)\\to\\dots\\to {\\mathcal Fl}^k_U(F)\\to \\dots\\ .$$\nAll pieces of the construction are functorial. Hence, the association $F\\mapsto {\\mathcal Fl}_U(F)$ is functorial in $F$. The inclusion $F\\to {\\mathcal Fl}_U^0(F)$ gives the natural transformation ${\\tt id}\\to {\\mathcal Fl}_U$.\n\n\\subsection{}\n\n\\begin{lem}\\label{ufla32}\nFor any sheaf $F\\in {\\tt Sh}_{\\tt Ab}(U)$ the sheaf $(p_U)_*\\circ p_U^*(F)$ is flasque and flabby.\n\\end{lem}\n\\begin{proof}\nFor $W\\subseteq U$ we have \n\\begin{equation}\\label{uwdhqiwduq}\n(p_U)_*\\circ p_U^*(F)(W)\n\\cong\\prod_{w\\in W} F_w\\ .\n\\end{equation}\nIt is now obvious  that\n$(p_U)_*\\circ p_U^*(F)(U)\\to (p_U)_*\\circ p_U^*(F)(W)$\nis surjective. \nA flasque sheaf is flabby by Lemma \\ref{wgszguwqsws}. \n\\end{proof}\n\n\n\\subsection{}\\label{system14}\n\nWe now consider a sheaf $F\\in {\\tt Sh}_{\\tt Ab}\\mathbf{X}$. For $(U\\to \\mathbf{X})$ let $F_U\\in {\\tt Sh}_{\\tt Ab}(U)$ denote its restriction to $(U)$.\nWe apply the previous construction to all objects $(U\\to X)\\in \\mathbf{X}$ and the sheaves $F_U$. Then we get a collection of complexes of sheaves\n${\\mathcal Fl}_U(F_U)$ for all $(U\\to X)\\in \\mathbf{X}$.\nLet $f\\colon V\\to U$ be a morphism in $\\mathbf{X}$. We shall construct a functorial morphism\n$f^*{\\mathcal Fl}_U(F_U)\\to {\\mathcal Fl}_V(F_V)$. \n\nLet $G\\in{\\tt Sh}(U)$, $H\\in{\\tt Sh}(V)$, and $f^*G\\to H$ be a morphism of sheaves. We\nconsider the diagram \n$$\\xymatrix{\\hat V\\ar[r]^{\\hat f}\\ar[d]^{p_V}&\\hat U\\ar[d]^{p_U}\\\\V\\ar[r]^f&U}\\ .$$ \nIt induces the transformation, natural in $G$ and $H$,\n\\begin{eqnarray*}\nf^*\\circ (p_U)_*\\circ p_U^*(G)&\\to&(p_V)_*\\circ \\hat f^*\\circ p_U^*(G)\\\\\n&\\cong&(p_V)_*\\circ p_V^*\\circ f^*(G)\\\\\n&\\to&(p_V)_*\\circ p_V^*(H)\n\\end{eqnarray*}\n\nWe now construct the map $f^*{\\mathcal Fl}_U(F_U)\\to {\\mathcal Fl}_V(F_V)$ of complexes inductively.\nAssume that we have already constructed the morphisms\n$f^*({\\mathcal Fl}^i_U(F_U))\\to {\\mathcal Fl}_V^i(F_V)$ for all $i\\le k$ compatible with the differential.\nUsing that $f^*$ is right exact (Lemma \\ref{lem:lrexact}), we have an induced morphism\n\\begin{equation*}\nf^*{\\tt coker}({\\mathcal Fl}_U^{k-1}(F_U)\\to {\\mathcal Fl}_U^k(F_U))\\to {\\tt coker}({\\mathcal Fl}^{k-1}_V(F_V)\\to\n {\\mathcal Fl}_V^k(F_V)).\n\\end{equation*}\nThe construction above gives a morphism\n$f^*{\\mathcal Fl}_U^{k+1}(F_U)\\to {\\mathcal Fl}_V^{k+1}(F_V)$, again compatible with the differential of the complexes.\n\nIn this way we get the morphism\n$f^*{\\mathcal Fl}_U(F_U)\\to {\\mathcal Fl}_V(F_V)$.\nBy an inspection of the construction we check that for a second morphism\n$g\\colon W\\to V$ in $\\mathbf{X}$ the morphisms\n$g^*f^*\\mathcal{F}_U(F_U)\\to g^*\\mathcal{F}_V(F_V)\\to \\mathcal{F}_W (F_W)$ and\n$(f\\circ g)^*\\mathcal{F}_U(F_U)\\to \\mathcal{F}_W(F_W)$\ncoincide.\n\nThe collections of resolutions $F_U\\to {\\mathcal Fl}_U(F_U)$, $(U\\to X)\\in \\mathbf{X}$, determines a resolution $F\\to {\\mathcal Fl}(F)$ in $C^+({\\tt Sh}_{\\tt Ab} \\mathbf{X})$.\n \n\n\\subsection{}\n\n\\begin{lem}\\label{tutswassoll1}\nThe association $F\\mapsto (F\\to {\\mathcal Fl}(F))$  is a functorial flabby resolution.\n\\end{lem}\n\\begin{proof}\nThe local constructions $F_U\\mapsto {\\mathcal Fl}_U(F_U)$ are functorial in $F_U$.\nThe connecting maps $f^*{\\mathcal Fl}_U(F_U)\\to {\\mathcal Fl}_V(F_V)$ are compatible with this functoriality. It follows that the construction $F\\to {\\mathcal Fl}(F)$ is functorial in $F$. \n\nThe restrictions ${\\tt Sh}\\mathbf{X}\\to {\\tt Sh}(U)$ detect flabbiness and exact sequences (see  \\ref{desc_sheaves_on_U}). Therefore the local statements \\ref{lem:flab_exact} and \\ref{ufla32}\nimply that the sequence \n$0\\to F\\to {\\mathcal Fl}(F)$ is a quasi-isomorphism, and that the  sheaves ${\\mathcal Fl}^k(F)$ are flabby for all $k\\ge 0$.\n\\end{proof}\n\n\n\\begin{ddd}\\label{system100}\nWe call $F\\to {\\mathcal Fl}(F)$ the functorial flabby resolution of $F$.\n\\end{ddd}\nNote that it actually produces resolutions by flasque sheaves.\n\n\\subsection{}\n\nLet $f\\colon \\mathbf{X}\\to \\mathbf{Y}$ be a map of locally compact stacks which has local sections.\nLet ${\\mathcal Fl}_\\mathbf{X}$ and ${\\mathcal Fl}_\\mathbf{Y}$ denote the flabby resolution functors for $\\mathbf{X}$\nand $\\mathbf{Y}$ according to Definition \\ref{system100}.\n\\begin{lem}\\label{ewkuwejahh}\nWe have a natural isomorphism of functors\n$f^*\\circ {\\mathcal Fl}_\\mathbf{Y}\\cong {\\mathcal Fl}_\\mathbf{X}\\circ f^*$.\n\\end{lem}\n\\begin{proof}\n  For $(U\\to X)\\in \\mathbf{X}$ we have by  \\ref{lem:identify_star_sharp} a natural isomorphism\n$f^*F_U\\cong F_{f_\\sharp U}$.\nIt gives natural isomorphisms  ${\\mathcal Fl}_U((f^*F)_U)\\cong {\\mathcal Fl}_{f_\\sharp\n    U}(F_{f_\\sharp U})$ and thus ${\\mathcal Fl} _\\mathbf{X}(f^*F)_U\\cong\n  (f^*{\\mathcal Fl}_\\mathbf{Y})_U$. \nFinally this collection of\n  isomorphisms gives a natural isomorphism\n  $${\\mathcal Fl}_\\mathbf{X}(f^*F)\\cong f^*{\\mathcal Fl}_\\mathbf{Y}(F)\\ . $$\n\\end{proof} \n\n\\subsection{}\n\n\\begin{lem}\\label{flat-preserv}\nThe functorial flabby resolution functor preserves flatness.\n\\end{lem}\n\\begin{proof}\nConsider a space $U$, $p:\\hat U\\to U$ as above and a flat sheaf $F\\in {\\tt Sh}_{\\tt Ab}(U)$.\nThen ${\\tt coker}(F\\to p_*p^*(F))$ is flat as shown in the proof of \\cite[Lemma 3.1.4]{MR1299726}.\nThis implies inductively that the sheaves ${\\mathcal Fl}_U^k(F)$ are flat for all $k\\ge 0$.\nThe result for the functorial flabby resolution functor on ${\\tt Sh}_{\\tt Ab}\\mathbf{X}$ now follows from the fact that the restriction functors\n${\\tt Sh}_{\\tt Ab} \\mathbf{X}\\to {\\tt Sh}_{\\tt Ab}(U)$ detect flatness (see \\ref{flatdetect}).\n\\end{proof}\n\n\n\\subsection{}\n\nWe can extend\n the flabby resolution functor \\ref{system100}\nto a quasi-isomorphism preserving functor\n$${\\mathcal Fl}\\colon C^+({\\tt Sh}_{\\tt Ab}\\mathbf{X})\\to C^+({\\tt Sh}_{\\tt Ab}\\mathbf{X})$$ by applying ${\\mathcal Fl}$ to a complex term-wise and forming the total complex of the resulting double  complex.\n\n\n\n\n\n\n\\section{A model for the push-forward}\\label{iowefefwewqfqfefewf}\n\n\n\n\n\\subsection{}\\label{afixas}\n\nLet $f\\colon G\\to X$ be a morphism of  locally compact stacks which has local sections. \n Following \\cite[Sec.~2.4]{bss} we construct a nice model for the functor $Rf_*\\circ f^*\\colon D^+({\\tt Sh}_{\\tt Ab} \\mathbf{X})\\to D^+({\\tt Sh}_{\\tt Ab} \\mathbf{X})$.  We choose an atlas $a\\colon A\\to G$. Then by Proposition \\ref{lem:representability} the composition $f\\circ a\\colon  A\\to G\\to X$ is representable. \nThen we can define the functor\n$${}^pC_A\\colon C^+(\\Pr_{\\tt Ab} \\mathbf{G})\\to C^+(\\Pr_{\\tt Ab} \\mathbf{X})$$\nas in \\cite[Sec.~2.4]{bss} (the subscript ${}^p$ indicates that it acts\nbetween categories of presheaves). \n\n\\subsection{}\\label{recall-ca}\n\nWe recall the definition ${}^pC_A$.\nFor $(U\\to X)$ consider the Cartesian diagram\n$$\\xymatrix{G_U\\ar[d]\\ar[r]&G\\ar[d]^f\\\\U\\ar[r]&X}\\ .$$\nThen for $F\\in \\Pr_{\\tt Ab}\\mathbf{G}$ we have\n\\begin{equation}\\label{eq:def_of_CA}\n  {}^pC^k_A(F)(U\\to X)=F((\\underbrace{A\\times_G\\dots\\times_GA}_{k+1 \\:factors})\\times_G G_U\\to G)\\ .\n  \\end{equation}\nThe differential \n${}^pC^k_A(F)(U\\to X)\\to {}^pC^{k+1}_A(F)(U\\to X)$ is induced as usual as an\nalternating sum \nby the projections $$(\\underbrace{A\\times_G\\dots\\times_GA}_{k+2 \\:factors})\\to (\\underbrace{A\\times_G\\dots\\times_GA}_{k+1 \\:factors})\\ .$$\n\n\n\n\n\nWe extend the functor ${}^pC_A$ to sheaves by the formula\n$$C_A:=i^\\sharp\\circ {}^pC_A\\circ i\\ .$$\n\n\n\\subsection{}\\label{reww1}\n\n\n\n\nThe functor $$i^\\sharp\\colon C^+(\\Pr_{\\tt Ab}\\mathbf{X})\\to C^+({\\tt Sh}_{\\tt Ab}\\mathbf{X})$$ is exact by\n\\ref{lechejwwcc}.\nThe functor ${}^p C_A$ is exact, see \\cite[2.4.8]{bss}.\nSince flabby sheaves are $i$-acyclic the functor $i\\circ {\\mathcal Fl}:C^+({\\tt Sh}_{\\tt Ab}\\mathbf{X})\\to C^+(\\Pr_{\\tt Ab}\\mathbf{X})$\npreserves quasi-isomorphisms.\n\nTherefore the composition $$i^\\sharp\\circ {}^pC_A\\circ i\\circ {\\mathcal Fl}=C_A\\circ {\\mathcal Fl}:C^+({\\tt Sh}_{\\tt Ab}\\mathbf{G})\\to C^+({\\tt Sh}_{\\tt Ab} \\mathbf{X})$$ preserves quasi-isomorphisms and descends\nto the homotopy categories\n\\footnote{By abuse of notation we use the same symbol}\n$$C_A\\circ {\\mathcal Fl}\\colon hC^+({\\tt Sh}_{\\tt Ab}\\mathbf{G})\\to hC^+({\\tt Sh}_{\\tt Ab} \\mathbf{X})\\ .$$\nAfter identification of the homotopy categories with the derived categories\nwe have  by \\cite[2.41]{bss} that\n$$C_A\\circ {\\mathcal Fl}\\cong Rf_*\\colon D^+({\\tt Sh}_{\\tt Ab} \\mathbf{G})\\to D^+({\\tt Sh}_{\\tt Ab} \\mathbf{X})\\ .$$\n\n\n\n\n\\subsection{}\\label{reww2}\n\nSince $f\\colon G\\to X$ has local sections the functor $f^*$ is exact.\nIt therefore descends to \n$$f^*\\colon hC^+({\\tt Sh}_{\\tt Ab}\\mathbf{X})\\to hC^+({\\tt Sh}_{\\tt Ab} \\mathbf{G})\\ .$$\nThe composition\n$$C_A\\circ {\\mathcal Fl}\\circ f^*\\colon hC^+({\\tt Sh}_{\\tt Ab}\\mathbf{X})\\to hC^+({\\tt Sh}_{\\tt Ab} \\mathbf{X})$$\nthus represents\n$$Rf_*\\circ f^*\\colon D^+({\\tt Sh}_{\\tt Ab} \\mathbf{X})\\to D^+({\\tt Sh}_{\\tt Ab} \\mathbf{X})\\ .$$\n\n\n\n\n\\subsection{}\\label{gghaaase}\nWe now study the dependence of $C_A$ on the choice of the atlas $A\\to G$.\nLet us consider a diagram \n\\begin{equation}\\label{afauniz}\n\\xymatrix{A^\\prime\\ar[rr]^\\phi\\ar[dr]^{a'}&&A\\ar[dl]^a\\\\&G&}\\ ,\n\\end{equation} \nwhere $a^\\prime$ satisfies the same assumptions as $a$ (see \\ref{afixas}).\nThe map $\\phi$ induces maps\n$$\\xymatrix{(A^\\prime\\times_G\\dots\\times_GA^\\prime)\\times_GG_U \\ar[rr]^{\\phi^{k+1}\\times {\\tt id}_{G_U}} \\ar[dr]&&(A\\times_G\\dots\\times_GA)\\times_GG_U \\ar[dl]\\\\&G&}$$\nand therefore\n\\begin{eqnarray*}\n{}^pC^k_A(F)(U\\to X)&=&F((\\underbrace{A\\times_G\\dots\\times_GA}_{k+1 \\:factors})\\times_G G_U\\to G)\\\\\n&\\to&\nF((\\underbrace{A^\\prime\\times_G\\dots\\times_GA^\\prime}_{k+1 \\:factors})\\times_G G_U\\to G)\\\\\n&=&\n{}^pC^k_{A^\\prime}(F)(U\\to X)\\ .\n\\end{eqnarray*}\nThis map is natural in $F$ and preserves the cosimplicial structures.\nIn other words, the diagram (\\ref{afauniz}) induces a natural transformation\n$${}^p C_{\\phi}\\colon  {}^pC_A\\to {}^pC_{A^\\prime}\\ .$$\nComposing with $i^\\sharp$ and $i\\circ {\\mathcal Fl}$ we \nget a morphism of functors\n$$C_\\phi\\colon C_A\\circ {\\mathcal Fl}\\to C_{A^\\prime}\\circ {\\mathcal Fl}\\colon hC^+({\\tt Sh}_{\\tt Ab}\\mathbf{G})\\to hC^+({\\tt Sh}_{\\tt Ab} \\mathbf{X})\\ .$$\nBoth $C_A\\circ {\\mathcal Fl}$ and $C_{A^\\prime}\\circ {\\mathcal Fl}$ represent $Rf_*$.\nUsing the explicit constructions and the proof of \\cite[Lemma 2.36]{bss}\none checks that the diagram\n$$\\xymatrix{H^0(C_A\\circ {\\mathcal Fl})(F)\\ar[rr]^{\\hspace{1cm}H^0(C_\\phi)}\\ar[dr]&&H^0(C_{A^\\prime}\\circ {\\mathcal Fl})(F)\\ar[dl]\\\\&f_*(F)&}\n$$\ncommutes.\nTherefore, on the level of derived categories,  $C_\\phi:C_A\\circ {\\mathcal Fl}\n\\to C_{A^\\prime}\\circ {\\mathcal Fl}$ is the canonical isomorphism between two realizations of $Rf_*$.\n\n\n\n\n\n\\subsection{}\\label{sec:let-qcolon-hto}\n\nLet $q\\colon H\\to G$ be a representable morphism with local sections.\nConsider the pullback diagram\n$$\\xymatrix{B\\ar[r]^b\\ar[d]^l&H\\ar[d]^q\\\\A\\ar[r]^a&G\\ar[d]^f\\\\&X}\\ .\n$$\nThen $b:B\\to H$ is an atlas, and  we can form the functor\n$C_B\\colon C^+(\\Pr_{\\tt Ab} \\mathbf{H})\\to C^+(\\Pr_{\\tt Ab}\\mathbf{X})$.\n\n\nObserve that\n$$B\\times_H\\dots \\times_HB\\cong (A\\times_G\\dots\\times_GA)\\times_G H\\ .$$\nFor $(U\\to X)$ consider the diagram\n$$\\xymatrix{H_U\\ar[r]\\ar[d]&H\\ar[d]^q\\\\G_U\\ar[d]\\ar[r]&G\\ar[d]^f\\\\U\\ar[r]&X}\\ .\n$$\nObserve further that\n$$(B\\times_H\\dots \\times_HB)\\times_HH_U\\cong (A\\times_G\\dots\\times_GA)\\times_G G_U\\times_G  H\\ .$$\nFor a presheaf $F\\in \\Pr \\mathbf{H}$ and $(V\\to G)\\in \\mathbf{G}$ we have\n${}^pq_*(F)(V)=F(V\\times_G H)$.\nWe now have the following identity\n\\begin{eqnarray*}\n{}^pC_A^k\\circ {}^pq_*(F)(U\\to X)&\\cong&{}^pq_*(F)(\\underbrace{(A\\times_G\\dots\\times_GA)}_{k+1 factors}\\times_GG_U\\to G)\\\\\n&\\cong &F(((\\underbrace{A\\times_G\\dots\\times_GA}_{k+1 factors})\\times_GG_U)\\times_GH\\to H)\\\\\n&\\cong&\nF((\\underbrace{B\\times_H\\dots\\times_HB}_{k+1 factors})\\times_HH_U\\to H)\\\\\n&\\cong&{}^pC^k_B(F)(U\\to X)\n\\end{eqnarray*}\nThis isomorphism is functorial in $F$ and induces a natural isomorphism\n$${}^pC_A\\circ {}^pq_*\\cong {}^p C_{q^*A}\\ ,$$\nwhere we write $q^*A:=B$.\n\n\nThe functor ${}^pq_*$ preserves sheaves  \\cite[Lemma 2.13]{bss}. Therefore we get the identity \n$$i\\circ i^\\sharp \\circ{}^p q_*\\circ i={}^p q_*\\circ i$$\n and thus an  isomorphism\n\\begin{equation}\\label{zughj293}C_A\\circ q_*\\cong i^\\sharp\\circ {}^p C_A\\circ i\\circ i^\\sharp\\circ {}^pq_*\\circ i\\cong   i^\\sharp\\circ {}^p C_A\\circ  {}^pq_*\\circ i \\cong i^\\sharp\\circ {}^pC_{q^*A}\\circ i\\cong C_{q^*A}\\ .\\end{equation}\n\n\n\n \n \n\n \n\n\n\n\\subsection{}\n\nConsider a Cartesian diagram\n$$\\xymatrix{H\\ar[d]^g\\ar[r]^v&G\\ar[d]\\\\Y\\ar[r]^u&X}$$\nwhere $u$ has local sections.  We extend the diagram to\n$$\\xymatrix{B\\ar[d]\\ar[r]&A\\ar[d]\\\\H\\ar[r]^v\\ar[d]^g&G\\ar[d]^f\\\\Y\\ar[r]^u&X}\\ .$$\nThe map $B\\to H$ is again an atlas.  \n\\begin{lem}\\label{pulcomghjdf}\nWe have a natural isomorphism of functors\n$$ u^*\\circ C_A\\cong C_B\\circ v^*\\ .$$\n\\end{lem}\n\\begin{proof}\nWe first find a natural isomorphism\n$${}^pu^* \\circ {}^p C_A\\cong {}^pC_B\\circ {}^pv^*.$$\nLet $(U\\to Y)\\in \\mathbf{Y}$ and $F\\in \\Pr_{\\tt Ab}\\mathbf{G}$. Then we have\n$${}^pu^*\\circ {}^pC_A(F)(U)\\cong {}^pC_A(F)(u_\\sharp U)\\ .$$\n We have a diagram\n$$\\xymatrix{H_U\\cong G_{u_\\sharp U}\\ar[r]\\ar[d]&H\\ar[r]^v\\ar[d]^g&G\\ar[d]\\\\U\\ar[r]&Y\\ar[r]^u&X}\\ .$$\nWe calculate\n\\begin{eqnarray*}\n(A\\times_G\\dots\\times_GA)\\times_G G_{u_\\sharp U}&\\cong&\n(A\\times_G\\dots\\times_GA)\\times_G H\\times_H G_{u_\\sharp U}\\\\\n&\\cong&v_\\sharp (B\\times_H\\dots\\times_HB)\\times_H H_{U}\n\\end{eqnarray*}\nThis implies that\n\\begin{eqnarray*}\n{}^pu^*\\circ C_A(F)(U)&\\cong&{}^pC_A(F)(u_\\sharp U)\\\\\n&\\cong&F((A\\times_G\\dots\\times_GA)\\times_G G_{u_\\sharp U})\\\\\n&\\cong&F(v_\\sharp ((B\\times_H\\dots\\times_HB)\\times_H H_{U}))\\\\\n&\\cong&({}^pv^*F)((B\\times_H\\dots\\times_HB)\\times_H H_{U})\\\\\n&\\cong&{}^pC_B\\circ {}^pv^*(F)(U)\n\\end{eqnarray*}\nSince $u$ and $v$ have local sections, by \\ref{prexact} the functors\n${}^pu^*$ and ${}^pv^*$ commute with $i\\circ i^\\sharp$, and this isomorphism\ninduces\n$$u^*\\circ C_A\\cong C_B\\circ v^*$$\n(compare with the calculation (\\ref{zughj293})). \n\\end{proof} \n\n\n\\subsection{}\n\n The isomorphisms of Lemma \\ref{pulcomghjdf} and Lemma \\ref{ewkuwejahh}\n induce an isomorphism\n\\begin{equation}\\label{uiidwqdwqdwqd}\nu^*\\circ C_A\\circ {\\mathcal Fl}\\cong C_B\\circ u^*\\circ {\\mathcal Fl}\\cong  C_B\\circ {\\mathcal Fl}\\circ v^*\\ .\n\\end{equation}\n\nOn the other hand, by Lemma \\ref{lem:pullpush} we have an isomorphism\n$$u^*\\circ Rf_*\\cong Rg_*\\circ v^*\\ .$$\n\\begin{lem}\\label{uiqehewqdqwdwqdqd}\nThe following diagram of natural isomorphisms of functors \n$$D^+({\\tt Sh}_{\\tt Ab}\\mathbf{G})\\to D^+({\\tt Sh}_{\\tt Ab}\\mathbf{H})$$ commutes.\n$$\\xymatrix{u^*\\circ C_A\\circ {\\mathcal Fl}\\ar[d]^\\cong\\ar[r]^\\cong&C_B\\circ {\\mathcal Fl}\\circ v^*\\ar[d]^\\cong\\\\\nu^*\\circ Rf_*\\ar[r]^\\cong&Rg_*\\circ v^*}\n$$\n\\end{lem}\n\\begin{proof}\nIt is easy to check that this commutativity holds true on the level of zeroth cohomology sheaves.\nSince all functors are the derived versions of their zeroth cohomology functors\nthe required commutativity follows.\n\\end{proof}\n\n\\begin{kor}\\label{gdashdgasd}\nThe following diagram of natural isomorphisms commutes\n$$\\xymatrix{u^*\\circ C_A\\circ {\\mathcal Fl}\\circ f^*\\ar[d]^\\cong\\ar[r]^\\cong&C_B\\circ {\\mathcal Fl}\\circ g^*\\circ u^*\\ar[d]^\\cong\\\\\nu^*\\circ Rf_*\\circ f^*\\ar[r]^\\cong&Rg_*\\circ g^*\\circ  u^*}\n$$\n\\end{kor}\n\n\n\n\n\n\n\n\n\n \n \n\n\\section{Zig-zag diagrams and limits}\n\n \n\\subsection{}\\label{system11}\n\nWe define the unbounded derived category $D(\\mathcal{A})$ of an abelian category\nas the homotopy category $hC(\\mathcal{A})$ of  complexes (with no restrictions) in $\\mathcal{A}$.\n\n\n\\begin{ddd}\\label{system104}\nAn abelian category $\\mathcal{A}$ with the following properties \n\\begin{enumerate}\n\\item $\\mathcal{A}$ is cocomplete,\n\\item filtered colimits are exact,\n\\item $\\mathcal{A}$ has a generator, i.e.~there is an object $Z$ such that for every\n  object $F$ with proper subobject $F'\\subset F$, ${\\tt Hom}(Z,F')\\to {\\tt Hom}(Z,F)$ is\n  not surjective.\n\\end{enumerate}\nis called a Grothendieck abelian category.\n\\end{ddd}\n\nIn this section, we will consider a Grothendieck category in which countable\nproducts exist, e.g.~a complete Grothendieck category.\nThe category ${\\tt Sh}_{\\tt Ab} \\mathbf{X}$ of sheaves of abelian groups on a site $\\mathbf{X}$ is a complete\nGrothendieck abelian category \\cite[Chapter I, Thm. 3.2.1]{MR1317816}. \n\\begin{lemma}\\label{lem:diagramGroth}\n  If $Z$ is a small category and $\\mathcal{A}$ is a Grothendieck abelian category in\n  which countable products exists,\n  then the diagram category $\\mathcal{A}^Z$ is again a Grothendieck abelian category\n  in which countable products exist.\n\\end{lemma}\n  This is proved in \\cite[1.4.3]{MR1317816}.\n\n\n\\subsection{}\\label{ztoou6}\n\nWe consider the category $C(\\mathcal{A})$ of complexes in a Grothendieck abelian category  $\\mathcal{A}$.\nIt is known that $C(\\mathcal{A})$ has a model category structure\n(see \\cite[Theorem 2.2]{MR1814077} where this fact is attributed to Joyal, \\cite[Thm. 2.3.12]{MR1650134} for the example of the category of modules over a ring, and \\cite{MR1780498} for a proof in general). This model structure is given by the following data:\n\\begin{enumerate}\n\\item The weak equivalences are the quasi-isomorphisms.\n\\item The cofibrations are the degree-wise injections.\n\\item The fibrations are defined by the right lifting property.\n\\end{enumerate}\nBy $hC(\\mathcal{A})$ we denote the homotopy category of $C(\\mathcal{A})$.\nThe category $hC(\\mathcal{A})$ is triangulated with\nthe shift functor $T\\colon hC(\\mathcal{A})\\to hC(\\mathcal{A})$  given\nby the shift of complexes $T(X)=X[1]$. The class of distinguished triangles is generated\nby the mapping cone sequences on $C(\\mathcal{A})$, \n$$\\dots\\to A\\stackrel{f}{\\to} B\\to C(f)\\to T(A)\\dots\\ .$$\n \n\nThe extension of a morphism in $[f]\\in hC(\\mathcal{A})$ with chosen representative $f\\in C(\\mathcal{A})$ \nto a triangle can thus naturally be  defined using the mapping cone  $C(f)$.\n \n \n\n \n \n\\subsection{}\\label{ztoou}\nLet $\\mathcal{A}$ be a Grothendieck abelian category, and consider\n  a small category $Z$. Then we have an equivalence\n$C(\\mathcal{A})^Z\\cong C(\\mathcal{A}^Z)$. Because $\\mathcal{A}^Z$ is a Grothendieck category by Lemma\n  \\ref{lem:diagramGroth},  \nwe can equip the category of $Z$-diagrams $C(\\mathcal{A})^Z$\nwith the injective model category structure.\nBy translation of \\ref{ztoou6} we get the following description.\n\\begin{enumerate}\n\\item The weak equivalences are the level-wise quasi-isomorphisms.\n\\item The cofibrations are the level-wise injections.\n\\item The fibrations are defined by the right lifting property.\n\\end{enumerate}\n\n\\subsection{}\\label{hcddef}\n\nWe consider the category $U$ pictured by\n$$\\xymatrix{\\bullet&\\bullet\\ar[l]\\ar[r]&\\bullet&\\bullet\\ar[l]\\ar[d]\\\\\\bullet\\ar[u]&&&\\bullet}.$$\n\n\nWe let $\\mathcal{D}(\\mathcal{A})\\subset C^+(\\mathcal{A})^U$ be the subcategory of objects of the form\n\\begin{equation}\\label{square1}\n\\xymatrix{Y_0&Y_1\\ar[l]^\\sim\\ar[r]&Y_2&Y_3\\ar[l]^\\sim\\ar[d]\\\\X\\ar[u]&&&X[-2]}\\ \n\\end{equation}\nwith bounded below complexes $Y_i,X$.\nA morphism in the category $\\mathcal{D}(\\mathcal{A})$ is given by maps $Y_i\\to Y_i^\\prime$,\n$i=0,1,2,3$, and $X\\to X^\\prime$ which are compatible with the structure\nmaps. A quasi-isomorphism in this category is a morphism which is a\nquasi-isomorphism level-wise. \n\n\n\n\n\n\\subsection{}\\label{system15}\n\nWe let $Z$ be the category pictured by\n$$\\xymatrix{\\vdots\\ar[dr]&\\vdots\\\\\\bullet \\ar[r]\\ar[dr]&\\bullet\\\\\n\\bullet\\ar[r]\\ar[dr]&\\bullet\\\\\n\\bullet\\ar[r]\\ar[dr]&\\bullet\\\\\n\\bullet\\ar[r]&\\bullet}\\ .\n$$\n\nLet $C(\\mathcal{A})^Z$ be the category of $Z$-diagrams of complexes in $\\mathcal{A}$.  \nWe define a functor\n$$R_1\\colon \\mathcal{D}(\\mathcal{A})\\to  C(\\mathcal{A})^Z$$\nwhich maps the diagram (\\ref{square1}) to the $Z$-diagram\n$$\\xymatrix{\\vdots\\ar[dr]&\\vdots\\\\Y_3[4] \\ar[r]^{}\\ar[dr]^{}&Y_2[4]\\\\\nY_1[2]\\ar[r]^{}\\ar[dr]^{}&Y_0[2]\\\\\nY_3[2]\\ar[r]^{\n&Y_2[2] \n}\\ .\n$$\nThe maps are induced by the shifted maps of the diagram  (\\ref{square1}),\nand the composition $Y_3[2k+2]\\to X[2k]\\to Y_0[2k]$.\nThe functor $R_1$  preserves quasi-isomorphisms, since those are defined\nlevel-wise.\n\n\\subsection{}\n\n We now define a triangulated functor\n$$\\lim\\colon h(C(\\mathcal{A})^Z)\\to hC(\\mathcal{A})$$\nby a direct construction \non the level of complexes.  \n\nConsider a $Z$-diagram $X\\in C(\\mathcal{A})^Z$\n$$\\xymatrix{C_3 \\ar[r]^{c_3}\\ar[dr]^{d_3}&B_3\\\\\nC_2\\ar[r]^{c_2}\\ar[dr]^{d_2}&B_2\\\\\nC_1\\ar[r]^{c_1}\\ar[dr]^{d_1}&B_1\\\\\nC_0\\ar[r]^{c_0}&B_0}\\ .\n$$\nWe define the morphism in $C(\\mathcal{A})$\n$$\\phi_X\\colon \\prod_{i\\ge 0} C_i \\to \\prod_{i\\ge 0 }B_i$$\nwhich maps\n$(x_i)_{i\\ge 0}$ to $(c_i(x_i)-d_{i+1}(x_{i+1}))_{i\\ge 0}$.\nThen we define $\\lim(X)$ as a shifted cone of $\\phi_X$:  $$\\lim(X):=C(\\phi_X)[-1]\\in C(\\mathcal{A})\\ .$$  \nSince quasi-isomorphisms in $C(\\mathcal{A})^Z$ are defined level-wise, the\nfunctorial  construction  $X\\to \\lim X$ preserves quasi-isomorphisms and thus\ndescends to a functor  \n$$\\lim\\colon h(C(\\mathcal{A})^Z)\\to h C(\\mathcal{A})\\ .$$\nNote that $\\lim$ commutes with the shift and sum, so that it is  a triangulated functor.\n\n\\subsection{}\n\nWe now consider the composition\n$\\lim\\circ R_1\\colon \\mathcal{D}(\\mathcal{A})\\to hC(\\mathcal{A})$.\nThe composition of the maps (or their inverses, respectively) in the diagram (\\ref{square1}) gives rise to a morphism \n$D[-2]\\colon X\\to X[-2]$ in $hC(\\mathcal{A})$.  We consider the  sequence\n\\begin{equation}\\label{seq31o}\nX^\\bullet \\colon X\\stackrel{D}{\\leftarrow} X[2]\\stackrel{D[2]}{\\leftarrow} X[4]\\leftarrow \u00b8\\dots\\ .\n\\end{equation} \nin $hC(\\mathcal{A})$. As already explained in \\ref{system12}, for such a diagram in the triangulated category $hC(\\mathcal{A})$ the homotopy limit ${\\tt holim}(X^\\bullet)\\in hC(\\mathcal{A})$ is a well-defined isomorphism class of objects. It is \ngiven by the mapping cone shifted by $-1$ of the morphism\n$$\\prod_{i\\ge 0}X[2i]\\to \\prod_{i\\ge 0}X[2i]$$\nwhich maps $(x_i)_{i\\ge 0}$ to $(x_i-D[2i]x_{i+1})_{i\\ge 0}$\n(see \\cite[Sec.~1.6]{MR1812507}).\n\n\\begin{lem}\\label{system17}\nFor a diagram $W\\in \\mathcal{D}(\\mathcal{A})$ of the form \\eqref{square1}\nwe have a non-canonical isomorphism\n$${\\tt holim}(X^\\bullet)\\cong \\lim\\circ R_1(W)\\ .$$\n\\end{lem}\n\\begin{proof}\nWe use the dual statement of \\cite[Lemma 1.7.1]{MR1812507}.\nFor $i\\ge 1$ let  $C_{2i-1}=Y_3[2i]$, $C_{2i}:=Y_1[2i]$, $B_{2i-1}:=Y_2[2i]$\nand $B_{2i}:=Y_{0}[2i]$. Note that we have morphisms\n$v_i\\colon C_{i}\\to B_i$ in $C(\\mathcal{A})$ which become isomorphisms in\n$hC(\\mathcal{A})$. Moreover, we have maps $w_{2i}\\colon C_{2i}\\to B_{2i-1}$ coming\nfrom the map $Y_1\\to Y_2$ of \\eqref{square1}, and morphisms $w_{2i+1}\\colon\nC_{2i+1}\\to B_{2i}$ coming from $Y_3[2]\\to X\\to Y_0$ of \\eqref{square1}.\nWe consider the diagram in $hC(\\mathcal{A})$, using the invertibility of $v_i$ in\n$hC(\\mathcal{A})$,\n$$\n\\begin{CD}\n  \\prod_{i\\ge 1} C_{i} @>{\\prod v_i-\\prod w_i}>{\\phi_{R_1(W)}}>\n\\prod_{i\\ge 1}B_{i}\n\\\\\n@VV{{\\tt id}}V @VV{\\prod_{i\\ge 1} v_i^{-1}}V \\\\\n    \\prod_{i\\ge 1} C_i @>>\n\\prod_{i\\ge 1} C_i\\ ,\n\\end{CD}\n$$\nwhose vertical maps are isomorphism. By definition, the mapping cone of the\nupper horizontal map is $\\lim\\circ R_1(W)$. Because the vertical maps are\nisomorphisms in $hC(\\mathcal{A})$, this is isomorphic to the mapping cone of the lower horizontal map, which gives the homotopy limit of the sequence\n$$ Y_3[2]\\stackrel{}{\\leftarrow}Y_1[2]\\stackrel{}{\\leftarrow}Y_3[4]\\leftarrow Y_1[4]\\leftarrow Y_3[6]\\dots\n\\ .$$\n\nWe can expand this sequence to\n\\begin{multline}\\label{longsetw}\nX\\stackrel{}{\\leftarrow}Y_3[2]\\stackrel{}{\\leftarrow}\nY_2[2]\\stackrel{}{\\leftarrow}Y_1[2]\\stackrel{}{\\leftarrow}Y_0[2]\\stackrel{}{\\leftarrow}X[2]\\\\\n\\leftarrow Y_3[4]\\stackrel{}{\\leftarrow}Y_2[4]\\leftarrow Y_1[4]\\leftarrow Y_0[4]\\leftarrow X[4]\\leftarrow Y_3[6]\\dots\\ ,\n\\end{multline} \nand because the sequence (\\ref{seq31o}) is just another contraction of\n\\eqref{longsetw}, by \\cite[Lemma 1.7.1]{MR1812507} its homotopy limit\n${\\tt holim}(X^\\bullet)$ is then also isomorphic to \n$\\lim\\circ R_1(W)$. \n\\end{proof}\n\n\n \n\n\\section{The functorial periodization}\\label{funct_per}\n\n\n\\subsection{}\\label{hdwidhwqdiwqdwdw}\n\nLet $X$ be a locally compact stack. Define \n$C^+({\\tt Sh}^{flat}_{\\tt Ab}\\mathbf{X})\\subseteq C^+({\\tt Sh}_{\\tt Ab}\\mathbf{X})$ to be the full subcategory of\nbounded below complexes of flat sheaves. \n\\begin{lem}\\label{flat-inclu}\nThis inclusion induces an equivalence of homotopy categories\n$$hC^+({\\tt Sh}^{flat}_{\\tt Ab}\\mathbf{X})\\stackrel{\\sim}{\\to} hC^+({\\tt Sh}_{\\tt Ab}\\mathbf{X})\\ .$$\n\\end{lem}\n\\begin{proof}\nWe first construct a functorial flat resolution functor\n$$R:{\\tt Sh}_{\\tt Ab}\\mathbf{X}\\to C^b({\\tt Sh}^{flat}_{\\tt Ab}\\mathbf{X})\\ .$$\n Note that a torsion free sheaf is flat.\nIf $F\\in {\\tt Sh}_{\\tt Ab}\\mathbf{X}$, then let $\\hat F\\in \\Pr\\mathbf{X}$ denote the underlying presheaf of sets.\nLet $\\mathbb{Z}\\hat F\\in \\Pr_{\\tt Ab}\\mathbf{X}$ be the presheaf of free abelian groups generated by $\\hat F$, and $\\mathbb{Z} F:=i^\\sharp \\mathbb{Z}\\hat F$ be its sheafification. Then we have a natural evaluation $\\mathbb{Z}\\hat F\\to F$, which extends uniquely to $e:\\mathbb{Z} F\\to F$ since $F$ is a sheaf. We define\n$R(F)$ to be the complex $\\ker(e)\\to \\mathbb{Z} F$, where $\\mathbb{Z} F$ is in degree zero.\nThe natural map $R(F)\\to F$ is a quasi-isomorphism. Moreover,\n$\\mathbb{Z} F$ and its subsheaf $\\ker(e)$ are torsion-free, hence flat.\n\nWe extend $R$ to a functor\n$R:C^+({\\tt Sh}_{\\tt Ab}\\mathbf{X})\\to C^+({\\tt Sh}^{flat}_{\\tt Ab}\\mathbf{X})$ by applying $R$ objectwise and taking the total complex of the resulting double complex.\n\nThe inclusion $C^+({\\tt Sh}^{flat}_{\\tt Ab}\\mathbf{X})\\to C^+({\\tt Sh}_{\\tt Ab}\\mathbf{X})$\nand $R: C^+({\\tt Sh}_{\\tt Ab}\\mathbf{X})\\to C^+({\\tt Sh}^{flat}_{\\tt Ab}\\mathbf{X})$ induce mutually inverse\nfunctors of the homotopy categories.\n\\end{proof}\n\n \n\n\\subsection{}\n\nLet $f\\colon G\\to X$ be a topological gerbe with band $U(1)$ over a locally compact stack.\nRecall the associated geometry introduced in \\ref{ddef2}. {Using the functorial version we\n get the diagram\n\\begin{equation}\\label{system40}\\xymatrix{&T^2\\times G\\ar[dl]^{p}\\ar[dr]^m &\\\\G\\ar[dr]^f&&G\\ar[dl]^f\\\\&X&}\\end{equation}\nwhich $2$-functorially depends on the gerbe $G\\to X$.}\nThe map $p\\colon T^2\\times G\\to G$ is the projection onto the second factor,\nand $m:=p\\circ \\phi$.\n \n\n\\subsection{}\\label{system21}\n\n \n\nObserve that $p$ is a trivial oriented fiber bundle with fiber $T^2$.\nLet $$0\\to \\underline{\\Z}_{{\\tt Site}(T^2\\times G)}\\to {\\mathcal Fl}(\\underline{\\Z}_{{\\tt Site}(T^2\\times G)})$$ be the\nfunctorial flat and flabby resolution of $\\underline{\\Z}_\\mathbf{G}$ \nconstructed in \\ref{system14}, see also \\ref{flat-preserv} for flatness. By\n$$K^\\cdot\\colon 0\\to  K^0\\to K^1\\to K^2\\to 0$$\nwe denote the truncation of ${\\mathcal Fl}(\\underline{\\Z}_{{\\tt Site}(T^2\\times G)})$ after the\nsecond term, i.e. with $$K^2:=\\ker({\\mathcal Fl}^2(\\underline{\\Z}_{{\\tt Site}(T^2\\times G)})\\to {\\mathcal Fl}^3(\\underline{\\Z}_{{\\tt Site}(T^2\\times G)}))\\ .$$\nThe complex $K^\\cdot$ is still a flat and $p_*$-acyclic resolution of $\\underline{\\Z}_{{\\tt Site}(T^2\\times G)}$ (Lemma \\ref{shortrest12}).  Let $$T\\colon C^+({\\tt Sh}_{\\tt Ab}{\\tt Site}(T^2\\times G))\\to C^+({\\tt Sh}_{\\tt Ab}{\\tt Site}(T^2\\times G))$$ be the functor given on objects by   \n$$T_{K^\\cdot}(F):=F\\otimes K^\\cdot\\ .$$\n\n \n\n\n\n\\subsection{}\\label{rcubd}\n \n{We consider the commutative diagram \\ref{system40}}.\nSince $f\\circ p\\cong f\\circ m$ {(recall that we actually can assume equality)}  we have by Lemma  \\ref{feriueewwwwzzz} and Corollary \\ref{uefhewiufuwefzzz} isomorphisms of functors\n$m^*\\circ f^*\\cong p^*\\circ f^*$ and $f_*\\circ m_*\\cong f_*\\circ p_*$.\nWe fix an atlas $A\\to G$ and define\n$X\\colon C^+({\\tt Sh}^{flat}\\mathbf{X})\\to C^+({\\tt Sh}\\mathbf{X})$ by\n$$X:=C_A\\circ f^*\\circ {\\mathcal Fl}\\ .$$ \nSince $f$ has local sections   we have\n $f^*\\circ {\\mathcal Fl}\\cong {\\mathcal Fl}\\circ f^*$ by Lemma \\ref{ewkuwejahh}.\nIt now follows from \\ref{reww2} that $X\\cong C_A\\circ {\\mathcal Fl}\\circ f^*$ preserves quasi-isomorphisms.  \n It therefore descends to the homotopy categories and induces the functor $Rf_*\\circ f^*$\n$$D^+({\\tt Sh}_{\\tt Ab}\\mathbf{G})\\stackrel{Lemma\\  \\ref{flat-inclu}}{\\cong} hC^+({\\tt Sh}^{flat}_{\\tt Ab}\\mathbf{G})\\stackrel{X}{\\to} hC^+({\\tt Sh}_{\\tt Ab}\\mathbf{X})\\cong D^+({\\tt Sh}_{\\tt Ab}\\mathbf{X})\\ .$$\n\n\n\\subsection{}\\label{rcubd1}\n\nWe further form $B:=m^*A\\times_{T^2\\times G} p^*A$.\nIt comes with a natural morphism $B\\to m^*A$ over $T^2\\times G$\nwhich induces a transformation\n$C_{m^*A}\\to C_B$. \nUsing the unit ${\\tt id}\\to m_*\\circ m^*$, the inclusion\n${\\tt id}\\to T_{K^\\cdot}$, and the isomorphisms $m^*\\circ f^*\\cong p^*\\circ f^*$,\nand using that by \\ref{sec:let-qcolon-hto} $C_A\\circ m_*\\cong C_{m^*A}$, we\ndefine a natural transformation\n\\begin{eqnarray*}\nX&= &C_A\\circ f^*\\circ {\\mathcal Fl}\\\\\n&\\to&C_A\\circ m_*\\circ m^*\\circ  f^*\\circ {\\mathcal Fl}\\\\\n&\\to&C_A\\circ m_*\\circ T_{K^\\cdot}\\circ  m^* \\circ f^*\\circ {\\mathcal Fl}\\\\\n&\\cong&C_{m^*A}\\circ T_{K^\\cdot} \\circ m^* \\circ f^*\\circ {\\mathcal Fl}\\\\\n&\\cong&C_{m^*A}\\circ T_{K^\\cdot} \\circ p^* \\circ f^*\\circ {\\mathcal Fl}\\\\\n&\\to&C_{m^*A}\\circ {\\mathcal Fl}\\circ T_{K^\\cdot} \\circ p^* \\circ f^*\\circ {\\mathcal Fl}\\\\\n&\\stackrel{}{\\to}&C_{B}\\circ {\\mathcal Fl}\\circ T_{K^\\cdot} \\circ p^* \\circ f^*\\circ {\\mathcal Fl}\\\\\n&=:&Y_0\n\\end{eqnarray*}\n\n\nUsing the other projection $B\\to p^*A$ we define\n\\begin{eqnarray*}\nY_1&:=&C_{p^*A}\\circ {\\mathcal Fl}\\circ T_{K^\\cdot}\\circ p^*\\circ f^*\\\\\n&\\stackrel{\\sim}{\\to}&C_{B}\\circ {\\mathcal Fl}\\circ T_{K^\\cdot} \\circ p^* \\circ f^*\\\\\n&\\stackrel{\\sim}{\\to}&C_{B}\\circ {\\mathcal Fl}\\circ T_{K^\\cdot} \\circ p^* \\circ f^*\\circ\n{\\mathcal Fl}\\\\\n&=&Y_0 \\ .\n\\end{eqnarray*}\nUsing the identity $C_{p^*A}\\cong C_A\\circ p_*$ we define\n\\begin{eqnarray*}\nY_1&=&C_{p^*A}\\circ {\\mathcal Fl}\\circ T_{K^\\cdot}\\circ p^*\\circ f^*\\\\\n&\\cong&C_{A}\\circ p_* \\circ {\\mathcal Fl}\\circ T_{K^\\cdot}\\circ p^*\\circ f^*\\\\\n&\\to&C_{A}\\circ {\\mathcal Fl}\\circ p_* \\circ {\\mathcal Fl}\\circ T_{K^\\cdot}\\circ p^*\\circ f^*\\\\\n&=:&Y_2\n\\end{eqnarray*}\nNote that $p_*\\circ T_K$ is an exact functor by Lemma \\ref{ggtre1} and\ncalculates \n$Rp_*$ by Corollary \\ref{corol:calculate_Rf}. Since $p_*\\circ {\\mathcal Fl}\\circ\nT_K$ represents the same functor\nthe map $p_*\\circ T_K\\to p_*\\circ {\\mathcal Fl}\\circ T_K$ induces a quasi-isomorphism which is preserved by $C_A\\circ {\\mathcal Fl}$.\nThe natural transformation\n$T_{p_*K^\\cdot}\\xrightarrow{\\sim} p_*\\circ T_{K^\\cdot}\\circ p^*$\nis an isomorphism, if applied to complexes of flat sheaves by \\ref{system81}.\nBy Lemma \\ref{prexact} the pull-back $f^*$ preserves flatness.\n \nThese two facts explain the quasi-isomorphisms in\n \\begin{eqnarray*}\nY_3&:=&C_{A}\\circ {\\mathcal Fl}\\circ T_{p_*K^\\cdot}\\circ f^*\\\\\n   & \\stackrel{\\sim}{\\to} & C_{A}\\circ {\\mathcal Fl}\\circ p_*  \\circ T_{K^\\cdot}\\circ p^*\\circ f^*\\\\\n&\\stackrel{\\sim}{\\to}&C_{A}\\circ {\\mathcal Fl}\\circ p_* \\circ {\\mathcal Fl}\\circ T_{K^\\cdot}\\circ p^*\\circ f^*\\\\\n&=&Y_2\\ .\\end{eqnarray*}\nUsing the projection $T_{p_*K}\\stackrel{[-2]}{\\to}{\\tt id}$ of (\\ref{iofejoiwfwefwef}) we\ndefine the natural transformation\n\\begin{eqnarray}\\label{fret-wq}\nY_3&=&C_{A}\\circ {\\mathcal Fl}\\circ T_{p_*K^\\cdot}\\circ f^*\\\\\n&\\to&C_{A}\\circ {\\mathcal Fl}\\circ f^*[-2]\\nonumber\\\\\n&\\cong&C_A\\circ f^*\\circ {\\mathcal Fl}[-2]\\nonumber\\\\\n&=&X[-2]\\nonumber\\ .\n\\end{eqnarray}\nObserve that all functors $Y_i$ preserve quasi-isomorphisms, using that $f^*$,\n$p^*$, $C_A\\circ {\\mathcal Fl}$, $p_*\\circ T_K$ (and by Lemma \\ref{system81} therefore\nalso $T_{p_*K}$) do so.\n\n\\subsection{}\\label{system16}\n\nThe construction \\ref{rcubd}, \\ref{rcubd1} gives a quasi-isomorphism preserving functor\n$$R_0\\colon C^+({\\tt Sh}^{flat}_{\\tt Ab}\\mathbf{X})\\to \\mathcal{D}({\\tt Sh}_{\\tt Ab}\\mathbf{X})$$ (see \\ref{hcddef} for the definition of the target).\nBy composition with the functor $R_1$ (see \\ref{system15}) we get a functor\n$$R:=R_1\\circ R_0\\colon C^+({\\tt Sh}^{flat}_{\\tt Ab}\\mathbf{X})\\to C({\\tt Sh}_{\\tt Ab}\\mathbf{X})^Z\\ .$$\nIt preserves quasi-isomorphisms and therefore descends to (again using Lemma \\ref{flat-inclu})\n$$R\\colon D^+({\\tt Sh}_{\\tt Ab}\\mathbf{X})\\to h(C({\\tt Sh}_{\\tt Ab}\\mathbf{X})^Z)\\ .$$\n\n\\subsection{}\\label{aabgfg}\n\\newcommand{{\\tt Stacks}}{{\\tt Stacks}}\nThe construction of the functor $R_0$ explicitly depends on the choice of an \natlas $A\\to G$.  These choices form a subcategory $\\mathcal{Z}\\subset {\\tt Stacks}\/G$.\nThe choice of $A\\to G$ enters the definition via the functor $C_A$.\nFor the moment let us indicate the dependence on $A$ in the notation and write\n$R_0^A$ for the functor $R_0$ defined with the choice $A$. \n\nObserve, that\n$A\\to m^*A$, $A\\to p^*A$ and $A\\to m^*A\\times_{T^2\\times G}p^*A$ are functors\n${\\tt Stacks}\/G\\to {\\tt Stacks}\/(T^2\\times G)$.\nThe construction \\ref{gghaaase} shows that for a given $F\\in D^+({\\tt Sh}_{\\tt Ab}\\mathbf{X})$\nthe association $A\\to R_0^A(F)$ extends to a functor \n$$R_0^{\\dots}(F)\\colon \\mathcal{Z}^{op}\\to \\mathcal{D}({\\tt Sh}_{\\tt Ab}\\mathbf{X})\\ .$$\nThe components $X\\cong C_A\\circ {\\mathcal Fl}\\circ f^*$ and $Y_i\\cong C_*\\circ {\\mathcal Fl}\\circ\\dots$ (where $*\\in \\{A,p^*A,m^*A,m^*A\\times_{T^2\\times G}p^*A\\}$)  all involve a flabby resolution functor in front of $C_*$. If $A\\to A^\\prime$ is a morphism in $\\mathcal{Z}$, then the transformation\n$C_{A^\\prime}\\circ{\\mathcal Fl}\\to C_A\\circ {\\mathcal Fl}$ (or the similar transformations for\nthe other subscripts) produce quasi-isomorphisms by \\ref{gghaaase}.\n\n\n\nIt follows that the functor $R_0^{\\dots}(F)\\colon \\mathcal{Z}^{op}\\to \\mathcal{D}({\\tt Sh}_{\\tt Ab}\\mathbf{X})$\nmaps all morphisms to quasi-isomorphisms. We now consider the composition\n$R^{\\dots}(F):=R_1\\circ R_0^{\\dots}(F)\\colon \\mathcal{Z}^{op}\\to h(C({\\tt Sh}_{\\tt Ab}\\mathbf{X})^Z)$.\n\nFor two objects $A,B\\in \\mathcal{Z}$ we consider the diagram\n$$\\xymatrix{&A\\times B\\ar[dl]^{s}\\ar[dr]^t&\\\\A&&B}\\ ,$$\nwhere the fiber product is taken in ${\\tt Stacks}\/G$.\nWe consider the isomorphism\n$$R(A,B):=R^t\\circ (R^s)^{-1}\\colon R^A(F)\\to R^B(F)$$ in $h(C({\\tt Sh}_{{\\tt Ab}}(X))^Z)$.\nUsing the commutativity of the squares in the diagram\n$$\\xymatrix{&&A\\times B\\times C\\ar[dl]\\ar[dr]\\ar[d]\\\\&A\\times B\\ar[dl]\\ar[dr]&A\\times C\\ar[dll]\\ar[drr]&B\\times C\\ar[dr]\\ar[dl]\\\\A&&B&&C}$$\nwe check that\n$$R(A,B)\\circ R(B,C)=R(A,C)\\ .$$\nThis has the following consequence:.\n\\begin{lem}\\label{lem:indofchoice}\nThe functor $R\\colon D^+({\\tt Sh}_{\\tt Ab}\\mathbf{X})\\to hC(({\\tt Sh}_{\\tt Ab}\\mathbf{X})^Z)$ is independent of the choice of the atlas $A\\to G$ up to canonical isomorphism. \n\\end{lem}\n\n\nConsider an  automorphism $\\phi\\colon A\\to A$ in $\\mathcal{Z}$ and observe that it induces the identity on the level of cohomology, i.e. $H^*(R^\\phi)={\\tt id}$.\nIt is an interesting question whether $R^\\phi$ is the identity. \n\n\n\n\n\n\n\n\n  \n\n\n\n\n\n\n\\subsection{}\n\n\\begin{ddd}\\label{system18}\nWe define the periodization functor \n$$P_G:=\\lim\\circ R\\colon D^+({\\tt Sh}_{\\tt Ab}\\mathbf{X})\\to h(C(({\\tt Sh}_{\\tt Ab}\\mathbf{X})^Z))\\to\nhC({\\tt Sh}_{\\tt Ab}\\mathbf{X})\\ .$$  \n\\end{ddd}\nBy Lemma \\ref{lem:indofchoice} it is well\ndefined up to canonical isomorphism.\n\n\\subsection{}\\label{system19}\n\nLet $F\\in D^+({\\tt Sh}_{\\tt Ab}\\mathbf{X})$. By \\ref{reww2} $X(F)=C_A\\circ f^*\\circ {\\mathcal Fl}(F)$ represents $Rf_*\\circ f^*(F)$.\nThe composition \n$D[-2]\\colon X\\to X[-2]$ of the maps (or their inverses, respectively) in\nthe diagram $R_0^A(F)\\in \\mathcal{D}({\\tt Sh}_{\\tt Ab}\\mathbf{X})$ represents the map $D_G\\colon Rf_*\\circ f^*(F)\\to Rf_*\\circ f^*(F)[-2]$ defined in \nDefinition \\ref{defofd}. By Lemma \\ref{system17} we see that $P_G(F)$ (according to \\ref{system18})\nis isomorphic to our former Definition \\ref{system188} of the isomorphism class $P_G(F)$.\n\n\n\n\n\\section{Properties of the periodization functor}\n\n\\subsection{}\n\nThe domain and the target of $P_G$ are triangulated categories. Distinguished\ntriangles in both categories are all triangles which are isomorphic to\nmapping cone sequences\n$$\\dots\\to C(f)[-1]\\to X\\stackrel{f}{\\to} Y\\to C(f)\\to\\dots\\ .$$\n\\begin{lem}\nThe functor $P_G\\colon D^+({\\tt Sh}_{\\tt Ab}\\mathbf{X})\\to hC({\\tt Sh}_{\\tt Ab}\\mathbf{X})$ is triangulated.\n\\end{lem}\n\\begin{proof}\nWe must show that it is additive, preserves the shift,  and maps distinguished triangles to distinguished triangles.\nIt follows from the explicit constructions that the functors $\\lim$ and $R_1$ are additive and preserve the shift. The functorial flabby resolution ${\\mathcal Fl}$ on sheaves is additive. On complexes of sheaves it is defined as the level-wise application of the flabby resolution functor composed with the total complex construction. Therefore it also commutes with the shift. All other functors  involved in the construction of $R_0$ (e.g. $C_A$, $q^*$, $T_{K^\\cdot}$) are additive and commute with the shift, too.\n\n\nSince the distinguished triangles in \n$D^+({\\tt Sh}_{\\tt Ab}\\mathbf{X})$, $h(C({\\tt Sh}_{\\tt Ab}\\mathbf{X})^Z)$, and $hC({\\tt Sh}_{\\tt Ab}\\mathbf{X})$ are defined as\ntriangles which are isomorphic to mapping cone sequences, and the latter only\ndepend on the additive structure and the shift, we see that\n$\\lim$ and $R$ preserve triangles.\n\\end{proof}\n\n\n\n\\subsection{}\n\n\n\\begin{lem}\\label{zweipo}\nFor  $F\\in D^+({\\tt Sh}_{\\tt Ab}\\mathbf{X})$ the object $P_G(F)\\in hC({\\tt Sh}_{\\tt Ab}\\mathbf{X})$ is\ntwo-periodic.\n\\end{lem}\n\\begin{proof}\nThe isomorphism $P_G(F)[2]\\to P_G(F)$ is given by the isomorphism $W$ in \n  \\ref{system3}.\n\\end{proof}\nThe two periodicity will be analyzed in more detail in Subsection \\ref{sec:periodicity}.\n\n\n\\subsection{}\\label{sec:naturality}\n\nLet $u\\colon Y\\to X$ be a map of topological stacks which admits local sections.\nThen we consider a Cartesian diagram\n\\begin{equation}\\label{system211}\n  \\begin{CD} H @>{v}>> G\\\\\n    @VVgV @VV{f}V\\\\\n    Y @>{u}>> X .\n\\end{CD}\n\\end{equation}\n\n\\begin{lem}\\label{system22}\nThe diagram (\\ref{system211}) induces an isomorphism \n$ u^*\\circ P_G\\xrightarrow{\\sim}  P_H\\circ u^*$.\n\\end{lem}\n\\begin{proof}{\nBy taking the pull-back of  \n(\\ref{system40}) along $u$ we get  the extension}\nof  the Cartesian diagram above to \n$$\n\\xymatrix{T^2\\times H\\ar@{=>}[d]^{n,q}\\ar[r]^{w}&T^2\\times\n  G\\ar@{=>}[d]^{m,p}\\\\H\\ar[r]^v\\ar[d]^g&G\\ar[d]^f\\\\Y\\ar[r]^u&X}\\ .$$\n{Note that there is no $2$-isomorphism between $n$ and $q$ or $m$ and $p$, respectively.}\nSince $u$ has local sections the functor $u^*\\colon{\\tt Sh}_{\\tt Ab}\\mathbf{X}\\to {\\tt Sh}_{\\tt Ab}\\mathbf{Y}$  is exact by Lemma\n\\ref{prexact}. It therefore  extends to functors \n $u^*\\colon \\mathcal{D}({\\tt Sh}_{\\tt Ab}\\mathbf{X})\\to \\mathcal{D}({\\tt Sh}_{\\tt Ab}\\mathbf{Y})$ and \n$u^*\\colon  C({\\tt Sh}_{\\tt Ab} \\mathbf{X})^Z\\to C({\\tt Sh}_{\\tt Ab} \\mathbf{Y})^Z$ which both preserve quasi-isomorphisms.\nWe therefore also have corresponding functors on the derived categories which will all be denoted by $u^*$. \nIn the following we are going to show that there are natural isomorphisms\n\\begin{enumerate}\n\\item $u^*\\circ R_1\\cong R_1\\circ u^*$\n\\item $u^*\\circ \\lim \\cong \\lim \\circ u^*$\n\\item $u^*\\circ R_0\\cong R_0\\circ u^*$\n\\end{enumerate}\nof functors on the level of homotopy categories.\n \nIn fact it  follows from an inspection of the construction of $R_1$ that already\n$u^*\\circ R_1\\cong R_1\\circ u^*$ on the level of functors\n$\\mathcal{D}({\\tt Sh}_{\\tt Ab}\\mathbf{X})\\to C({\\tt Sh}_{\\tt Ab} \\mathbf{Y})^Z$, i.e. before descending to the homotopy category.\nAssertion (1) follows.\n\nSince $u^*\\colon C({\\tt Sh}_{\\tt Ab}\\mathbf{X})^Z\\to C({\\tt Sh}_{\\tt Ab}\\mathbf{Y})^Z$ preserves products and mapping cones \nwe again have  $u^*\\circ \\lim\\cong \\lim\\circ u^*$ before going to the homotopy categories.\nThis implies (2).\n\nIn order to see (3), using $v$ we construct a canonical isomorphism \n$$u^*\\circ R^A_0\\cong R^{C}_0\\circ u^*\\colon C^+({\\tt Sh}^{flat}_{\\tt Ab}\\mathbf{X})\\to \\mathcal{D}({\\tt Sh}_{\\tt Ab}\\mathbf{Y})\\ ,$$\nwhere we indicate the dependence of the functor $R_0$ on the choices by a superscript as in \\ref{aabgfg}. The atlas $C\\to H$ is given by the \ndiagram\n$$\\xymatrix{C\\ar[d]\\ar[r]&A\\ar[d]\\\\H\\ar[r]^v\\ar[d]^g&G\\ar[d]^f\\\\Y\\ar[r]^u&X}\\ ,$$\nwhere the upper square is also Cartesian.\n \n\nThe isomorphism (3) is induced by a collection of isomorphisms indexed by the objects of the diagram $U$ (\\ref{hcddef})\nwhich induce a morphism of diagrams in $h\\mathcal{D}({\\tt Sh}_{\\tt Ab}\\mathbf{Y})$.\n\n\n\n\n\n\n\n \n\nFirst we have\n\\begin{equation}\\label{eq:uXcommute}\n  \\begin{split}\n    u^*\\circ X&=u^*\\circ C_A\\circ f^*\\circ {\\mathcal Fl}\\\\\n    &\\cong C_C\\circ v^*\\circ f^*\\circ {\\mathcal Fl}\\\\\n    &\\cong C_C\\circ g^*\\circ u^*\\circ {\\mathcal Fl}\\\\\n    &\\cong C_C\\circ g^*\\circ {\\mathcal Fl}\\circ u^*\\\\\n    &=  X\\circ u^* \\,\n\\end{split}\n\\end{equation}\nwhere we use Lemma \\ref{pulcomghjdf}, $v^*\\circ f^*\\cong g^*\\circ u^*$ (see Lemma \\ref{uefhewiufuwefzzz}) and the\nfact that the flabby resolution functor commutes with the pull-back by $u$,\nsince $u$ has local sections (Lemma \\ref{ewkuwejahh}). \n\nLet $D:=n^*C\\times_{T^2\\times H} q^*C$. We write\n$K^\\cdot_{T^2\\times G}$ for the complex formerly denoted by $K^\\cdot$.\n\nNext we observe that there is a canonical isomorphism\n$w^*K_{T^2\\times G}^\\cdot\\cong K^\\cdot_{T^2\\times H}$. In fact \n$K_{T^2\\times G}^\\cdot$ and  $K^\\cdot_{T^2\\times H}$\nare given by truncations of the complexes \n${\\mathcal Fl}(\\underline{\\Z}_{{\\tt Site}(T^2\\times G)})$ and ${\\mathcal Fl}(\\underline{\\Z}_{{\\tt Site}(T^2\\times H)})$. The\nisomorphism is induced by the fact that $w^*$ commutes with the flabby\nresolution functor, and the isomorphism \n\\begin{equation*}\nw^*\\underline{\\Z}_{{\\tt Site}(T^2\\times G)}\\cong \\underline{\\Z}_{{\\tt Site}(T^2\\times H)}.\n\\end{equation*}\nThis implies by Lemma \\ref{tens-pres} that $w^*\\circ  T_{K_{T^2\\times G}^\\cdot}\\cong  T_{K_{T^2\\times H}^\\cdot}\\circ w^*$. In order to increase readability of the formulas we will omit the double subscript from now on and write $T_{K^\\cdot}$ for both functors.\nUsing this observation, Lemma \\ref{pulcomghjdf},  and the other previously used isomorphisms, we get  \n\\begin{eqnarray*}u^*\\circ Y_0&\\cong&\nu^*\\circ C_{B}\\circ {\\mathcal Fl}\\circ T_{K^\\cdot} \\circ p^* \\circ f^*\\circ {\\mathcal Fl}\\\\\n&\\cong&C_D\\circ w^*\\circ {\\mathcal Fl}\\circ T_{K^\\cdot} \\circ p^* \\circ f^*\\circ {\\mathcal Fl}\\\\\n&\\cong&C_D\\circ {\\mathcal Fl}\\circ w^*\\circ  T_{K^\\cdot} \\circ p^* \\circ f^*\\circ {\\mathcal Fl}\\\\\n&\\cong&C_D\\circ {\\mathcal Fl}\\circ T_{K^\\cdot} \\circ w^* \\circ p^* \\circ f^*\\circ {\\mathcal Fl}\\\\\n&\\cong&C_D\\circ {\\mathcal Fl}\\circ T_{K^\\cdot} \\circ q^* \\circ v^*\\circ f^*\\circ {\\mathcal Fl}\\\\\n&\\cong&C_D\\circ {\\mathcal Fl}\\circ T_{K^\\cdot} \\circ q^* \\circ g^*\\circ u^*\\circ {\\mathcal Fl}\\\\\n&\\cong&C_D\\circ {\\mathcal Fl}\\circ T_{K^\\cdot} \\circ q^* \\circ g^*\\circ {\\mathcal Fl}\\circ u^*\\\\\n&\\cong&Y_0\\circ u^*\n\\end{eqnarray*}\nIn a similar manner we get\n\\begin{eqnarray*}\nu^*\\circ Y_1&\\cong&u^*\\circ C_{p^*A}\\circ {\\mathcal Fl}\\circ T_{K^\\cdot}\\circ p^*\\circ f^*\\\\\n&\\cong&C_{q^*C}\\circ w^*\\circ {\\mathcal Fl}\\circ T_{K^\\cdot}\\circ p^*\\circ f^*\\\\\n&\\vdots&\\\\\n&\\cong&Y_1\\circ u^*\\\\\nu^*\\circ Y_2&\\cong&Y_2\\circ u^*\\\\\nu^*\\circ Y_3&\\cong&Y_3\\circ u^*\n\\end{eqnarray*}\nFor these isomorphisms, we use in particular Lemma \\ref{lem:pullpush} to get\n$v^*p_*\\cong q_*w^*$, and moreover Lemma \\ref{tens-pres} to get the chain of isomorphisms\n\\begin{equation*}\n  v^*(F\\otimes p_*K)\\cong v^*F\\otimes v^*p_*K \\cong v^*F\\otimes q_*w^*K\\cong\n  v^*F\\otimes q_*K \\cong T_{q_*K}(v^*F),\n\\end{equation*}\nwhich gives the isomorphism $v^*\\circ T_{p_*K}\\cong T_{q_*K}\\circ v^*$.\n\nBy a tedious check of the commutativity  of many little squares \nwe see that\nthese maps indeed define an isomorphism of functors\n$u^*\\circ R^A_0\\cong R^C_0\\circ v^*$.\nAs an example of these checks, let us indicate some details of the argument\nfor the map\n$Y_3\\to X[-2]$. For $F\\in D^+({\\tt Sh}_{\\tt Ab}\\mathbf{X})$ we have\nthe maps \n$\\phi:Y_3(F)\\to X[-2](F)$\nand $\\psi:Y_3(u^*F)\\to X[-2](u^*F)$ given by (\\ref{fret-wq}). We must show that\n$$\\xymatrix{u^*Y_3(F)\\ar[r]^\\cong\\ar[d]^{u^*\\phi}&Y_3(u^*F)\\ar[d]^{\\psi}\\\\u^*X[-2](F)\\ar[r]^\\cong&X[-2](u^*F)}$$\ncommutes.\n This indeed follows from the sequence of commutative diagrams\n\\begin{equation}\\label{eq:biggcom}\n  \\begin{CD}\n    u^*Y_3 @= u^*C_A{\\mathcal Fl} T_{p_*K}f^* @>{T_{p_*K}\\xrightarrow{[2]}{\\tt id}}>> u^*C_A{\\mathcal Fl} f^*[-2]@=      u^*X[-2]\\\\\n&& @VV{\\cong}V  @VV{\\cong}V \\\\\n&& C_Bv^*{\\mathcal Fl} T_{p_*K}f^*       @>{T_{p_*K}\\xrightarrow{[2]}{\\tt id}}>>  C_B v^* {\\mathcal Fl} f^*[-2]\\\\ \n&& @VV{\\cong}V @VV{\\cong}V \\\\\n &&   C_B {\\mathcal Fl} v^* T_{p_*K}f^*     @>{T_{p_*K}\\xrightarrow{[2]}{\\tt id}}>> C_B {\\mathcal Fl} v^*f^*[-2]\\\\\n&& @VV{\\cong}V @VV{\\cong}V \\\\\nY_3u^* @= C_B {\\mathcal Fl} T_{q_*K}g^*u^* @>{T_{q_*K}\\xrightarrow{[2]}{\\tt id}}>> C_B {\\mathcal Fl}\ng^*u^*[-2] @=X [-2]u^* \n  \\end{CD}\n\\end{equation}\nwhere for the last we use that $w$ preserves the orientation of the fiber $T^2$.\n\\end{proof}\nThe following statement directly follows from the constructions.\n\\begin{lem}\\label{citebaleczhbwd}\nThe isomorphism of Lemma \\ref{system22}\nbehaves functorially under compositions of diagrams of the form (\\ref{system211}).\\end{lem}\n\n\n\\subsection{}\\label{sec:nat_of_eval}\n\nLet $F\\in D^+({\\tt Sh}_{\\tt Ab}\\mathbf{X})$. Recall that\n$P_G(F)$ is the homotopy limit of\na $Z$-diagram consisting of sheaves $Y_0[2i]$, $Y_1[2i]$, $Y_2[2i]$, $Y_3[2i]$.\n  For all $i\\ge 0$\nwe construct an evaluation transformation\n$$e_i\\colon P_G(F)\\to Rf_*\\circ f^*(F)[2i]$$  \nas the composition of the \ncanonical map from the limit to $Y_3[2i+2]$  with the structure map\nto $X[2i]$ and  the identification $X[2i](F)\\cong  Rf_*\\circ f^*[2i](F)$.\n To be precise we consider $Rf_*f^*(F)\\in D({\\tt Sh}_{\\tt Ab}\\mathbf{X})$ via the inclusion\n$D^+({\\tt Sh}_{\\tt Ab}\\mathbf{X})\\to D({\\tt Sh}_{\\tt Ab}\\mathbf{X})$.\nIn the situation of \\ref{sec:naturality} an inspection of the\nproof of Lemma \\ref{system22} together with Corollary \\ref{gdashdgasd} shows that we have a\ncommutative diagram in $D({\\tt Sh}_{\\tt Ab}\\mathbf{X})$\n\\begin{equation}\\label{eq:ei_natural}\n  \\begin{CD}\n    u^* P_G(F) @>{\\cong}>{v^*}> P_H(u^*F)\\\\\n    @VV{u^*e_i}V @VV{e_i}V\\\\\n    u^* Rf_*f^*(F)[2i] @>{\\cong}>{v^*}> Rg_*g^*(u^*F)[2i] \\quad .\n  \\end{CD}\n\\end{equation}\n\nNote, however, that the morphism in the bottom line is only defined on\n$D^+({\\tt Sh}_{{\\tt Ab}}\\mathbf{X})$ (or equivalently on its image in $D({\\tt Sh}_{{\\tt Ab}}\\mathbf{X})$), and\nwe do not know whether we can extend it to the full unbounded derived\ncategory. Fortunately, we do not have to do this for the purposes of the\npresent paper.\n\n\n  \n\n\\subsection{}\n\nConsider the special case of the diagram (\\ref{system211})\nwhere $Y=X$, $u={\\tt id}_X$, $H=G$, and $v$ is an automorphism of the gerbe $G$.\nLemma \\ref{system22} provides an  automorphism\n$v^*\\colon P_G\\to P_G$ of periodization functors.\n\n\\subsection{}\n\nLet us illustrate this automorphism by an example.\nWe consider the trivial $U(1)$-gerbe $G \\to S^2$ over $S^2$ and let\n$\\phi\\in {\\tt Aut}(G\/S^2)$ be classified by $1\\in H^2(S^2;\\mathbb{Z})\\cong\n\\mathbb{Z}$. \nIt induces an automorphism of the cohomology\n$H^*(S^2;P_G(\\underline{F}_{S^2}))$, where $\\underline{F}_{S^2}$ is the sheaf represented by a discrete abelian group $F$.\nWe have a Cartesian diagram\n$$\\xymatrix{G\\ar[r]\\ar[d]^g&\\mathcal{B} U(1)\\ar[d]\\\\S^2\\ar[r]^f&{*}}\\ .$$\nSince $f^*\\underline{F}_*\\cong \\underline{F}_{S^2}$ we have\n\\begin{eqnarray*}\nH^*(S^2;P_G(\\underline{F}_{S^2}))&\\cong& H^*(S^2;P_G(f^*\\underline{F}_{*}))\\\\\n&\\stackrel{Lemma \\:\\ref{system22}}{\\cong}&H^*(S^2;f^*P_{\\mathcal{B} U(1)}(\\underline{F}_{*}))\\\\\n&\\stackrel{Lemma \\:\\ref{projefoa}}{\\cong}&H^*(S^2;\\underline{\\Z})\\otimes H^*(*;P_{\\mathcal{B} U(1)}(\\underline{F}_{*}))\\\\\n&\\cong&\\mathbb{Z}[w]\/(w^2)\\otimes H^*(*;P_{\\mathcal{B} U(1)}(\\underline{F}_{*}))\\ ,\n\\end{eqnarray*}\nwhere $H^*(*;P_{\\mathcal{B} U(1)}(\\underline{F}_{*}))$ has been calculated in examples in Proposition \\ref{system30}.\nIf $F=\\mathbb{Q}$ or $\\mathbb{Q}\/\\mathbb{Z}$, then $H^{ev}(*;P_{\\mathcal{B} U(1)}(\\underline{F}_{*}))\\cong \\mathbb{Q}$ or $\\dots \\cong \\mathbb{A}^\\mathbb{Q}_f$, respectively. If $F=\\mathbb{Z}$, then \n$H^{odd}(*;P_{\\mathcal{B} U(1)}(\\underline{\\Z}_{*}))\\cong \\mathbb{A}^\\mathbb{Q}_f\/\\mathbb{Q}$.\n\\begin{lem}\nIn all these cases the action of $\\phi^*$ is given by \n$$\\phi^*(1\\otimes \\lambda+ w\\otimes \\mu)=1\\otimes \\lambda+w\\otimes\n(\\lambda+\\mu)\\ ,$$ \nwhere $\\lambda,\\mu\\in \\mathbb{Q}$, $\\mathbb{A}^\\mathbb{Q}_f$, or $\\mathbb{A}^\\mathbb{Q}_f\/\\mathbb{Q}$, respectively.\n\\end{lem}\n\\begin{proof}\nWe will use the  description of\n$H^*(S^2,P_G(\\underline{F}_{S^2}))$ given in  Corollary \\ref{lim1seq}.\nIn Lemma \\ref{system5} have already calculated the automorphism on\n$H^*(S^2, Rg_*g^*\\underline{F}_{S^2})\\cong  F[w][[z]]\/(w^2)$ induced by the diagram\n$$\\xymatrix{G\\ar[rr]^\\phi\\ar[dr]_g&&G\\ar[dl]^g\\\\&S^2}\\ .$$  \nIt is given by $z\\mapsto z+w$, $w\\mapsto w$.\nThe operation induced by $D_G$ is $\\frac{d}{dz}$, and\n  the periodized cohomology  is given as the\nkernel (in the cases $F=\\mathbb{Q}$ and $F=\\mathbb{Q}\/\\mathbb{Z}$) or cokernel (in the case $F=\\mathbb{Z}$) of $\\prod_{i\\ge 0} {\\tt id}[2i] -\\prod_{i\\ge 0} D_G[2i]$ on $\\prod_{i\\ge 0} F[w][[z]]\/(w^2)[2i]$.\nRecall from \\ref{sec:calculate_period_cohom}\nthat the \nclass \n$a\\in H^0(S^2,P_G(\\underline{\\mathbb{Q}}_{S^2}))\\cong \\mathbb{Q}[w]\/(w^2)$ is\nrepresented by $(a,az,az^2\/2,\\dots, az^k\/k!\\dots )$, which is mapped by $\\phi^*$ to\n$(a,a(w+z),a(w+z)^2\/2,\\dots)$.\nWe must  read off a representative of this class in the form above.\nIf $a=w$ then $w(w+z)^k\/k!=wz^k\/k!$ and therefore\n$\\phi^*w=w$. On the other hand, if $a=1$, then \n$a(w+z)^k\/k!=z^k\/k!+w z^{k-1}\/(k-1)!$, so that $\\phi^*(1)=1+w$.\n\nExactly the same argument applies if $F=\\mathbb{Q}\/\\mathbb{Z}$. Finally, the cohomology with\ncoefficients $F=\\mathbb{Z}$ is the cokernel (up to shift of degree) of the map induced\nby the inclusion $\\mathbb{Q}\\hookrightarrow \\mathbb{A}_f^\\mathbb{Q}$, which implies the assertion also for\n$F=\\mathbb{Z}$. \n\\end{proof}\n\n\\section{Periodicity}\\label{sec:periodicity}\n\n\\subsection{}\n\nWe consider a topological $U(1)$-gerbe $f\\colon G\\to X$ over a locally compact stack.\nLet $F\\in D^+({\\tt Sh}_{\\tt Ab}\\mathbf{X})$.\nIn Lemma \\ref{zweipo} we have argued that $P_G(F)\\in D({\\tt Sh}_{\\tt Ab}\\mathbf{X})$ is two-periodic.\nThe periodicity is implemented by a certain isomorphism\n$W:P_G(F)[2]\\to P_G(F)$ which may depend on additional choices, see also the discussion in \\ref{choicesiasg}. In the present subsection we show that there is a canonical two-periodicity isomorphism.\n\n\n\\subsection{}\\label{ewjdgednaa}\n\nThe gerbe $G\\to X$ gives rise {in a $2$-functorial way} to\nthe diagram  (see \\ref{ddef2} for details)\n\\begin{equation}\\label{system53}\n\\xymatrix{\\tilde G\\ar[dr]^s\\ar[d]^r\\ar[rr]^\\phi&&\\tilde G\\ar[dl]^s\\ar[d]^r\\\\G\\ar[dr]_f&X\\times T^2\\ar[d]^p&G\\ar[dl]^f\\\\&X&\n}\\ .\n\\end{equation}\nThis diagram induces the desired periodization isomorphism\nas the following composition of natural transformations\n\\begin{multline}\\label{prechgh}\nW\\colon P_G(F)\\stackrel{\\text{unit}}{\\rightarrow} Rp_*p^*P_G(F)\\stackrel{\\text{Lemma\n    \\ref{system22}}}{\\rightarrow}  Rp_*P_{\\tilde\n    G}(p^*F)\\\\\n\\xrightarrow{\\phi^*}Rp_*P_{\\tilde G}(p^*F) \\cong \nRp_*p^*P_{G}(F)\\xrightarrow{\\int_p} P_G(F)[-2]\\ .\\end{multline}\n\n\\begin{prop}\\label{dhewud82d}\nThe transformation (\\ref{prechgh})\n$$W\\colon P_G(F)\\to P_G(F)[-2]$$ \nis a canonical choice for the isomorphism in  \nProposition \\ref{system3}.\n\\end{prop}\n\n\n\\subsection{}\nTo start the proof of Proposition \\ref{dhewud82d},\nrecall the definition\n$$D_G\\colon Rf_*f^*(F)\\to Rf_*f^*(F)[-2]$$ as the composition\n$$Rf_*f^*(F)\\xrightarrow{\\text{unit}} Rf_*Rr_*R\\phi_*\\phi^* r^*f^*(F)\\stackrel{!}{\\cong} Rf_*Rr_*r^*f^*(F)\\xrightarrow{\\int_r} Rf_*f^*(F)[-2]\\ ,$$\nwhere at the marked isomorphism $\"!\"$ we use the natural isomorphisms\n\\ref{keykey} and \\ref{uefhewiufuwefzzz} associated to the identity  $f\\circ r= f\\circ r\\circ\\phi$\n\n\nRecall from  \\ref{sec:nat_of_eval}\nthe definition\nof the natural evaluation transformation $e_i\\colon P_G(F)\\to Rf_*f^*(F)[2i]$ for all $i\\ge 0$.\n\n\\begin{lem}\\label{uzasdda}\nThe following diagram commutes:\n$$\\xymatrix{P_G(F)\\ar[d]^{e_{i+1}}\\ar[r]^W&P_G(F)\\ar[d]^{e_i}\\\\\nRf_*f^*(F)[2i+2]\\ar[r]^{D_G}&Rf_*f^*(F)[2i]}\\ .$$\n\\end{lem}\n\\begin{proof}\nWe split this square in parts.  First we observe that in $D({\\tt Sh}_{\\tt Ab} \\mathbf{X})$\n$$\n\\begin{CD}\n  P_G(F)\n  @>{\\text{unit}}>> Rp_*p^*P_G(F) @>{Rp_*r^*}>\\cong> Rp_*P_{\\tilde\n  G}(p^*F)\\\\\n   @VV{e_{i+1}}V @VV{Rp_*p^*e_{i+1}}V @VV{Rp_*e_{i+1}}V \\\\\n    Rf_*f^*(F)[2i+2] @>{\\text{unit}}>> Rp_*p^*\n    Rf_*f^*(F)[2i+2] @>{Rp_*r^*}>\\cong> Rp_*\n    Rs_*s^*p^*(F)[2i+2]\\\\\n    @VV=V && @VV{\\cong}V \\\\\n    Rf_*f^*(F)[2i+2] @>{Rf_*f^*\\text{unit}}>> Rf_*f^*Rp_*p^*(F)  @>{\\cong}>>\n  Rf_*Rr_*r^*f^*(F)[2i+2] \n\\end{CD}\n$$\ncommutes (use Lemma   \\ref{lem:pullpush} for the upper left and the lower and \\ref{sec:nat_of_eval} for the upper right rectangle).\n\n\nIn the next step we observe that\n$${\\small\n\\begin{CD}\n Rp_*P_{\\tilde G}(p^*F) @>{{\\tt id}}>>  Rp_*P_{\\tilde G}(p^*F)\n      @>{Rp_*\\phi^*}>{\\cong}>   \n Rp_*P_{\\tilde\n      G}(p^*F)\\\\\n @VV{Rp_*e_{i+1}}V && @VV{Rp_*e_{i+1}}V \\\\\n Rp_*Rs_*s^*p^*(F)[2i+2] @>{\\text{unit}}>> Rp_*Rs_*R\\phi_*\\phi^*s^*p^*(F)[2i+2]\n      @>\\cong>> Rp_*Rs_*s^*p^*(F)[2i+2] \\\\\n @VV{\\cong}V @VV{\\cong}V @VV{\\cong}V\\\\\nRf_*Rr_*r^*f^*(F)[2i+2] @>{\\text{unit}}>> Rf_*Rr_*R\\phi_*\\phi^*r^*f^*(F)[2i+2]\n      @>\\cong>> Rf_*Rr_*r^*f^*(F)[2i+2] \n\\end{CD}\n}$$\ncommutes, where we use for the upper rectangle again \\ref{sec:nat_of_eval},  and $p\\circ s\\circ \\phi=p\\circ s$, $p\\circ s=f\\circ r$,  $f\\circ r\\circ \\phi=f\\circ r$ and\nLemma \\ref{lem:pullpush} for the remaining squares. \n\nIn the last step we observe the commutativity of\n\\begin{equation*}\n  \\begin{CD}\n    Rp_*P_{\\tilde G}(p^*F) @>{(r^*)^{-1}}>{\\cong}> Rp_*p^*P_{G}(F) @>{\\int_p}>>\n    P_G(F)[-2]\\stackrel{T^{-2}}{\\cong} P_G(F)\\\\ \n    @VV{Rp_*e_{i+1}}V @VV{Rp_*p^*e_{i+1}}V @VV{T^{-2}e_{i+1}\\cong e_i}V\\\\\n    Rp_*Rs_*s^*p^*(F)[2i+2] @>{(r^*)^{-1}}>{\\cong}> Rp_*p^*Rf_*f^*(F)[2i+2] @>{\\int_p}>>\n    Rf_*f^*(F)[2i]\\\\\n    @VV{\\cong}V && @VV{=}V\\\\\n    Rf_*Rr_*r^*f^*(F)[2i+2] & @>{Rf_*(\\int_r)}>> & Rf_*f^*(F)[2i].\n  \\end{CD}\n\\end{equation*}\nAgain, for the commutativity of the upper left rectangle we use\n\\eqref{eq:ei_natural} of \\ref{sec:nat_of_eval}. For the upper right corner we\nuse the fact that $\\int_p$ is a natural\ntransformation between the functors $Rp_*p^*$ and ${\\tt id}$ on $D({\\tt Sh}_{\\tt Ab}\\mathbf{X})$.  For the lower rectangle we use Lemma \\ref{system300}.\n\\end{proof}\n\n\\subsection{}\n\nWe now finish the proof of Proposition \\ref{dhewud82d}.\nWe have an exact triangle\n$$\\dots\\to P_G(F)\\stackrel{\\prod_{i\\ge 0}e_i}{\\to} \\prod_{i\\ge 0} Rf_*f^*(F)[2i]\\stackrel{\\alpha}{\\to }\\prod_{i\\ge 0} Rf_*f^*(F)[2i]\\stackrel{[1]}{\\to}\\dots$$\nwhere (using the language of elements) the map $\\alpha$ is given by\n$$\\alpha(x_i)_{i\\ge 0}=(x_i-D_Gx_{i+1})_{i\\ge 0} .$$ \nBy Lemma \\ref{uzasdda} we have a morphism of exact triangles\n$$\\xymatrix{ P_G(F)\\ar[d]^W\\ar[r]^{\\hspace{-0.9cm}\\prod_{i\\ge 0}e_i} & \\prod_{i\\ge 0} Rf_*f^*(F)[2i]\\ar[d]^\\beta\\ar[r]^\\alpha&\\prod_{i\\ge 0} Rf_*f^*(F)[2i]\\ar[d]^\\beta\\\\\nP_G(F)[-2]\\ar[r]^{\\hspace{-0.9cm}\\prod_{i\\ge 0}e_i}& \\prod_{i\\ge 0} Rf_*f^*(F)[2i-2]\\ar[r]^\\alpha&\\prod_{i\\ge 0} Rf_*f^*(F)[2i-2]\n}\\ ,$$\nwhere the map $\\beta$ is given by\n$\\beta(x_i)_{i\\ge 0}:=(D_Gx_i)_{i\\ge 0}$. \nIn Lemma \\ref{uwzgfwehjcfbsdac} we have shown that $W$ is an isomorphism.\n\\hspace*{\\fill}$\\Box$ \\\\[0cm]\\noindent\n\n \n\n\n\n\n\n\n\n\n\\chapter{$T$-duality}\\label{system4000}\n\n\\section{The universal $T$-duality diagram}\n\n\\subsection{}\n\nTopological $T$-duality intends to model the underlying topology of string theoretic $T$-duality on the level of targets and quantum field theory.  In the special case of targets modeled by a gerbe on top of a $T^n$-principal bundle over a space,  topological $T$-duality is by now a well-defined mathematical concept, see \\cite{math.AT\/0701428}, \\cite{math.GT\/0501487} and the literature cited therein. In the case of $T$-principal bundles it was extended to orbifolds in \\cite{MR2246781}.\nIn the present paper we propose a definition of $T$-duality in the case of $T$-bundles over arbitrary stacks.\nThis framework includes arbitrary $T$-actions on spaces. The  special case\nof an almost free action (i.e.~every orbit is either free or a fixed point)\nhas been treated with completely different methods in \\cite{pande-2006}.\n\n\\subsection{}\n\nThe notion of a $T$-duality diagram has first been introduced in \\cite{math.GT\/0501487}. \nIn the present paper we first produce a universal $T$-duality diagram over the stack $\\mathcal{B} U(1)=[*\/U(1)]$.\nThen we proceed to define a $T$-duality diagram over a general stack as one which is locally isomorphic to the universal one.\n\n\\subsection{}\\label{system1003}\n\nThe universal $T$-duality diagram is a diagram of stacks\n\\begin{equation}\\label{univtdza1}\n\\xymatrix{&p_{univ}^*G_{univ}\\ar[dl]\\ar[dr]\\ar[rr]^{u_{univ}}&&\\hat p_{univ}^* \\hat G_{univ}\\ar[dl]\\ar[dr]&\\\\G_{univ}\\ar[dr]^{f_{univ}}&&F_{univ}\\ar[dl]^{p_{univ}}\\ar[dr]^{\\hat p_{univ}}&&\\hat G_{univ}\\ar[dl]^{\\hat f_{univ}}\\\\&E_{univ}\\ar[dr]^{\\pi_{univ}}&&\\hat E_{univ}\\ar[dl]^{\\hat \\pi_{univ}}&\\\\&&B_{univ}&&}\\ .\n\\end{equation}\nIn the following we explain the stacks and the maps.\n\\begin{itemize}\n\\item $B_{univ}:=\\mathcal{B} U(1)$\n\\item $E_{univ}:=*$ and $\\pi_{univ}$ is the map which classifies the trivial $U(1)$-bundle over the point $*$.\n\\item $G_{univ}:=\\mathcal{B} U(1)$, and $f_{univ}$ is the unique map.\n\\item $\\hat E_{univ}:=\\mathcal{B} U(1)\\times U(1)$, and $\\hat \\pi_{univ}$ is the projection onto the first factor.\n\\item $\\hat f_{univ}\\colon \\hat G_{univ}\\to \\hat E_{univ}$ is a gerbe with band $U(1)$ classified by $z\\otimes v\\in H^2(\\mathcal{B} U(1);\\mathbb{Z})\\otimes H^1(U(1);\\mathbb{Z})\\cong H^3(\\mathcal{B} U(1)\\times U(1);\\mathbb{Z})$, where\n$z\\in H^2(\\mathcal{B} U(1);\\mathbb{Z})$ and $v\\in H^1(U(1);\\mathbb{Z})$ are the standard generators.\n\\item $F_{univ}:=E_{univ}\\times_{B_{univ}}\\hat E_{univ}\\cong U(1)$, and $p_{univ},\\hat p_{univ}$ are the canonical projections.\n\\item Since $H^2(F_{univ};\\mathbb{Z})\\cong 0\\cong H^3(F_{univ};\\mathbb{Z})$, the pull-back\n$\\hat p^*_{univ}\\hat G_{univ}$ can be identified with the trivial gerbe\n$p_{univ}^*G_{univ}\\cong U(1)\\times\\mathcal{B} U(1)$ by a unique isomorphism class of maps\nrepresented by $u_{univ}$. \n\\end{itemize}\nLet us fix once and for all a universal $T$-duality diagram (i.e. a choice of $u_{univ}$ in its isomorphism class {and $2$-isomorphisms filling the faces}).\n\n\n\\subsection{}\n\nLet $B$ be a topological stack and consider a diagram \n\\begin{equation}\\label{tzfrzrzr}\n\\xymatrix{&p^*G\\ar[dl]\\ar[dr]\\ar[rr]^u&&\\hat p^* \\hat G\\ar[dl]\\ar[dr]&\\\\G\\ar[dr]^f&&F\\ar[dl]^p\\ar[dr]^{\\hat p}&&\\hat G\\ar[dl]^{\\hat f}\\\\&E\\ar[dr]^\\pi&&\\hat E\\ar[dl]^{\\hat \\pi}&\\\\&&B&&}\n\\end{equation}\nof topological stacks where the squares are Cartesian, $f\\colon G\\to E$ and $\\hat f\\colon \\hat G\\to \\hat E$ are topological  $U(1)$-gerbes,  and $u$ is an isomorphism of gerbes over $F$.\n\nAn isomorphism between two such diagrams over $B$ is first of all a large commutative diagram in stacks, but we furthermore require that the horizontal morphisms are morphisms of $U(1)$-banded gerbes in all places \nwhere this condition makes sense.\n\n\n\\begin{ddd}\\label{eruihfrvc}\nThe diagram (\\ref{tzfrzrzr}) is called a $T$-duality diagram if for every object $(U\\to B)\\in \\mathbf{B}$ there exists a covering\n$(U_i\\to U)_{i\\in I}\\in {\\tt cov}_\\mathbf{B}(U)$ such that for all $i\\in I$ the pull-back of\nthe diagram (\\ref{tzfrzrzr}) along the map $U_i\\to U \\to B$ is isomorphic to the pull-back of the  universal $T$-duality diagram (\\ref{univtdza1}) along a map $U_i\\to B_{univ}$.\n\\end{ddd}\n\n\n\n\n\\subsection{}\n\nIn the following we describe the concept of $T$-duality. \nLet $B$ be a topological stack. A pair $(E,G)$ over $B$ consists of a $T$-principal bundle\n$\\pi\\colon E\\to B$ and a $U(1)$-gerbe $f\\colon G\\to E$. \n\\begin{ddd}\\label{def:admits_dual}\nWe say that a pair $(E,G)$ admits a $T$-dual, if it appears as a part of a $T$-duality diagram\n\\ref{tzfrzrzr}. In this case the pair $(\\hat E,\\hat G)$ is called a $T$-dual of $(E,G)$.\n\\end{ddd}\n\nThis is our proposal for the mathematical concept of $T$-duality for pairs of $T$-principal bundles and gerbes.\nUsing the $T^n$-bundle variant of the universal $T$-duality diagram one can easily generalize this definition to the higher-dimensional case. But note that, in contrast to the case of\none-dimensional fibers, a unique isomorphism $u_{univ}$ does not exist for\n$T^n$ if one uses the exact parallel setup. \nThis explains why suitable\nmodifications are necessary in \\cite{math.GT\/0501487}. In particular, the universal base space is not simply\nthe $n$-fold product of copies of  $B_{univ}$ used in the one-dimensional case.\n\n\\subsection{}\n\nIn the following we show that the concept of topological $T$-duality as defined above really coincides with the former definitions.\n\\begin{lem}\\label{lem:identify_T_duality}\n Definitions \\ref{eruihfrvc} and \\ref{def:admits_dual} reduce to the notion of $T$-duality as used in\n  \\cite{math.GT\/0501487}, \\cite{MR2130624}, if $B$ is a locally acyclic space.\n\\end{lem}\n\\begin{proof}\nBy Definition \\ref{eruihfrvc} a $T$-duality triple over a space $B$ is given by the following data:\n\\begin{enumerate}\n\\item  locally trivial $U(1)$-principal bundles $E,\\hat E$ over $B$,\n\\item $U(1)$-banded gerbes $G$, $\\hat G$ over $E$ or $\\hat E$,\n  respectively,\n\\item  an isomorphism $u$ between the pullbacks of\n  $G$ and $\\hat G$ to the correspondence space $E\\times_B \\hat E$.\n\\end{enumerate} \nEvery point $b\\in B$ admits an acyclic neighborhood $b\\in U\\subseteq B$.\nThe bundles $E$ and $\\hat E$ are trivial over $U$, i.e. we have $E_{|U}\\cong U\\times U(1)\\cong \\hat E_{|U}$.\nSince  $H^3(U\\times U(1);\\mathbb{Z})\\cong 0$, the restrictions of the gerbes $G_{|E_{|U}}$ and $\\hat G_{|\\hat E_{|U}}$ are trivial, too. The Definition  \\ref{eruihfrvc} requires that the isomorphism of trivial gerbes $u_{|E_{|U}\\times_U\\hat E_{|U}}$ is \n classified by the\n  generator of $H^2(E_{|U}\\times_U\\hat E_{|U};\\mathbb{Z})$ (note that $E_{|U}\\times_U\\hat E_{|U}\\cong U\\times U(1)\\times U(1)$).\nThis reformulation of the definition of a $T$-duality triple over a locally acyclic space $B$\nis exactly the definition of a $T$-duality triple in  \\cite{math.GT\/0501487}.\n\n\n   \n\nIn the approach of  \\cite{MR2130624}  to $T$-duality\nwe start with a pair $(E,G)$. We characterize $T$-dual pairs by topological conditions.\nWe then analyze the classifying space of pairs and observe that the universal pair has a \nunique $T$-dual pair which gives rise to the $T$-duality transformation.\n\nIt turns out that the classifying space of pairs in \\cite{MR2130624} is equivalent to the classifying space of $T$-duality triples in \\cite{math.GT\/0501487}, and that the universal pair and its dual\nare  parts of the  universal $T$-duality triple. This shows that the approaches of  \\cite{MR2130624}\nand \\cite{math.GT\/0501487} are equivalent.\n\\end{proof}\n\n\n\n\n\n\n\n \n\n\n\n\n\n\\section{$T$-duality and periodization diagrams}\n\n\\subsection{}\n\nRecall that the construction of the periodization functor\n$P_G$ was based on the diagrams introduced in \\ref{ddef2}. In the present subsection we\nrelate these diagrams to $T$-duality. \n\n\\subsection{}\n\nThe double of the universal $T$-duality diagram (\\ref{univtdza1}) is (by\ndefinition) the \nbig universal periodization diagram\n\\begin{equation}\\label{univtdza11}\n{\\small \\xymatrix{&{\\tt pr}_0^*p_{univ}^*G_{univ}\\ar[dl]^{\\tilde\n      {\\tt pr}_0}\\ar[dr]\\ar[r]^{{\\tt pr}_0^*u_{univ}}& {\\tt pr}_{\\hat E}^*\\hat\n    G_{univ}\\ar[r]^{{\\tt pr}_1^*u_{univ}^{-1}}\\ar[d]&{\\tt pr}_1^*p_{univ}^* G_{univ}\n    \\ar[dl] \\ar[dr]_{\\tilde\n      {\\tt pr}_1}&\\\\p_{univ}^*G_{univ}\\ar[dd]^{f_{univ}^*p_{univ}}\\ar[dr]^{p_{univ}^*{f_{univ}}}&&F_{univ}\\times_{\\hat\n      E_{univ}}F_{univ}\\ar[dl]^{{\\tt pr}_0}\\ar[dr]^{{\\tt pr}_1}&&p_{univ}^*\n    G_{univ}\\ar[dd]_{f_{univ}^*p_{univ}}\\ar[dl]_{p_{univ}^*{f_{univ}}}\\\\&F_{univ}\\ar[dr]^{p_{univ}}&&F_{univ}\\ar[dl]^{p_{univ}}&\\\\G_{univ}\\ar[rr]^{f_{univ}}&&E_{univ}&&G_{univ}\\ar[ll]^{f_{univ}}}\n}\n\\end{equation}\nNote that all  squares are Cartesian, with the exception \nof the central square\n$$\\xymatrix{&F_{univ}\\times_{\\hat E_{univ}}  F_{univ}\\ar[dl]\\ar[dr]&\\\\F_{univ}\\ar[dr]&&F_{univ}\\ar[dl]\\\\&E_{univ}&}$$\nwhich does not commute.\n The same remark applies to similar diagrams we introduce later.\n\n\n\\subsection{}\n\nWe form the diagram\\footnote{{This diagram does not commute. It is a short-hand for a square of the form (\\ref{system40}) with a $2$-isomorphism\nbetween $f_{univ}\\circ q_{univ}$ and $f_{univ}\\circ m_{univ}$. We will adopt a similar convention for other diagrams written in this short-hand form below.}}\n\n\n\n\\begin{equation}\\label{ajksdhbxaiudkj}\n\\xymatrix{{\\tt pr}_0^*p^*_{univ}G_{univ}\\ar@\/^1pc\/[r]^{q_{univ}}\\ar@\/_1pc\/[r]_{m_{univ}}&G_{univ}\\ar[r]^{f_{univ}}&E_{univ}}\\ ,\n\\end{equation}\nwhere $$m_{univ}:= f_{univ}^*p_{univ}\\circ \\tilde {\\tt pr}_1\\circ {\\tt pr}_1^* u_{univ}^{-1}\\circ {\\tt pr}_0^*u_{univ}\\ ,\\quad \nq_{univ}:= f_{univ}^*p_{univ}\\circ \\tilde {\\tt pr}_0\\ .$$\n\\begin{ddd}\nThe diagram (\\ref{ajksdhbxaiudkj}) is called the small universal periodization diagram.\n\\end{ddd}\n\n\n\\subsection{}\n\nLet $f\\colon G\\to X$ be a topological gerbe with band $U(1)$ over a stack $X$. Then we consider the pull-back of the small universal periodization diagram to $X$ via the projection $r\\colon X\\to E_{univ}\\cong *$. We form the tensor product with the gerbe $G$ (see \\cite[6.1.9]{math.AT\/0701428} for some details on such tensor products) and obtain the diagram \n\\begin{equation}\\label{jhsdjkhwud}\n\\xymatrix{\\tilde H\\ar@\/^1pc\/[r]^{q}\\ar@\/_1pc\/[r]_{m}&H \\ar[r]^f& X}\\ ,\n\\end{equation}\nwhere \n$$\\tilde H:= {\\tt pr}_X^*G\\otimes {\\tt pr}^*_{F_{univ}\\times_{\\hat E_{univ}}F_{univ}}{\\tt pr}^*_0 p_{univ}^*G_{univ}\\ , \\quad H:=G\\otimes r^*G_{univ}\\ ,$$\n$${\\tt pr}_X\\colon X\\times F_{univ}\\times_{\\hat E_{univ}}F_{univ}\\to X\\ ,$$ \n$${\\tt pr}_{F_{univ}\\times_{\\hat E_{univ}}F_{univ}}\\colon X\\times F_{univ}\\times_{\n\\hat E_{univ}}F_{univ}\\to F_{univ}\\times_{\\hat E_{univ}} F_{univ}$$ are the\nprojections, and $m$, $q$ are induced by the corresponding universal maps\n$m_{univ}$ or $q_{univ}$, respectively.\n\\begin{ddd}\nThe diagram (\\ref{jhsdjkhwud}) is called the small periodization diagram of $G\\to X$.\n\\end{ddd}\nIn fact we have defined a {$2$-functor}  from ${\\tt gerbes}\/X$ to a $2$-category of such small\nperiodization diagrams.  \nUsing the fact that $G_{univ} = \\mathcal{B} U(1)$ we have a canonical identification\n$H\\cong G$. Furthermore, $F_{univ}\\times_{\n\\hat E_{univ}}F_{univ}\\cong T^2$, and we can identify\n$\\tilde H \\to X\\times F_{univ}\\times_{\\hat E_{univ}}F_{univ}$ with $G\\times T^2 \\to X\\times T^2$.\n\\begin{lem}\nWith these identifications  the small periodization diagram\n(\\ref{jhsdjkhwud}) is isomorphic to  the diagram (\\ref{system40})\nused in the definition of $P_G$.\n\\end{lem}\n\\begin{proof}\n  This follows directly from the definitions of these maps.\n\\end{proof}\n\n\n\\subsection{}\\label{system52}\n\nThe $T$-duality diagram (\\ref{tzfrzrzr}) gives rise to the big double $T$-duality diagram\n \\begin{equation}\\label{univtdzfe1}\n\\xymatrix{&{\\tt pr}_0^*p^*G\\ar[dl]^{\\tilde {\\tt pr}_0}\\ar[dr]\\ar[r]^{{\\tt pr}_0^*u}& {\\tt pr}_{\\hat E}^*\\hat G\\ar[r]^{{\\tt pr}_1^*u^{-1}}\\ar[d]&{\\tt pr}_1^*p^* G \\ar[dl] \\ar[dr]_{\\tilde {\\tt pr}_1}&\\\\p^*G\\ar[dd]^{f^*p}\\ar[dr]^{p^*f}&&F\\times_{\\hat E}F\\ar[dl]^{{\\tt pr}_0}\\ar[dr]^{{\\tt pr}_1}&&p^* G\\ar[dd]^{f^*p}\\ar[dl]_{p^*f}\\\\&F\\ar[dr]^p&&F\\ar[dl]^{p}&\\\\G\\ar[rr]^f&&E&&G\\ar[ll]^f}\n\\end{equation}\nNote that the middle square does not commute.\nWe have \n$$F\\times_{\\hat E}F\\cong (E\\times_B\\hat E)\\times_{\\hat E}(\\hat E\\times_B E)\\cong \nE\\times_B\\hat E\\times_B E\\stackrel{\\sim}{\\leftarrow} E\\times_B\\hat E\\times U(1)\\ ,$$\nwhere the last arrow is given by $(e,\\hat e,eu)\\leftarrow (e,\\hat e,u)$.\nUnder this identification ${\\tt pr}_0(e,\\hat e,u)=(e,\\hat e)$ and ${\\tt pr}_1(e,\\hat e,u)=(eu,\\hat e)$.\nWe can correct this non-commutativity as follows.\nLet $c:F\\times_{\\hat E} F\\to F\\times_{\\hat E} F$ be the isomorphism, which under the above\nidentification is given by\n$c(e,\\hat e,u):= (eu^{-1},\\hat e,u)$.\nNote that ${\\tt pr}_1\\circ c={\\tt pr}_0$.\nFurthermore note that ${\\tt pr}_{\\hat E}={\\tt pr}_{\\hat E}\\circ c:F\\times_{\\hat E} F\\to \\hat E$.\nTherefore we get a canonical morphism $\\hat c$ satisfying\n$\\overline{{\\tt pr}_{\\hat E}}=\\overline{{\\tt pr}_{\\hat E}}\\circ \\hat c$\nin the diagram\n$$\\xymatrix{{\\tt pr}^*_{\\hat E}\\hat G\\ar[d]\\ar[r]^{\\hat c}&{\\tt pr}^*_{\\hat E}\\hat G\\ar[d]\\ar[r]^{\\overline{{\\tt pr}_{\\hat E}}}&\\hat G\\ar[d]\\\\F\\times_{\\hat E} F\\ar[r]^c&F\\times_{\\hat E} F\\ar[r]^{{\\tt pr}_{\\hat E}}&\\hat E}\\ .$$\nIf we plug this in the big double $T$-duality diagram, then we get the big commutative $T$-duality diagram diagram\n\\begin{equation}\\label{univtdzfe11wdqdqw}\n\\xymatrix{&{\\tt pr}_0^*p^*G\\ar[dl]^{\\tilde {\\tt pr}_0}\\ar[dr]\\ar[r]^{{\\tt pr}_0^*u}& {\\tt pr}_{\\hat E}^*\\hat G  \\ar[d]                         \\ar[r]^{\\hat c}         &{\\tt pr}^*_{\\hat E} \\hat G                          \\ar[r]^{{\\tt pr}_1^*u^{-1}}\\ar[d]&{\\tt pr}_1^*p^* G \\ar[dl] \\ar[dr]_{\\tilde {\\tt pr}_1}&\\\\p^*G\\ar[dd]^{f^*p}\\ar[dr]^{p^*f}&&F\\times_{\\hat E}F\\ar[dl]^{{\\tt pr}_0}                      \\ar[r]^c& F\\times_{\\hat E}F              \\ar[dr]^{{\\tt pr}_1}                   &&p^* G\\ar[dd]^{f^*p}\\ar[dl]_{p^*f}\\\\&F\\ar[dr]^p&&&F\\ar[dl]^{p}&\\\\G\\ar[rr]^f&&E  \\ar@{=}[r]  &E                                                                                    &&G\\ar[ll]^f}\n\\end{equation}\n\n\n>From this we derive the diagram \n\\begin{equation}\\label{sdsdh}\n\\xymatrix{{\\tt pr}_0^*p^*G\\ar@\/^1pc\/[r]^{q_T}\\ar@\/_1pc\/[r]_{m_T}&G\\ar[r]^f&E}\\ ,\n\\end{equation}\nwhere $$q_T:=f^*p\\circ \\tilde {\\tt pr}_0\\ ,\\quad m_T:= f^*p\\circ \\tilde {\\tt pr}_1\\circ {\\tt pr}_1^*u^{-1}\\circ \\hat c\\circ {\\tt pr}_0^* u\\ .$$\n\\begin{ddd}\nThe diagram (\\ref{sdsdh}) is called the small double $T$-duality diagram associated to  (\\ref{tzfrzrzr}).\n\\end{ddd}\n\\subsection{}\nThe following fact is an immediate consequence of the definitions.\n\\begin{prop}\\label{eukdnwe}\nThe small double $T$-duality diagram (\\ref{sdsdh})  is locally isomorphic to the small periodization diagram  (\\ref{jhsdjkhwud})\n of $G\\to E$.\n\\end{prop}\n \n\n\\section{Twisted cohomology and the $T$-duality transformation}\\subsection{}\n\\newcommand{\\mathfrak{p}}\\newcommand{\\fq}{\\mathfrak{q}}\\newcommand{\\fu}{\\mathfrak{u}}{\\mathfrak{p}}\\newcommand{\\fq}{\\mathfrak{q}}\\newcommand{\\fu}{\\mathfrak{u}}\n\\newcommand{\\mathfrak{I}}{\\mathfrak{I}}\nLet $E$ be a topological stack. In order to write out operations on twisted cohomology effectively\nwe introduce some notation for operations on $D^+({\\tt Sh}_{\\tt Ab} E)$ or $D({\\tt Sh}_{\\tt Ab} E)$.\nIf $p:F\\to E$ is a map of topological stacks, then we let\n$\\mathfrak{p}}\\newcommand{\\fq}{\\mathfrak{q}}\\newcommand{\\fu}{\\mathfrak{u}^*:{\\tt id}\\to Rp_*p^*$ denote the unit.\nIf $p$ is an oriented fiber bundle, then\nwe let $\\mathfrak{p}}\\newcommand{\\fq}{\\mathfrak{q}}\\newcommand{\\fu}{\\mathfrak{u}_!:Rp_*p^*\\to {\\tt id}$ denote the integration map.\nIf $\\pi:E\\to B$ is a second map, then we write $\\pi_* \\mathfrak{p}}\\newcommand{\\fq}{\\mathfrak{q}}\\newcommand{\\fu}{\\mathfrak{u}^*$,\n$\\pi_*\\mathfrak{p}}\\newcommand{\\fq}{\\mathfrak{q}}\\newcommand{\\fu}{\\mathfrak{u}_!$ or simply also $\\mathfrak{p}}\\newcommand{\\fq}{\\mathfrak{q}}\\newcommand{\\fu}{\\mathfrak{u}^*$ and $\\mathfrak{p}}\\newcommand{\\fq}{\\mathfrak{q}}\\newcommand{\\fu}{\\mathfrak{u}_!$ for the induced transformations\n$R\\pi_*\\pi^*\\to R\\pi_*Rp_*p^*\\pi^*$ and \n$R\\pi_*Rp_*p^*\\pi^*\\to R\\pi_*\\pi^*$.\n\nIf $$\\xymatrix{G\\ar[rr]^v\\ar[d]&&H\\ar[d]\\\\E\\ar[rr]^u\\ar[dr]^\\pi&&F\\ar[dl]^{\\hat \\pi}\\\\&B&}$$\nis a diagram with $U(1)$-gerbes $H\\to F$ and $G\\to E$ such that the square is Cartesian,\nthen we write\n$P(v)$ for the transformation\n$u^*\\circ P_H\\to P_G\\circ u^*$,\nand we use the same symbol for the induced transformation\n$R \\pi_* u^* P_H \\hat \\pi^*\\to R\\pi_*P_G u^* \\hat \\pi^*$.\n\nIn a commutative diagram\n$$\\xymatrix{&F\\ar[dl]^p\\ar[dr]^{\\hat p}&\\\\\nE\\ar[dr]^\\pi && \\hat E\\ar[dl]^{\\hat \\pi} \\\\\n&B&}$$\nwe will use the symbol $\\mathfrak{I}$ or, if necessary,\n$\\mathfrak{I}_{\\pi\\circ p=\\hat \\pi\\circ \\hat p}$\nin order to denote the transformation\n$$R\\pi_*Rp^*p^*\\pi^*\\stackrel{\\sim}{\\to} R\\hat \\pi_*R\\hat p_*\\hat p^*\\hat \\pi^*\\ .$$\n\n \n\n\n\n\n\n\n\n\n \n \n\n\n\n\n \n\n\n\n\n\n\n\n\n\n\n\n\\subsection{}\n\nWe consider a topological gerbe $f\\colon G\\to E$ with band $U(1)$ over a locally compact stack.\nIn  \\cite{bss}  we define the $G$-twisted cohomology of $E$ with coefficients in $F\\in D^+({\\tt Sh}_{\\tt Ab}{\\bf E})$ by $$H^*(E,G;F):=H^*(E;Rf_*f^*(F))\\ .$$ \n\n\\subsection{}\n\nAssume now that $f\\colon G\\to E$ is a part of a $T$-duality diagram\n\\begin{equation}\\label{tzfrzrzr111}\n\\xymatrix{&p^*G\\ar[dl]^q\\ar[dr]\\ar[rr]^u&&\\hat p^* \\hat G\\ar[dl]\\ar[dr]^{\\hat q}&\\\\G\\ar[dr]^f&&F\\ar[dl]^p\\ar[dr]^{\\hat p}&&\\hat G\\ar[dl]^{\\hat f}\\\\&E\\ar[dr]^\\pi&&\\hat E\\ar[dl]^{\\hat \\pi}&\\\\&&B&&}\\ .\n\\end{equation}\nThen we define the transformation\n\\begin{equation}\\label{system41}\nJ:= \\hat \\fq_!\\circ \\mathfrak{I}\\circ (\\fu^{-1})^*\\circ \\fq^*:R\\pi_*Rf_*f^*\\pi^*\\to R\\hat \\pi_*R\\hat f_*\\hat f^*\\hat \\pi^*\\ .\n\\end{equation}\nNote that here $\\mathfrak{I}=\\mathfrak{I}_{\\pi f q u^{-1}=\\hat \\pi  f  \\hat q }$.\n\n\n \n\nConsider a sheaf $F\\in D^+({\\tt Sh}_{\\tt Ab}\\mathbf{B})$. Note that, by definition,\n$H^*(E,G;\\pi^*F)=H^*(B;R\\pi_*Rf_*f^*\\pi^*F)$.\n\\begin{ddd}\nFor $F\\in D^+({\\tt Sh}_{\\tt Ab}{\\bf E})$ the $T$-duality \n  transformation is defined as the map\n $$T\\colon H^*(E,G;\\pi^*F)\\to H^{*-1}(\\hat E,\\hat G;\\hat \\pi^*F)$$\n induced by the natural transformation  (\\ref{system41}).\n\\end{ddd}\n \\subsection{}\\label{system50}\n\nLet us calculate the effect of the $T$-duality transformation in a simple example.\nThere is a unique isomorphism class of $T$-duality diagrams over the point $B=*$.\nIn this case $E=U(1)$ and $G=U(1)\\times \\mathcal{B} U(1)$.\nWe consider a discrete abelian group $F$. Then we have\n$$H^*(E,G;\\pi^*\\underline{F}_{B})\\cong \\mathbb{Z}[[z]] [v]\/(v^2)\\otimes F\\ ,\n H^*(\\hat E,\\hat\nG;\\hat\\pi^*\\underline{F}_{B})\\cong \\mathbb{Z}[[z]] [\\hat v]\/(\\hat v^2)\\otimes F\\ ,$$ where \n$\\deg(v)=1=\\deg(\\hat v)$ and $\\deg(z)=2$. \n\n\n\nTo explicitly calculate the effect of $T$ in this case, observe that the\n  cohomology of $Rf_*Rq_*q^*f^* \\underline{F}$ is $\\mathbb{Z}[[z]]\\otimes\n  \\Lambda(v,\\hat v)\\otimes F$ with $v$ and $\\hat v$ the generators of the two\n $S^1$-factors $E$ and $\\hat E$ in $F$. \nThe automorphism $u$ induces\n  in cohomology, i.e.~on $\\mathbb{Z}[[z]]\\otimes\n  \\Lambda(v,\\hat v)\\otimes F$, the algebra homomorphism given by $z\\mapsto z+v\\hat v$,\n  $v\\mapsto v$, $\\hat v\\mapsto \\hat v$. It follows that\n  \\begin{equation*}\n    \\begin{split}\n      T(z^n\\otimes f) &= \\int_{F\/\\hat E}(z^n\\otimes f+nz^{n-1}v\\hat v\\otimes\n      f)=nz^{n-1}\\hat v\\otimes f\\\\\n      T(z^nv\\otimes f)&= \\int_{F\/\\hat E} z^nv\\otimes f = z^n\\otimes f.\n  \\end{split}\n\\end{equation*}\nWe see that the $T$-duality transformation is not an isomorphism.\n\n\n\n\n\n\n\n\\subsection{}\n\nOur main motivation for introducing the periodization functor is the\nconstruction of twisted sheaf cohomology\nwhich admits a $T$-duality isomorphism.  Let $G\\to E$ be a topological gerbe with band $U(1)$ over a locally compact stack $E$.\n\\begin{ddd}\nWe define the periodic $G$-twisted cohomology of $E$ with coefficients in $F\\in D^+({\\tt Sh}_{\\tt Ab}{\\bf E})$ by\n$$H^*_{per}(E,G;F):=H^*(E;P_G(F))\\ .$$\n\\end{ddd}\n\nNote that here we use the sheaf theory operations for the unbounded derived category, see Subsection \\ref{system4001} for details.\n\\subsection{}\n\nAssume again that $f\\colon G\\to E$ is part of a $T$-duality diagram (\\ref{tzfrzrzr111}).\nWe define a natural transformation\n\\begin{equation}\\label{system42}\nJ\\colon R\\pi_*\\circ P_G\\circ \\pi^*\\to R\\hat \\pi_*\\circ  P_{\\hat G}\\circ \\hat \\pi^*\n\\end{equation}\nby \n$$J:=\\hat \\mathfrak{p}}\\newcommand{\\fq}{\\mathfrak{q}}\\newcommand{\\fu}{\\mathfrak{u}_!\\circ\\mathfrak{I}\\circ  P(u)^{-1} \\circ \\mathfrak{p}}\\newcommand{\\fq}{\\mathfrak{q}}\\newcommand{\\fu}{\\mathfrak{u}^*\\ .$$\n\n\nIt again involves sheaf theory operations in the unbounded derived category.\n\nConsider a sheaf $F\\in D^+({\\tt Sh}_{\\tt Ab}\\mathbf{B})$.  \nNote that by definition $H^*_{per}(E,G;\\pi^*F) = H^*(B,R\\pi_*P_G(\\pi^*(F)))$.\n\\begin{ddd}\\label{system51}\nFor $F\\in D^+({\\tt Sh}_{\\tt Ab}{\\bf E})$ the $T$-duality \n transformation in periodic twisted cohomology \n $$T\\colon H^*_{per}(E,G;\\pi^*F)\\to H_{per}^{*-1}(\\hat E,\\hat G;\\hat \\pi^*F)$$\nis the map induced by the natural transformation  (\\ref{system42}).\n\\end{ddd}\n\n\\subsection{}\n\nAs an illustration let us calculate the action of the $T$-duality transformation\nin the example started in \\ref{system50}. \nThe sequence $\\mathcal{S}_G(\\underline{F})$ for $F=\\mathbb{Z},\\mathbb{Q},\\mathbb{Q}\/\\mathbb{Z}$ either has trivial $\\lim$ or trivial\n$\\lim^1$. Therefore in this special case the morphism $T$ calculated in\n\\ref{system50} defines uniquely an\nendomorphism of $H^*_{per}(E,G;\\pi^*\\underline{F}_B)$ (we identify $E\\cong \\hat E$). For example if  $F=\\mathbb{Q}$, then\nwe read off directly from \\ref{system50} that (with $H^{0}_{per}(E,G;\\pi^*\\underline{\\mathbb{Q}}) \\cong \\mathbb{Q}[v]\/v^2$) the $T$-duality morphism is\n\\begin{equation*}\n  T\\colon \\mathbb{Q}[v]\/v^2\\to  \\mathbb{Q}[v]\/v^2\\ ,\\quad  T(v)=1\\ ,\\quad  T(1)=v\\ .\n\\end{equation*}\nIn particular, we see in this example that\nnow we get an isomorphism. \n\n\n\\subsection{}\n\nIn the remainder of the present subsection we show the following theorem.\n\n\\begin{theorem}\\label{system66}\nThe $T$-duality transformation in twisted periodic cohomology \\ref{system51}\nis an isomorphism.\n\\end{theorem}\n\\begin{proof}\nThe opposite of the $T$-duality diagram (\\ref{tzfrzrzr111}) is obtained by\nreflecting it in the middle vertical, and by replacing $u$ by its inverse.\nWe let\n$T^\\prime\\colon H^*_{per}(\\hat E,\\hat G;\\hat \\pi^*F)\\to H_{per}^{*-1}( E, G;\\pi^*F)$\nbe the associated $T$-duality transformation.\n\nBoth, the  $T$-duality diagram and its opposite\ncan be recognized as subdiagrams of the \n (slightly extended) big commutative $T$-duality diagram \n\\begin{equation}\\label{univtdzfe11zuzuqwd}\n\\xymatrix{&{\\tt pr}_0^*p^*G\\ar[ddl]^{\\tilde {\\tt pr}_0}\\ar[ddr]\\ar[r]^{{\\tt pr}_0^*u}& {\\tt pr}_{\\hat E}^*\\hat G \\ar[dr] \\ar[dd]                         \\ar[rr]^{\\hat c}    &     &{\\tt pr}^*_{\\hat E} \\hat G             \\ar[dl]             \\ar[r]^{{\\tt pr}_1^*u^{-1}}\\ar[dd]&{\\tt pr}_1^*p^* G \\ar[ddl] \\ar[ddr]_{\\tilde {\\tt pr}_1}&\\\\\n&\\hat p^*\\hat G\\ar[dd]\\ar[rr]&&\\hat G\\ar[dd]&&\\hat p^*\\hat G\\ar[ll]\\ar[dd]\\ar[dr]^{u^{-1}}&\n\\\\p^*G\\ar[ru]^u\\ar[dd]^{f^*p}\\ar[dr]^{p^*f}&&F\\times_{\\hat E}F\\ar[dl]^{{\\tt pr}_0}\\ar[dr] ^s                     \\ar[rr]^{\\hspace{-1cm}c}&& F\\times_{\\hat E}F              \\ar[dr]^{{\\tt pr}_1}         \\ar[dl]            &&p^* G\\ar[dd]^{f^*p}\\ar[dl]_{p^*f}\\\\&F\\ar[dr]^p\\ar[rr]^{\\hat p}&&\\hat E&&F\\ar[ll]^{\\hat p}\\ar[dl]^{p}&\\\\G\\ar[rr]^f&&E  \\ar@{=}[rr]  &&E                                                                                    &&G\\ar[ll]^f}\n\\end{equation}\nWe now calculate the composition $T^\\prime\\circ T$.\nThe compatibility of the integration with pull-back in the Cartesian diagram\n$$\\xymatrix{F\\ar[d]^{\\hat p}&F\\times_{\\hat E}F\\ar[d]^{{\\tt pr}_1}\\ar[l]_{{\\tt pr}_0 }\\\\\\hat E&F\\ar[l]^{\\hat p}}$$\n is employed in the equality marked by $!$ below. The equality $\\hat p\\circ {\\tt pr}_0\\circ c^{-1}=\\hat p\\circ {\\tt pr}_0$ is used in the equality $!!$. Finally we use ${\\tt pr}_0\\circ c={\\tt pr}_1$ at $!!!$.\nWe have\n\\begin{eqnarray*}\nJ^\\prime\\circ J&=&\\mathfrak{p}}\\newcommand{\\fq}{\\mathfrak{q}}\\newcommand{\\fu}{\\mathfrak{u}_!\\circ\\mathfrak{I}\\circ  P(u) \\circ \\hat \\mathfrak{p}}\\newcommand{\\fq}{\\mathfrak{q}}\\newcommand{\\fu}{\\mathfrak{u}^*\\circ \\hat \\mathfrak{p}}\\newcommand{\\fq}{\\mathfrak{q}}\\newcommand{\\fu}{\\mathfrak{u}_!\\circ\\mathfrak{I}\\circ  P(u)^{-1} \\circ \\mathfrak{p}}\\newcommand{\\fq}{\\mathfrak{q}}\\newcommand{\\fu}{\\mathfrak{u}^*\\\\\n&\\stackrel{!}{=}&\\mathfrak{p}}\\newcommand{\\fq}{\\mathfrak{q}}\\newcommand{\\fu}{\\mathfrak{u}_!\\circ\\mathfrak{I}\\circ  P(u) \\circ \\mathfrak{pr_1}_!\\circ \\mathfrak{I}\\circ \\mathfrak{pr_0}^* \\circ \\mathfrak{I}\\circ  P(u)^{-1} \\circ \\mathfrak{p}}\\newcommand{\\fq}{\\mathfrak{q}}\\newcommand{\\fu}{\\mathfrak{u}^*\\\\&\\stackrel{!!}{=}&\\mathfrak{p}}\\newcommand{\\fq}{\\mathfrak{q}}\\newcommand{\\fu}{\\mathfrak{u}_!\\circ\\mathfrak{I}\\circ  P(u) \\circ \\mathfrak{pr_1}_!\\circ \\mathfrak{I}\\circ P(\\hat c^{-1})\\circ (\\mathfrak{ c}^{-1})^*\\circ \\mathfrak{pr_0}^* \\circ \\mathfrak{I}\\circ  P(u)^{-1} \\circ \\mathfrak{p}}\\newcommand{\\fq}{\\mathfrak{q}}\\newcommand{\\fu}{\\mathfrak{u}^*\\\\\n&\\stackrel{!!!}{=}&\\mathfrak{p}}\\newcommand{\\fq}{\\mathfrak{q}}\\newcommand{\\fu}{\\mathfrak{u}_!\\circ \\mathfrak{pr_1}_!\\circ P({\\tt pr}_1^* u)\\circ  P(\\hat c^{-1})\\circ P({\\tt pr}_0^*u)^{-1}\\circ  \\mathfrak{pr_1}^* \\circ \\mathfrak{p}}\\newcommand{\\fq}{\\mathfrak{q}}\\newcommand{\\fu}{\\mathfrak{u}^*\\\\\n&=&\\mathfrak{p}}\\newcommand{\\fq}{\\mathfrak{q}}\\newcommand{\\fu}{\\mathfrak{u}_!\\circ \\mathfrak{pr_1}_!\\circ P( {\\tt pr}_1^* u\\circ  \\hat c^{-1}\\circ({\\tt pr}_0^*u)^{-1})\\circ  \\mathfrak{pr_1}^*\\circ \\mathfrak{p}}\\newcommand{\\fq}{\\mathfrak{q}}\\newcommand{\\fu}{\\mathfrak{u}^*\n\\end{eqnarray*}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nThis is exactly the transformation associated to the associated small double $T$-duality diagram \n(\\ref{sdsdh}) (actually its mirror). Since this is locally isomorphic to the small periodization diagram we see that locally $J^\\prime\\circ J$ coincides with $\\pi_* W$, where $W$ is as in Proposition  \\ref{dhewud82d}.  By Proposition \\ref{dhewud82d} this transformation is an isomorphism on periodic sheaves\nof the form $R\\pi_*P_G(\\pi^*F)$.\nTherefore $T\\circ T^\\prime$ is an isomorphism.\nWe can interchange the roles of $T$ and $T^\\prime$, hence $T \\circ T^\\prime$ is an isomorphism, too.\nThis implies the result.\n\\end{proof}\n\n\n\n\n\n\n\n \n\\chapter{Orbispaces}\\label{system5000}\n\n\\section{Twisted periodic delocalized cohomology of orbispaces}\n\n\\subsection{}\n\nLet us recall some notions related to orbispaces (compare \\cite{math.KT\/0609576}). Orbispaces as particular kind of topological stacks \nhave previously been introduced in \\cite[Sec.~2.1]{MR2246781} and \\cite[Sec.~19.3]{math.AG\/0503247}). In the present paper  we use the set-up of   \\cite{MR2246781} \nbut add the additional condition that an orbifold atlas should be separated. This condition is needed in order to show that the loop\nstack of an orbifold is again an orbifold. \n\n\n\n\n\\begin{enumerate}\n\\item A topological groupoid $A\\colon A^1\\Rightarrow A^0$ is called separated if the identity ${\\mathbf{1}}_A\\colon A^0\\to A^1$ of the groupoid is a closed map.\n\\item A topological groupoid $A^1\\Rightarrow A^0$ is called proper \nif $(s,r)\\colon A^1\\to A^0\\times A^0$ is a proper map. \n\\item A topological groupoid is called {\\'e}tale if\nthe source and range maps $s,r\\colon A^1\\to A^0$ are  {\\'e}tale.\n\\item A proper \\'etale  topological groupoid $A^1\\Rightarrow  A^0$ is called very proper if \nthere exists a continuous function $\\chi \\colon  A^0\\to [0,1]$ such that\n\\begin{enumerate}\n\\item $r\\colon {\\tt supp}(s^*\\chi)\\to A^0$ is proper\n\\item  $\\sum_{y\\in A^x} \\chi(s(y))=1$ for all $x\\in A^0$.\n\\end{enumerate}\n\\item A topological stack is called (very) proper (or {\\'e}tale, separated,\n  respectively),  if it admits an atlas $A\\to X$ such that the topological\n  groupoid \n$A\\times_XA\\Rightarrow A$ is (very) proper  (or {\\'e}tale, separated, respectively).\n \n\\item An orbispace atlas of a topological stack $X$ is an atlas $A\\to X$ such that $A\\times_XA\\Rightarrow A$ is a very proper  {\\'e}tale and separated  groupoid. \n\\item\nAn orbispace $X$ is a topological stack which admits an orbispace atlas.\n\\item If $X,Y$ are orbispaces, then a morphism of orbispaces $X\\to Y$\nis a representable morphism of stacks.\n\\item A locally compact orbispace is an orbispace $X$ which admits an orbispace atlas $A\\to X$ such that $A$ is locally compact. \n \\end{enumerate}\n\n\\subsection{}\n\nIf $X$ is a stack, then its inertia stack (sometimes called loop stack) $LX$\nis defined as the two-categorical equalizer \nof the diagram $$\\xymatrix{X\\ar@{=>}[r]^{{\\tt id}_X}_{{\\tt id}_X}&X}\\ .$$ \nIn \\cite[Sec 2.2]{math.KT\/0609576} we have introduced   an\nexplicit model of $LX$ and studied its properties. {\nThe loop stack $LX$ depends $2$-functorially on $X$. Indeed,  since $\\underline{{\\tt Hom}}_{\\tt\n  Cat}$ is a strict $2$-functor, the loop functor is\na strict functor between $2$-categories.} {As already\nmentioned before, later we will suppress the 2-morphisms in 2-commutative diagrams\nin 2-categories for better legibility.} If\n$X$ is a topological stack (orbispace), then \n$LX$ is a topological stack (orbispace), too (see \\cite[Lemma\n2.25]{math.KT\/0609576}, \\cite[Lemma 2.33]{math.KT\/0609576}).\n\n\\begin{lem}\nIf $X$ is a locally compact orbispace, then $LX$ is a locally compact orbispace, too.\n\\end{lem}\n\\begin{proof}\nLet $A\\to X$ be a locally compact orbispace atlas of $X$.\nThen we have the proper, separated and \\'etale topological  groupoid\n$A\\times_XA\\Rightarrow A$. Since the source map of this groupoid is  \\'etale, the space of\nmorphisms $A\\times_XA$ of this groupoid is locally compact, too.\n\nIn the proof of Lemma \\cite[Lemma 2.25]{math.KT\/0609576} we constructed an orbispace atlas \n$W\\to LX$ of $LX$, where $W$ was given by the pull-back of spaces\n$$\\xymatrix{W\\ar[r]\\ar[d]^w&A\\times_XA\\ar[d]^{({\\tt pr}_1,{\\tt pr}_2)}\\\\\nA\\ar[r]^{{\\tt diag}}&A\\times A}\\ .$$\nThis implies that $W$ is locally compact.\n\\end{proof}\n\n\n\\subsection{}\\label{system60}\n \n\nLet $G\\to X$ be a topological gerbe with band $U(1)$ over a locally compact  orbispace. \nThe truly interesting $G$-twisted cohomology of $X$ (with complex coefficients)\nis not the cohomology $H^*_{per}(X,G;\\underline{\\mathbb{C}})$ (see \\ref{system51}), \nbut a more complicated delocalized version $H^*_{deloc,per}(X,G)$,\nwhich we will define below (see \\cite[Sec.~1.3]{math.KT\/0609576} for an explanation).\n\n\n \nAs shown in \\cite[Sec.~2.5]{math.KT\/0609576} the gerbe gives rise to a principal bundle $\\tilde G^\\delta\\to LX$\nwith structure group $U(1)^\\delta$ in a functorial way,\n where $U(1)^\\delta$ denotes the group $U(1)$ with\nthe discrete topology. By $\\mathcal{L}\\in {\\tt Sh}_{\\tt Ab}\\mathbf{LX}$\nwe denote the sheaf of locally constant sections of the associated vector bundle $\\tilde G^\\delta\\times_{U(1)^\\delta}\\mathbb{C}\\to LX$.\n\nWe define the gerbe $G_L\\to LX$ as the pull-back\n$$\\xymatrix{G_L\\ar[d]^{f_L}\\ar[r]&G\\ar[d]^f\\\\LX\\ar[r]&X}\\ .$$\n\n\\begin{ddd}\\label{eiuwqd}\nWe define \n$$\\mathcal{L}_G:=P_{G_L}(\\mathcal{L})\\in D({\\tt Sh}_{\\tt Ab} \\mathbf{LX})\\ .$$\nThe $G$-twisted delocalized periodic cohomology of $X$ is defined as\n$$H^*_{deloc,per}(X,G):=H^*(LX;\\mathcal{L}_G)\\ .$$\n\\end{ddd}\n\n\\section[$T$-duality in twisted periodic delocalized cohomology]{The $T$-duality transformation in twisted periodic delocalized cohomology}\n\n\n\n\n\\subsection{}\n\n We consider a $T$-duality diagram \n\\begin{equation}\\label{tzfrzrddzr123}\n\\xymatrix{&p^*G\\ar[dl]\\ar[dr]\\ar[rr]^u&&\\hat p^* \\hat G\\ar[dl]\\ar[dr]&\\\\G\\ar[dr]^f&&F\\ar[dl]^p\\ar[dr]^{\\hat p}&&\\hat G\\ar[dl]^{\\hat f}\\\\&E\\ar[dr]^\\pi&&\\hat E\\ar[dl]^{\\hat \\pi}&\\\\&&B&&}\n\\end{equation}\n(see Definition  \\ref{eruihfrvc}), where $B$ is a  locally compact  orbispace. \n\n\n \n\nWe apply the loops functor $L\\colon orbispaces\\to orbispaces$ to the subdiagram\n$$\\xymatrix{&F\\ar[dr]^{\\hat p}\\ar[dl]_{p}&\\\\E\\ar[dr]^\\pi&&\\hat E\\ar[dl]_{\\hat \\pi}\\\\&B&}$$ and get\n$$\\xymatrix{&LF\\ar[dr]^{L\\hat p}\\ar[dl]_{Lp}&\\\\LE\\ar[dr]^{L\\pi}&&L\\hat E\\ar[dl]_{L\\hat \\pi}\\\\&LB&}\\ .$$ \nIn the first diagram the maps $p,\\hat p,\\pi,\\hat \\pi$ are all $U(1)$-principal bundles.\nThe maps $Lp,L\\hat p,L\\pi,L\\hat \\pi$ are not necessarily surjective. Thus in general the derived diagram of loop stacks is not part of a $T$-duality diagram. But it is so locally in a certain sense which we will explain in the following.\n\n\\subsection{}\n\nWe can extend the second diagram by the local systems (see \\ref{system60})\n\\begin{equation}\\label{tzfrzrddzr444}\n\\xymatrix{&Lp^*\\mathcal{L}\\ar[dl]\\ar[dr]\\ar[rr]^u&&L\\hat p^* \\hat \\mathcal{L}\\ar[dl]\\ar[dr]&\\\\\\mathcal{L}\\ar[dr]&&LF\\ar[dl]^{Lp}\\ar[dr]^{L\\hat p}&&\\hat \\mathcal{L}\\ar[dl]\\\\&LE\\ar[dr]^{L\\pi}&&L\\hat E\\ar[dl]^{L\\hat \\pi}&\\\\&&LB&&}\n\\end{equation}\nand the pull-backs of gerbes\n\\begin{equation}\\label{tzfrzrddzr}\n\\xymatrix{&Lp^*G_L\\ar[dl]\\ar[dr]\\ar[rr]^u&&L\\hat p^* \\hat G_L\\ar[dl]\\ar[dr]&\\\\G_L\\ar[dr]^{f_L}&&LF\\ar[dl]^{Lp}\\ar[dr]^{L\\hat p}&&\\hat G_L\\ar[dl]^{\\hat f_L}\\\\&LE\\ar[dr]^{L\\pi}&&L\\hat E\\ar[dl]^{L\\hat \\pi}&\\\\&&LB&&}\n\\end{equation}\nIn particular, we have an isomorphism\n\\begin{equation}\\label{ezfgbdmnwdas}\nu:Lp^*\\mathcal{L}_G\\stackrel{\\sim}{\\to} L\\hat p^* \\hat\\mathcal{L}_{\\hat G}\\ .\n\\end{equation}\n\n\n\n\\subsection{}\\label{system62}\n\n \n\n\n\n\n\n\n \nNote that $\\hat p\\colon F\\to \\hat E$ is a $U(1)$-principal bundle.\nIn \\cite[Lemma 2.34]{math.KT\/0609576} we have constructed\na map $h\\colon L\\hat E\\to U(1)^\\delta$ which measures the action of the automorphisms of the points of $\\hat E$\non the fibers of $\\hat p$. We get a decomposition into a disjoint union of open substacks\n$$L\\hat E\\cong \\bigsqcup_{u\\in U(1)} L\\hat E_u\\ ,$$\nwhere $L\\hat E_u:=h^{-1}(u)$. Here and in the following we use the simplified\nnotation $h^{-1}(u)$ for the pullback of $h\\colon L\\hat E\\to U(1)^{\\delta}$\nalong the inclusion $i_u\\colon *\\to U(1)$ with $ i_u(*):=u$.  By \\cite[Lemma 2.36]{math.KT\/0609576}, the map\n$L\\hat p\\colon LF\\to L\\hat E$ factors over the inclusion $J:L\\hat E_1\\to L\\hat\nE$, and the corresponding map $L\\hat p_1\\colon LF\\to L\\hat E_1$ is a\n$U(1)$-principal bundle. The integration \n$$\\mathfrak{L\\hat p_1}_!\\colon R(L\\hat p_1)_*\\circ L\\hat p_1^*\\to {\\tt id}$$\nis well-defined. The open inclusion $J$ induces\na natural transformation $\\mathfrak{J}_!\\colon RJ_*\\circ J^*\\to {\\tt id}$.\nWe can thus define\n$$\\mathfrak{L\\hat p}_!:=\\mathfrak{J}_! \\circ \\mathfrak{L\\hat p_1}_! \\colon  RL\\hat p_*\\circ L\\hat p^*  \\to {\\tt id}\\ .$$\n\n\\subsection{}\n\n\\begin{ddd}\\label{system61}\nThe local $T$-duality transformation   associated to the diagram (\\ref{tzfrzrddzr123}) is \ngiven by the composition\n\\begin{eqnarray*}\nT_{loc}:=\\mathfrak{L\\hat p}_!\\circ u\\circ \\mathfrak{L p}^*\\colon RL\\pi_*\\mathcal{L}_G\\to RL\\hat \\pi_*\\hat \\mathcal{L}_{\\hat G}\n\\ ,\\end{eqnarray*}\nwhere $u$ is induced by (\\ref{ezfgbdmnwdas}).\n\\end{ddd}\n\nNote that $H^*_{deloc,per}(E,G)\\cong H^*(LB;RL\\pi_*\\mathcal{L}_G)$.\nHence we can make the following definition.\n\\begin{ddd}\nThe $T$-duality transformation in twisted periodic delocalized cohomology associated to the\n$T$-duality diagram (\\ref{tzfrzrddzr123}) is the transformation\n$$T\\colon H^*_{deloc,per}(E,G)\\to H^*_{deloc,per}(\\hat E,\\hat G)$$ induced by the local\n$T$-duality transformation $T_{loc}$ defined in \\ref{system61}.\n\\end{ddd}\n\n\n\n\n\n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{The geometry of $T$-duality diagrams over orbispaces}\n\n\n \n\n\\subsection{}\nWe consider a $T$-duality diagram (\\ref{tzfrzrddzr123}) over a locally compact orbispace. \nAs explained in \\cite[Sec.~2.5]{math.KT\/0609576} (see also \\ref{system60})\n the gerbe $G\\to E$ naturally gives rise to a $U(1)^\\delta$-principal bundle $\\tilde G^\\delta\\to LE$.\nLet  $g\\colon LB_1\\to U(1)^\\delta$ be the function which describes the holonomy of the bundle $\\tilde G^\\delta\\to LE$ along the fibers of $LE\\to LB_{1}$ (see \\cite[2.6.3]{math.KT\/0609576}).\nIn the following we recall from \\cite{math.KT\/0609576} a cohomological description of the functions\n$g$ and $h$ (introduced in \\ref{system62}).\n \nLet $c_1\\in H^2(B;\\mathbb{Z})$ denote the first Chern class of the $U(1)$-principal bundle $\\pi\\colon E\\to B$, and let\n $d\\in H^3(E;\\mathbb{Z})$ denote the Dixmier-Douady class of the gerbe $f\\colon G\\to E$.\nBy integration over the fiber it gives rise to a class $\\int_\\pi d\\in H^2(B;\\mathbb{Z})$.\n In \\cite[2.4.11]{math.KT\/0609576} we have shown that a class $\\chi\\in H^2(B;\\mathbb{Z})$ gives rise to a function $\\bar\\chi\\colon LB\\to U(1)^\\delta$ in a natural way.  \n\n\\begin{prop}[Lemma 2.38 and Prop. 2.49  \\cite{math.KT\/0609576} ]\\label{auqewmdq}\nWe have the equalities\n\\begin{enumerate}\n\\item\n$$\\overline{c_1}=h\\colon LB\\to U(1)^\\delta\\ .$$\n\\item\n$$\\overline{\\int_\\pi d}_{|LB_1}=g\\colon  LB_1\\to U(1)^\\delta\\ .$$ \n\\end{enumerate}\n\\end{prop}\n \n\n\n\n\n\n\n\n\\subsection{}\n\n\n\n\n   \nWe now have functions $h,\\hat h\\colon LB\\to U(1)^\\delta$ associated to the $U(1)$-principal bundles\n$\\pi\\colon E\\to B$ and $ \\hat \\pi\\colon \\hat E\\to B$.  We define\n$$LB_{(1,*)}:=h^{-1}(1)\\ ,\\quad LB_{(*,1)}:=\\hat h^{-1}(1)\\ .$$ \nWe furthermore have functions (see \\ref{tzfrzrddzr123})\n$$g\\colon LB_{(1,*)}\\to U(1)^\\delta\\ ,\\quad \\hat g\\colon LB_{(*,1)}\\to U(1)^\\delta$$\nmeasuring the holonomy of $\\tilde G^\\delta\\to LE$ and $\\tilde {\\hat G}^\\delta\\to L\\hat E$ along the fibers.\n\n\\begin{prop}\\label{ueidkjqwdq}\nWe have the equalities\n$$\\hat g=h^{-1}_{|LB_{(*,1)}}\\ ,\\quad g=\\hat h^{-1}_{|LB_{(1,*)}}\\ .$$\n\\end{prop}\n\\begin{proof}\nLet $$d\\in H^3(E;\\mathbb{Z})\\ , \\quad \\hat d\\in H^3(\\hat E;\\mathbb{Z})$$\nbe the Dixmier-Douady classes of the gerbes $G_L\\to E$ and $\\hat G_L\\to \\hat E$.\nFurthermore let  $$c_1,\\hat c_1\\in H^2(B;\\mathbb{Z})$$ denote the first Chern classes of the\n $U(1)$-principal bundles $\\pi\\colon E\\to B$ and $\\hat \\pi\\colon \\hat E \\to B$.\n The theory of $T$-duality for orbispaces \\cite{MR2246781} gives the equalities\n$$c_1=-\\hat \\pi_!(\\hat d)\\ ,\\quad \\hat c_1 = - \\pi_!(d)\\ .$$\nHence the assertion follows from Proposition \\ref{auqewmdq}. \n\\end{proof}\n\n\\section[$T$-duality is an isomorphism]{The $T$-duality transformation in  twisted periodic delocalized cohomology is an isomorphism}\n\n\\subsection{}\n Let us consider a $U(1)$-principal bundle $\\pi\\colon E\\to B$ in locally compact orbispaces with first Chern class $c_1\\in H^2(B;\\mathbb{Z})$ and a topological $U(1)$-banded gerbe $f\\colon G\\to E$ with Dixmier-Douady class $d\\in H^3(E;\\mathbb{Z})$. In Definition \\ref{eiuwqd} we have introduced the object $\\mathcal{L}_G\\in D({\\tt Sh}_{\\tt Ab}\\mathbf{LE})$. Furthermore we have $U(1)^\\delta$-valued functions\n$h=\\overline{c_1}$ and $g=\\overline{\\pi_!(d)}$ on $LB$. Let $LB_1:=h^{-1}(1)$\n and note that $L\\pi\\colon LE\\to LB$ factors over the $U(1)$-principal bundle\n$L\\pi\\colon LE\\to LB_1$. We fix $u\\in U(1)^\\delta\\setminus \\{1\\}$ and consider the component\n$LB_{(1,u)}:=h^{-1}(1)\\cap g^{-1}(u)$.\n\\begin{lem}\\label{hjwegbd}\nWe have $R\\pi_*(\\mathcal{L}_G)_{|LB_{(1,u)}}\\cong 0$.\n\\end{lem}\n\\begin{proof}\nLet $(T\\to LB_{(1,u)})\\in \\mathbf{LB}_{(1,u)}$. \nAfter refining $T$  by a covering we can assume that there is a diagram \n$$\\xymatrix{\\mathcal{B} U(1)\\ar[d]^y&U(1)\\times \\mathcal{B}\n  U(1)\\ar[l]^z\\ar[d]^x&s^*G_L\\ar[d]\\ar[l]\\ar[r]&{G_L}_{(1,u)}\\ar[d]\\\\{*}&U(1)\\ar[l]^q\\ar[d]^q&T\\times\n  U(1)\\ar[l]^v\\ar[r]^s\\ar[d]^p&LE_{(1,u)}\\ar[d]^\\pi\\\\&{*}&T\\ar[l]^w\\ar[r]^t\n  &LB_{(1,u)}}$$  \nof Cartesian squares.\nWe get\n\\begin{eqnarray*}\nt^*R\\pi_*(\\mathcal{L}_G)&\\cong&Rp_* s^*(\\mathcal{L}_G)\\\\\n&= &Rp_* s^*(P_{G_L}(\\mathcal{L}))\\\\\n&\\cong&Rp_*P_{s^*G_L}(s^*\\mathcal{L})\\ .\n\\end{eqnarray*}\nLet $\\mathcal{H}\\in {\\tt Sh}_{\\tt Ab}({\\tt Site}(U(1)))$ be the locally constant sheaf over $U(1)$\nwith fiber $\\mathbb{C}$ and holonomy $u\\in U(1)\\setminus \\{1\\}$. Then we have $s^*\\mathcal{L}\\cong v^* \\mathcal{H}$.  \nWe calculate further\n\\begin{eqnarray*}\nRp_*P_{s^*G_L}(s^*\\mathcal{L})&\\cong&Rp_* P_{s^*G_L}(v^*\\mathcal{H})\\\\\n&\\cong&Rp_* v^*P_{U(1)\\times \\mathcal{B} U(1)}(\\mathcal{H})\\\\&\\cong&\nw^*Rq_*P_{U(1)\\times \\mathcal{B} U(1)}(\\mathcal{H})\\ .\n\\end{eqnarray*}\nIt remains to show that\n$$Rq_*P_{U(1)\\times \\mathcal{B} U(1) }(\\mathcal{H})\\cong 0\\ .$$ Recall from \\ref{system19} that the object\n$P_{U(1)\\times \\mathcal{B} U(1)}(\\mathcal{H})\\in D({\\tt Sh}_{\\tt Ab} {\\tt Site}(U(1)))$ is given (up to non-canonical isomorphism) by the ${\\tt holim}$ of a diagram\n$$0\\leftarrow Rx_* x^*(\\mathcal{H})\\stackrel{D}{\\leftarrow} Rx_* x^*(\\mathcal{H}) [2]\\stackrel{D}{\\leftarrow} Rx_* x^*(\\mathcal{H})[4]\\stackrel{D}{\\leftarrow} Rx_* x^*(\\mathcal{H})[6]\\dots\\ .$$\nThe functor $Rq_*$ commutes with this ${\\tt holim}$\\footnote{$Rq_*$ is a right-adjoint and commutes with products and mapping cones}. Therefore\n$Rq_*P_{U(1)\\times \\mathcal{B} U(1)}(\\mathcal{H})$ is given by the ${\\tt holim}$ of the diagram\n\\begin{multline*}\n  0\\leftarrow Rq_* Rx_* x^*(\\mathcal{H})\\stackrel{Rq_*(D)}{\\leftarrow} Rq_* Rx_*\n  x^*(\\mathcal{H})[2]\\\\\n \\stackrel{Rq_*(D)}{\\leftarrow} Rq_*Rx_*\n  x^*(\\mathcal{H})[4]\\stackrel{Rq_*(D)}{\\leftarrow} Rq_*Rx_* x^*(\\mathcal{H})[6]\\dots\\ .\n\\end{multline*}\nThe following calculation uses the projection formula twice, first by Lemma \\ref{gr213} for the non-representable map \n$x$ and a tensor product with a one-dimensional local system of complex vector\nspaces $\\mathcal{H}$, secondly using Lemma \n\\ref{projefoa} for the proper representable map $q$ and the tensor product\nwith the bounded below object $Ry_*(i^\\sharp\\mathbb{Z}_{{\\tt Site}([*\/U(1)])})\\in\nD^+({\\tt Sh}_{\\tt Ab}{\\tt Site}(U(1)))$ \n\\begin{eqnarray*}\nRq_*Rx_*x^*(\\mathcal{H})&\\cong&Rq_*Rx_*(\\underline{\\Z}_{{\\tt Site}(U(1)\\times \\mathcal{B} U(1)}\\otimes x^*(\\mathcal{H}))\\\\\n&\\cong&Rq_* (Rx_*(\\underline{\\Z}_{{\\tt Site}(U(1)\\times \\mathcal{B} U(1))})\\otimes \\mathcal{H})\\\\\n&\\cong&Rq_*(Rx_*(z^*\\underline{\\Z}_{{\\tt Site}(\\mathcal{B} U(1))})\\otimes \\mathcal{H})\\\\\n&\\cong&Rq_*(q^*(Ry_*\\underline{\\Z}_{{\\tt Site}(\\mathcal{B} U(1))})\\otimes \\mathcal{H})\\\\\n&\\cong&Ry_*\\underline{\\Z}_{{\\tt Site}(\\mathcal{B} U(1))}\\otimes Rq_*(\\mathcal{H})\\ .\n\\end{eqnarray*}\nSince the holonomy of $\\mathcal{H}$ along $U(1)$ is non-trivial, and the cohomology of\n$S^1$ with coefficients in a non-trivial flat line bundle is trivial,  we have\n$$Rq_*(\\mathcal{H})\\cong 0\\ .$$\n\\end{proof}\n\n\\subsection{}\n\nWe now consider a $T$-duality diagram  (\\ref{tzfrzrddzr123})\n where $B$ is a locally compact  orbispace.\n\n\\begin{theorem}\\label{system1006}\nThe local  $T$-duality transformation (Definition \\ref{system61}) \n$$T_{loc}\\colon RL\\pi_*(\\mathcal{L}_G)\\to RL\\hat \\pi_*(\\hat \\mathcal{L}_{\\hat G})[-2]$$ is an isomorphism in $D({\\tt Sh}_{\\tt Ab}\\mathbf{LB})$.\nIn particular, the $T$-duality transformation\n$$T\\colon H^*_{deloc,per}(E,G)\\to H^*_{deloc,per}(\\hat E,\\hat G)$$\nis an isomorphism.\n \\end{theorem}\n\\begin{proof}\nWe have functions\n$h,\\hat h\\colon LB\\to U(1)$ which define substacks\n$LB_{(1,*)}:=h^{-1}(1)$ and $LB_{(*,1)}:=\\hat h^{-1}(1)$.\nBy Proposition \\ref{ueidkjqwdq} we have $g=\\hat h^{-1}_{|LB_{(1,*)}}:LB_{(1,*)}\\to U(1)^\\delta$.\nBy Lemma \\ref{hjwegbd} the object\n$RL\\pi_*(\\mathcal{L}_G)\\in D({\\tt Sh}_{\\tt Ab} \\mathbf{LB})$ is supported on \n$$g^{-1}(1)=LB_{(1,*)}\\cap\nLB_{(*,1)}=: LB_{(1,1)}\\ .$$\nNote that\n$\\hat g=h^{-1}_{|LB_{(*,1)}}$, so that $RL\\hat \\pi_*\\hat \\mathcal{L}_{\\hat G}$ is supported on $LB_{(1,1)}$, too. Let $i\\colon LB_{(1,1)}\\to LB$ denote the inclusion. The following diagram is the pull-back of  (\\ref{tzfrzrddzr123}) via the map $LB_{(1,1)}\\to LB\\to B$  \n\\begin{equation}\\label{wedddbwed}\n{\\small \n\\xymatrix{&p_L^*(G_L)_{|LE_{|LB_{(1,1)}}}\\ar[dl]\\ar[dr]\\ar[rr]^{u_L}&&\\hat\n  p_L^* (\\hat G_L)_{|L\\hat\n    E_{|LB_{(1,1)}}}\\ar[dl]\\ar[dr]&\\\\(G_L)_{|LE_{|LB_{(1,1)}}}\\ar[dr]^{f_L}&&LF_{|LB_{(1,1)}}\\ar[dl]^{Lp}\\ar[dr]^{\\hat\n    Lp}&&(\\hat G_L)_{|L\\hat E_{|LB_{(1,1)}}}\\ar[dl]^{\\hat\n    f_L}\\\\&LE_{|LB_{(1,1)}}\\ar[dr]^{L\\pi_1}&&L \\hat\n  E_{|LB_{(1,1)}}\\ar[dl]^{L\\hat \\pi_1}&\\\\&&LB_{(1,1)}&&} \n}\n\\end{equation}\nWe consider $$\\mathcal{L}_1:=\\mathcal{L}_{|LE_{|LB_{(1,1)}}}\\ ,\\quad \\hat \\mathcal{L}_1:=\\hat \\mathcal{L}_{|L\\hat E_{|LB_{(1,1)}}}\\ .$$\nBecause we restrict to the subset $LB_{(1,1)}$ of trivial holonomy we have isomorphisms\n $$\\mathcal{L}_1\\cong L\\pi_1^*\\underline{\\mathbb{C}}_{\\mathbf{LB}_{(1,1)}}\\,\\quad \\hat \\mathcal{L}_1\\cong L\\hat  \\pi_1^* \\underline{\\mathbb{C}}_{\\mathbf{LB}_{(1,1)}}\\ .$$\nThe local $T$-duality transformation $T_{loc}$ is now locally equal to\nthe transformation $J$ defined in \\ref{system42} applied to the $T$-duality diagram\n(\\ref{wedddbwed}) and the sheaf $\\underline{\\mathbb{C}}_{\\mathbf{LB}_{(1,1)}}$. As in the proof of Theorem \\ref{system66} one shows, using the commutative double $T$-duality diagram, that $T_{loc}$ is an isomorphism.\n\nThe global second assertion can be deduced directly from Theorem \\ref{system66}. \nBy\nthe observation on the support of $RL\\pi_*(\\mathcal{L}_G)\\in D({\\tt Sh}_{\\tt Ab} \\mathbf{LB})$ made above\nwe get\n$$H^*_{deloc,per}(E,G)\\cong H^*_{per}(LB_{(1,1)};RL( \\pi_1)_*P_{( G_L)_{|L E_{|LB_{(1,1)}}}}(L\\pi_1^*\\underline{\\mathbb{C}}_{\\mathbf{LB}_{(1,1)}}))\\ ,$$\nand similarly\n$$H^*_{deloc,per}(\\hat E,\\hat G)\\cong H^*_{per}(LB_{(1,1)};RL(\\hat \\pi_1)_*P_{(\\hat G_L)_{|L\\hat E_{|LB_{(1,1)}}}}(L\\pi_1^*\\underline{\\mathbb{C}}_{\\mathbf{LB}_{(1,1)}}))\\ .$$\nWith these identifications the  $T$-duality transformation in twisted periodic delocalized cohomology is then equal to the $T$-duality transformation in twisted periodic cohomology for the diagram (\\ref{wedddbwed}) and the sheaf $\\underline{\\mathbb{C}}_{\\mathbf{LB}_{(1,1)}}\\in D^+({\\tt Sh}_{\\tt Ab} \\mathbf{LB}_{1,1})$.\n \\end{proof}\n\n\n\n \n\n\n\n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n \n\n\\chapter{Verdier duality for locally compact stacks}\\label{system3003}\n\n \n\n\\section{Elements of the theory of stacks on ${\\tt Top}$ and sheaf theory}\\label{wuiefwefwefqwd}\n\n\\subsection{}\n\n\nIn the present paper we consider stacks on the site ${\\tt Top}$. \nA prestack is a lax presheaf $X$ of groupoids on ${\\tt Top}$. The prefix \"lax\" indicates that for a pair of composable morphisms $u\\colon U\\to V$, $v\\colon V\\to W$ we have a natural transformation of functors $\\phi_{u,v}\\colon X(u)\\circ X(v)\\to X(v\\circ u)$ which is not necessarily the identity, and  which satisfies a compatibility condition for triples.\nA prestack is a stack  if it  satisfies the standard descent conditions on the level of objects and morphisms.\n A sheaf of sets can be considered as a stack in the canonical way. Via the Yoneda embedding\n ${\\tt Top}\\to {\\tt Sh}{\\tt Top}$ (note that the topology of ${\\tt Top}$ is sub-canonical, i.e. representable presheaves are sheaves) we consider topological spaces as stacks in the natural way.\n\n\\subsection{}\\label{staqwndwqodwqd}\n\nIn the following we collect some definitions and facts of the theory of stacks in topological spaces.\nStacks are objects of a two-category, and fibre products and more general limits in stacks are understood in the two-categorial sense. Note that two-categorial limits in stacks exists (see \\cite{math.KT\/0609576} for more information), and that the inclusion of spaces into stacks preserves those limits.\nA useful reference for stacks in topological spaces and manifolds is the\nsurvey \\cite{heinloth}.\n\n \n\\begin{enumerate}\n\\item\nA morphism of stacks $G\\rightarrow H$ is called representable, if\nfor each space $U$ and map $U\\rightarrow H$ the fibre product $U\\times_HG$ is equivalent to a space.\n\\item A representable map $G\\to H$ between stacks is called proper if for every map $K\\to H$ from a compact space the fibre product $K\\times_HG$ is a compact space.\n\\item\nA map $f:A\\to B$ of topological spaces has local sections if for each point $b\\in B$ in the image of $f$ there exists a neighbourhood $b\\in U\\subseteq B$ and a map $s:U\\to A$ such that $f\\circ u={\\tt id}_{U}$.\n\\item\nA representable morphism $G\\rightarrow H$  has local sections  if for every map $U\\to H$ from a space the induced map $U\\times_HG\\to U$ of spaces has local sections.\n\\item A representable map $G\\to H$ is surjective if for every map $U\\to H$ from a space\nthe induced map $U\\times_HG\\to U$ is a surjective map of spaces.\n\\item\nA map $A\\to X$ from a space $A$ to a stack $X$ is called an atlas of $X$, if\nit is surjective, representable\nand  admits local sections. A stack which admits an atlas is called a topological stack.\n\\item A morphism (not necessarily representable) between topological stacks $G\\to H$ is surjective  (or has local sections, respectively) if for an atlas $A\\to G$ the composition\n$A\\to G\\to H$ is surjective (or has local sections, respectively) (note that this composition is representable by Proposition \\ref{lem:representability} below).  \n\\item\n A composition of maps with local sections has local sections.\nThe corresponding  assertion is true for the following properties of maps:\n\\begin{enumerate}\n\\item representable\n\\item representable and proper\n\\item surjective.\n\\end{enumerate}\n\\item\nConsider a two-cartesian diagram of stacks\n \\begin{equation*}\n    \\begin{CD}\n      H @>>v> G\\\\\n      @VVgV @VVfV\\\\\n      Y @>>u> X\n    \\end{CD}\n  \\end{equation*}\nIf $u$ has local sections, then so has $v$. If $f$ is representable, then so is $g$.\n  \\end{enumerate}\n\n\n\\subsection{}\n\nThe inclusion of spaces into sheaves and of sheaves into stacks preserves small limits, where limits in stacks are understood in the two-categorical sense. This implies that a map of spaces $X\\to Y$ is representable.\nIn fact we have the following more general result.\n\n\n\n\n\n\n\\begin{prop}\\label{lem:representability}\n  Let $G$ be a topological stack and $X$ a space. Then every morphism $f\\colon\n  X\\to G$ is representable.\n\\end{prop}\nThe proof will be given in \\ref{jhsgbwqjswqs}\nand needs some preparations.\n\n\n\\subsection{}\n\n\n\nWe will need the notion of an open substack.\n  \\begin{definition}\\label{def:open_emb_of_stacks}\n    Let $G$ be a stack in topological spaces. A morphism $H\\to G$ of stacks is an\n    embedding of an open substack, if it is representable and for each  map $T\\to\n    G$ from a space $T$ the induced map of spaces $T\\times_GH\\to T$ is an open embedding\n    of topological spaces. \n  \\end{definition}\nNote that, via Yoneda, an open embedding of spaces is an open embedding of stacks.\n\n  \\begin{definition}\\label{didwedqwdqwdqwdd}\n   A morphism $U\\to G$ of topological stacks is locally an open embedding if $U\\cong \\bigsqcup_{i\\in I} U_i$ for a collection $(U_i)_{i\\in I}$ of topological stacks and $U_i\\to G$ is an\n    embedding of an open substack for every $i\\in I$.\n\\end{definition}\n\\newcommand{{\\mathrm{coeq}}}{{\\mathrm{coeq}}}\nLet us first characterize spaces as stacks which can be covered by a collection of spaces.\n\\begin{lem}\\label{kjhdkqwdqwwqd}\nLet $X$ be a stack in topological spaces for which there exists a morphism $U\\to X$ from a space which is surjective and locally an open embedding. Then $X$ is equivalent to a space.\n \\end{lem}\n\\begin{proof}\nLet $U\\cong \\sqcup_{i} U_i$ be such that $U_i\\to X$ is an open embedding for all $i$.\nThen we define the space $B$ as the coequalizer in spaces\n\\begin{equation}\\label{asdwdisdwqdqdqw}\nB:={\\mathrm{coeq}}(\\bigsqcup_{i,j} U_i\\times_X U_j\\rightrightarrows \\bigsqcup_i U_i)\\ .\n\\end{equation}\nSince $U_i\\to X$ is an open embedding we see that ${\\tt pr}_{U_i}\\colon U_i\\times_X U_j\\to U_i$\nis an open embedding. We can now refer to  \\cite[Prop. 16.1]{math.AG\/0503247} and deduce that\nthe equalizer in spaces $B$ is also the two-categorical equalizer in stacks of the diagram (\\ref{asdwdisdwqdqdqw}), which is of course equivalent to $X$. Note that the difficulty at this point is that the embedding of the category of spaces (viewed as a two-category)  into the two-category of stacks does not preserve general small colimits, as opposed to the case of limits.\n\nFor completeness we will give an argument.\nFirst note that ${\\tt pr}_{U_i}\\colon U_i\\times_X U_i\\stackrel{\\sim}{\\to} U_i$ is a homeomorphism. It thus follows from the groupoid structure of the coequalizer diagram that\n$U_i\\to B$ is injective for all $i$.  Since $\\bigsqcup_i U_i\\to B$\nis a topological quotient map it is open. Therefore  $\\bigsqcup_iU_i\\to B$ is a open covering.\nWe further conclude that the natural map\n$U_i\\times_X U_j\\to U_i\\times_BU_j$ is in fact a homeomorphism.\n\n\n\n\n\n\nThe claim is that $X$ is equivalent to $B$. We first construct a morphism $X\\to B$.\nLet $(T\\to X)\\in X(T)$. Then $(T_i:=T\\times_XU_i)_i$ is an open covering of $T$. \nUsing the identification $T_i\\times_TT_j\\cong T\\times_X(U_i\\times_X U_j)$ we get a diagram\n$$\\xymatrix{\\bigsqcup_{i,j}T_i\\times_TT_j\\ar@\/_-0.2cm\/[d]\\ar@\/^-0.2cm\/[d]\\ar[r]&U_i\\times_XU_j\\ar@\/_-0.2cm\/[d]\\ar@\/^-0.2cm\/[d]\\\\\\bigsqcup_i T_i\\ar[r]\\ar[d]&\\bigsqcup_i U_i\\ar[d]\\\\T\\ar@{.>}[r]&B}\\ ,$$\nwhere the horizontal maps are induced by the projections $T\\times_XU_i\\to U_i$, and\nthe left vertical is the representation of $T$ as a coequalizer.\nTherefore we obtain a unique factorization $(T\\to B)\\in B(T)$.\nThe construction is functorial in $T$ and therefore induces a morphism $X\\to B$. \n\nIn order to see that it has an inverse let $(T\\to B)\\in B(T)$ be given. Then we define the open covering\n$(T_i:=T\\times_BU_i)_i$ of $T$.\nThe compositions $$\\phi_i:T_i\\cong T\\times_B U_i\\stackrel{{\\tt pr}_{U_i}}{\\to} U_i\\to X$$ can be considered as a collection of objects $(\\phi_i\\in X(T_i))_i$.\nThe induced map\n\\begin{multline*}\n  T_i\\cap T_j\\cong T_i\\times_TT_j\\cong (T\\times_BU_i)\\times_T\n  (T\\times_BU_j)\\cong T\\times_B(U_i\\times_B U_j)\\\\\n\\stackrel{{\\tt pr}_{U_i\\times_B\n      U_j}}{\\to} U_i\\times_B U_j\\cong U_i\\times_X U_j\\to X\\times_XX\n\\end{multline*}\ncan be considered as a collection of isomorphisms $\\phi_{ij}:(\\phi_i)_{|T_i\\cap T_j}\\stackrel{\\sim}{\\to} (\\phi_j)_{|T_i\\cap T_j}$\n which satisfy the cocycle condition on triple intersections. Since $X$ is a stack we\n can therefore glue the local maps and get a map $(T\\to X)\\in X(T)$ which is unique\nup to unique isomorphism. This construction is again functorial in $T$ and provides \nthe map $B\\to X$.\n\nIt is easy to see that both maps $X\\to B$ and $B\\to X$ constructed above are mutually inverse.\n\\end{proof}\n\n \n\n\n\n\\subsection{}\\label{jhsgbwqjswqs}\n\n \nWe now show Proposition \\ref{lem:representability}\n\\begin{proof}   \nConsider a map $T\\to G$ from a space $T$.\n  We have to prove that the fiber product $T\\times_G X$ is\n  equivalent to a space. Using the assumption that $G$ is topological we choose \n  an atlas $A\\to G$ of $G$.\n  Because $A\\to G$ has local sections, we can find an open covering $\\bigsqcup_{i\\in I}U_i=: U\\to X$\n  such that $U\\times_G A\\to U$ \n  has a section $s\\colon U\\to U\\times_G A$.  We first want to show that $T\\times_G U$ is a space.\n  Since the structure map $A\\to G$ of an atlas is representable we know that $U\\times_GA$ and $T\\times_GA$ are spaces.\nTherefore,\n  $T\\times_GU\\times_G A\\cong  (T\\times_G A)\\times_A (U\\times_G A)$ is a space, too. The section\n  $s$ pulls back to a section $\\hat s:T\\times_GU\\to\n  T\\times_G U\\times_GA$ which implements $T\\times_G U$ as a subspace of\n  the space $T\\times_G U\\times_G A$.\n$$\\xymatrix{&T\\times_GU\\times_GA\\ar[rr]\\ar[dl]&&U\\times_GA\\ar[dl]\\ar[ddd]\\\\\nT\\times_GU\\ar@\/_-1cm\/[ur]^{\\hat s}\\ar[rr]\\ar[d]&&U\\ar[d]\\ar@\/_-1cm\/[ur]^s&\\\\\nT\\times_GX\\ar[rr]\\ar[dd]&&X\\ar[dd]&\\\\\n&&&A\\ar[dl]\\\\\nT\\ar[rr]&&G&}\\ .$$\nSince the map $U\\to X$ is surjective and locally an open embedding its pull-back\n$T\\times_GU\\to T\\times_GX$ is surjective and locally an open embedding, too.\nTherefore by  Lemma \\ref{kjhdkqwdqwwqd} the stack $T\\times_GX$\nis equivalent to a space. \n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{}\\label{hshsqiwhswqs}\n\nRecall that\na topological stack is called locally compact if it admits a locally compact atlas $A\\to G$ such that $A\\times_GA$ is a locally compact space.\nFurthermore recall that the site $\\mathbf{X}={\\tt Site}(X)$ associated to a locally compact stack $X$ is the full subcategory of locally compact spaces $U\\to X$ over $X$ such that the structure map has local sections. A morphism in this site $\\mathbf{X}$ is a diagram\n\\begin{equation}\\label{gsdhasddas}\n\\xymatrix{U\\ar[dr]\\ar[rr]&\\ar@{:>}[d]&V\\ar[dl]\\\\&X&}\n\\end{equation}\nconsisting of a morphism of spaces over $X$ and a two-morphism.\nThe topology on $\\mathbf{X}$ is given by the covering families of the objects $(U\\to X)$ induced by open covering of $U$.\n\nMuch of the general theory would work without the assumption of local compactness.\nBut local compactness is important in connection with the projection formula Lemma \\ref{system81}\nwhich is a crucial ingredient of the theory of integration. Since the latter is our main goal\nof the present section we generally adopt the restriction to locally compact stacks.\n\n\n\n\\subsection{}\n\nThe sheaf theory for topological stacks can be built in a parallel manner to the sheaf theory for smooth stacks developed in \\cite{bss}.\nThe transition goes via the following replacements of words:\n\\begin{enumerate}\n\\item For the definition of stacks the site of smooth manifolds ${\\tt Mf}^\\infty$ is replaced by the site of topological spaces ${\\tt Top}$. In the definition of the site of a locally compact stack manifolds are replaced by locally compact spaces.\n\\item The concept of a \\textit{smooth stack} is replaced by the concept of a  \\textit{locally compact  stack}.\n\\item The notion of a  \\textit{smooth map} is replaced by the notion of a \\textit{map which admits local sections}.\n\\end{enumerate}\n\n\nIn the present paper we freely use results in the general sheaf theory for topological stacks \nfrom \\cite[Sec.~2]{bss} in the case of stacks in topological spaces\nwhich are proved there for manifolds. It should be noted that with the\nconventions just made, all statements and proofs carry over verbatim\n \n\n\\subsection{}\\label{echejwwcc}\\label{lechejwwcc}\n\nLet $X$ be a locally compact stack. By\n $\\Pr\\mathbf{X}$ and ${\\tt Sh}\\mathbf{X}$ we denote the categories of presheaves and sheaves on $\\mathbf{X}$.\nThey are related by \n a pair of adjoint functors\n  $$i^\\sharp\\colon \\Pr\\mathbf{X}\\leftrightarrows{\\tt Sh}\\mathbf{X}:i\\ .$$\nThe sheafification functor $i^\\sharp$ is exact.\n \n   \n\n\\subsection{}\\label{lem:lrexact} \\label{sec:morph_of_sheaves}\n\nLet $f:X\\to Y$ be a morphism of locally compact stacks.\nIn induces a functor\n${}^pf_*:\\Pr\\mathbf{X}\\to \\Pr\\mathbf{Y}$ by\n$${}^pf_*F(V\\to Y):=\\lim F(U\\to X)\\ ,$$ where\nthe limit is taken over the category of diagrams\n\\begin{equation}\\label{dergetzwe}\n\\xymatrix{U\\ar[d]\\ar[r]&X\\ar[d]^f\\\\V\\ar[r]\\ar@{:>}[ur]&Y}\n\\end{equation}\nwith $(U\\to X)\\in \\mathbf{X}$. For details we refer to  \\cite[Sections 2.1, 2.2]{bss}.\nThis functor fits into an adjoint pair\n$${}^pf^*:\\Pr\\mathbf{Y}\\leftrightarrows\\Pr\\mathbf{X}:{}^pf_*\\ .$$\nThe functor \n${}^pf^*$ is given by\n$${}^pf^*G(U\\to X)=\\colim\\: G(V\\to Y)\\ ,$$\nwhere \nthe colimit is again taken over the category of diagrams\nwith $(V\\to Y)\\in \\mathbf{Y}$.\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n\nWe extend these functors to sheaves by\n$$f_*:=i^\\sharp\\circ  {}^pf_*\\circ i\\ ,\\quad f^*:=i^\\sharp\\circ {}^pf^*\\circ i$$\nand obtain an adjoint pair\n$$f^*:{\\tt Sh}\\mathbf{Y}\\leftrightarrows{\\tt Sh}\\mathbf{X}:f_*\\ .$$\nNote that\n${}^pf_*$ preserves sheaves (see \\cite[Lemma 2.13]{bss}).\nThe right-adjoint functor $f_*:{\\tt Sh}_{\\tt Ab}\\mathbf{X}\\to {\\tt Sh}_{\\tt Ab}\\mathbf{Y}$ is left exact and therefore admits a right-derived functor\n$$Rf_*:D^+({\\tt Sh}_{\\tt Ab}\\mathbf{X})\\to D^+({\\tt Sh}_{\\tt Ab}\\mathbf{Y})$$\nbetween the bounded below derived categories.\n\n\n\n\n\n\n\n\\subsection{}\n\nIf $g:Y\\to Z$ is a second morphism of locally compact stacks, then  we have natural isomorphisms\nof functors\n$$(g\\circ f)_*\\cong g_*\\circ f_*\\ ,\\quad f^*\\circ g^*\\cong (g\\circ f)^*$$\n(see \\ref{uefhewiufuwefzzz}). \nFurthermore, we have\n$$Rg_*\\circ Rf_*\\cong R(g\\circ f)_*$$ on the level of bounded below derived categories by Lemma\n\\ref{keykey}.\nThe relation $ f^*\\circ g^*\\cong (g\\circ f)^*$ descends to the derived categories if the pull-back functors are exact, e.g. if $f$ and $g$ have local sections (see \\ref{sharp}).\n These facts generalize corresponding results shown in \\cite{bss}.\n\n\n\n\n\n\n\\subsection{}\\label{sharp}\\label{system200}\\label{lem:map_on_pr_vers_map_on_sh} \\label{prexact}\\label{lem:identify_star_sharp}\nLet $f:G\\to H$ be a morphism between topological stacks which has local sections.\nIt induces a morphism between sites\n$f_\\sharp:\\mathbf{G}\\to \\mathbf{H}$ by composition. On objects it is given by\n$f_\\sharp(U\\to G):=(U\\to G\\to H)$\n(we will often use the short hand $U$ for $(U\\to G)$ and write $f_\\sharp U$).\nIn fact, since $U\\to \\mathbf{G}$ and $f$ have local sections, the composition $U\\to H$\nhas local sections. Furthermore, the map $U\\to H$ from a space to a topological\nstack is representable by Lemma \\ref{lem:representability}. \nOne checks that $f_\\sharp$ maps covering families to covering families and preserves the fiber products\nas in \\cite[1.2.2]{MR1317816}.\n\n\nIf $f:G\\to H$ has local sections, then \nthe functor\n$f^*:{\\tt Sh} \\mathbf{H}\\to {\\tt Sh} \\mathbf{G}$ is the pull-back  $f^*=(f_\\sharp)^*$ associated to a morphism of sites.  Explicitly it is given by \n$f^*F(U):=F(f_\\sharp U)$, compare Lemma \\cite[2.7]{bss}.\nIn  addition, the functor \n $f^*:{\\tt Sh}\\mathbf{H}\\to{\\tt Sh}\\mathbf{G}$ is exact (see \\cite[2.5.9]{bss})\nand preserves flat sheaves of abelian groups.\n\n\\begin{lem}\\label{derivedadj}\nIf  $f:X\\to Y$ is a morphism between locally compact stacks which has local sections, then we have the derived\nadjunction\n$$f^*\\colon D^+({\\tt Sh}_{\\tt Ab}\\mathbf{Y})\\leftrightarrows D^+({\\tt Sh}_{\\tt Ab}\\mathbf{X})\\colon Rf_*\\ .$$\n\\end{lem}\n\\begin{proof}\n Since $f^*$ is exact its right adjoint $f_*$ preserves injectives.\nIf $G\\in C^+({\\tt Sh}_{\\tt Ab}\\mathbf{X})$ is a complex of injectives and $F\\in C^+({\\tt Sh}_{\\tt Ab}\\mathbf{Y})$, then we have\n\\begin{multline*}\n  R{\\tt Hom}_{{\\tt Sh}_{\\tt Ab}\\mathbf{Y}}(F,Rf_*(G))\\cong {\\tt Hom}_{{\\tt Sh}_{\\tt Ab}\\mathbf{Y}}(F,f_*(G))\\\\\n\\cong\n  {\\tt Hom}_{{\\tt Sh}_{\\tt Ab}\\mathbf{X}}(f^*(F),G)\\cong R{\\tt Hom}_{{\\tt Sh}_{\\tt Ab}\\mathbf{X}}(f^*(F),G)\\ .$$\n\\end{multline*}\nThis implies the assertion. \\end{proof} \n\n\n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{} \n\n\\begin{lem}\\label{qwuidiuwqdwqdwqd}\nLet $X$ be a locally compact stack. If $C,B\\to X$ are maps from locally compact spaces, then  $C\\times_XB$ is locally compact.\n\\end{lem}\n\\begin{proof} By assumption $X$ is locally compact so that we can chose an atlas  $A\\to X$   such that\n$A$ and $A\\times_XA$ are locally compact. \nSince $A\\to X$ is surjective and has local sections, there\nexists an open covering $(B_i)$ of $B$ such that we have lifts\n$$\\xymatrix{&&A\\ar[d]\\\\B_i\\ar@{.>}[urr]\\ar[r]&B\\ar[r]&X}\\ .$$\nThen $(A\\times_X B_i)$ is an open covering of $A\\times_XB$.\nIn order to show that $A\\times_XB$ is locally compact it suffices\nto show that the space $A\\times_XB_i$ is locally compact. By \n$A\\times_XB_i\\cong (A\\times_XA)\\times_AB_i\\subseteq A\\times_XA\\times B_i$, this space is a closed\n(note that $A$ is Hausdorff) subspace of a locally compact space and hence itself locally compact.\n\nThe same argument shows that $C\\times_XA$ is locally compact.\nWe now write\n$C\\times_XB_i\\cong (C\\times_XA)\\times_A B_i\\subseteq  (C\\times_XA) \\times B_i$\nin order to see that $C\\times_XB_i$ is locally compact. Since\n$(C\\times_XB_i)$ is an open covering of $C\\times_XB$ we conclude that\n$C\\times_XB$ is locally compact.\n\\end{proof}\n \n\n \n\n\n\n\n\\subsection{}\n\nLet $f:X\\to Y$ be a morphism between locally compact stacks.\n\\begin{lem}\\label{obensharp}\\label{iwdoi2d23d}\nIf $f$ is representable,  then it induces a morphism\nof sites $f^\\sharp:\\mathbf{Y}\\to \\mathbf{X}$ given by $f^\\sharp(V\\to Y):=(X\\times_YV\\to X)$.\n\\end{lem}\n\\begin{proof}\nLet $B\\to X$ be a locally compact atlas.\nWe consider $(V\\to Y)\\in \\mathbf{Y}$ and form the   diagram of Cartesian squares  $$\\xymatrix{V\\times_YB\\ar[r]\\ar[d]&B\\ar[d]\\\\U\\ar[r]\\ar[d]&X\\ar[d]^f\\\\\nV\\ar[r]&Y}\\ .$$\nIn order to check that $(U\\to X)\\in \\mathbf{X}$ we must show that $U$ is locally compact. \nSince $B\\to X$ is surjective and has local sections we see  that\n $V\\times_YB\\to U$ is surjective and has local sections, too.\nSince $Y$ is locally compact we see by Lemma \\ref{qwuidiuwqdwqdwqd} that\n$V\\times_YB$ is locally compact.\nLet $u\\in U$ and $W\\subseteq U$ be a neighborhood of $u$ such that there exists a section \n$$\\xymatrix{&V\\times_YB\\ar[d]^\\pi\\\\W\\ar@{.>}[ur]^s\\ar[r]&U}\\ .$$\nLet $K\\subseteq \\pi^{-1}(W)$ be a compact neighborhood of $s(u)$.\nThen $s^{-1}(K)$ is a compact neighborhood of $u$.\nIndeed, $s^{-1}(K)$ is a closed subset of the compact set $\\pi(K)$.\n\nIt is easy to see  that $f^\\sharp$ maps covering families to covering families and preserves the fiber products required for a morphism of sites, see\n \\cite[1.2.2]{MR1317816}.\n\\end{proof}\n\n\n \n\n\n \n\n\n\nIf $f:X\\to Y$ is a representable morphism between locally compact stacks, then\n we have the relations $f^*=(f^\\sharp)_*:{\\tt Sh}\\mathbf{Y}\\to {\\tt Sh}\\mathbf{X}$\nand $f_*=(f^\\sharp)^*:{\\tt Sh} \\mathbf{X}\\to {\\tt Sh}\\mathbf{Y}$\n, see \\cite[Lemma 2.9]{bss}. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n\\subsection{}\\label{obstr1a}\\label{desc_sheaves_on_U}\n\n \nLet $X$ be a topological stack and $(U\\to X)\\in \\mathbf{X}$. Let $(U)$ denote the site whose objects \nand morphisms are the open subsets of $U$ and inclusions, and whose coverings are coverings by families of open subsets.\nWe have restriction functors\n$\\nu_U:{\\tt Sh}\\mathbf{X}  \\to {\\tt Sh}(U)$ and \n${}^p\\nu_U:\\Pr\\mathbf{X}\\to \\Pr(U)$. For $F\\in {\\tt Sh}\\mathbf{X}$ we also  write $\\nu_U(F)=:F_U$. We have the following assertions,\nmost of which are straightforward to prove. \n\n\\begin{enumerate}\n\\item Let $i^\\sharp$ and $i^\\sharp_U$ denote the sheafification functors on the sites $\\mathbf{X}$ and $(U)$. Then we have a natural isomorphism $$i_U^\\sharp\\circ {}^p\\nu_U\\cong \\nu_U\\circ i^\\sharp\\ ,$$ see\n\\cite[Lemma 2.4.7]{bss} \n\\item Let $F\\in {\\tt Sh}\\mathbf{X}  $. If $f\\colon U\\to V$ is a morphism (\\ref{gsdhasddas}) \nin $\\mathbf{X}  $, then we have a natural map $f^*F_V\\to F_U$.\n\\item There is a one-to one correspondence of sheaves $F\\in {\\tt Sh}\\mathbf{X}  $ on the one hand,  and of collections $(F_U)_{(U\\to X)\\in \\mathbf{X}  }$ of sheaves $F_U\\in {\\tt Sh}(U)$ together with functorial maps $f^*F_V\\to F_U$ for all morphisms $f\\colon U\\to V$ in $\\mathbf{X}$ on the other hand.\n\\item Let $F,G\\in {\\tt Sh}\\mathbf{X}  $. There is a one-to-one correspondence between compatible collections of morphisms $g_U\\colon F_U\\to G_U$ for  \nall $(U\\to X)\\in \\mathbf{X} $ and maps $g\\colon F\\to G$.\n\\item If $F,G\\in {\\tt Sh}\\mathbf{X}  $ or\n$F,G\\in D^+({\\tt Sh}_{\\tt Ab}\\mathbf{X}  )$, then\na map $F\\to G$ is an isomorphism if and only if the induced map $F_U\\to G_U$ is an isomorphism for all\n$(U\\to X)\\in \\mathbf{X}  $.\n\\item Let $f\\colon X\\to Y$ be a representable map of locally compact stacks, $(A\\to\n  Y)\\in \\mathbf{Y}  $  and $(B:=A\\times_YX\\to X)\\in \\mathbf{X}$.  Let $g\\colon B\\to A$ be the projection onto the first factor and $g_*:{\\tt Sh} (B)\\to {\\tt Sh}(A)$. Then\n  we have for $F\\in {\\tt Sh}\\mathbf{X}  $ or $G\\in D^+({\\tt Sh}_{\\tt Ab}\\mathbf{X})$\n$$(f_*F)_A\\cong g_*(F_B)\\ ,\\quad (Rf_*G)_A\\cong Rg_*(G_B)\\ .$$\nThe second isomorphism follows from the first using the fact that the restriction $\\nu_B$ preserves flabby or even injective sheaves (see Lemma \\ref{hwfwefewfw}).\n\\item If $f\\colon X\\to Y$ is a map of topological stacks which has local sections, $(B\\to X)\\in \\mathbf{X}  $, then we have $(B\\to X\\to Y)\\in \\mathbf{Y}  $ and for $F\\in{\\tt Sh}\\mathbf{Y}  $ \n$$(f^*F)_B\\cong F_B\\ .$$ \n\\item\nThe collection of restriction functors $(\\nu_U)_{(U\\to X)\\in \\mathbf{X}}$ detects flabby (flasque, flat) sheaves (see Definition \\ref{def:flabby}), i.e.\na sheaf $F\\in {\\tt Sh}_{\\tt Ab}\\mathbf{X}$ is flabby (flasque, flat)  if and only if $F_U\\in {\\tt Sh}_{\\tt Ab}(U)$ is flabby for all $(U\\to X) \\in \\mathbf{X}$ (compare \\ref{flatdetect} for the flat case). \n\n\\item The collection of restriction functors $(\\nu_U)_{(U\\to X)\\in \\mathbf{X}}$ detects exact sequences, i.e.\na sequence $F\\to G\\to H$ of sheaves of abelian groups on $\\mathbf{X}$ is exact if and only if\n$F_U\\to G_U\\to H_U$ is exact for all $(U\\to X)\\in \\mathbf{X}$. \n\\end{enumerate}\n\n\\begin{lem}\\label{hwfwefewfw}\nLet $(U\\to X)\\in \\mathbf{X}$.\nThe functor $\\nu_U:{\\tt Sh}_{\\tt Ab}\\mathbf{X}\\to {\\tt Sh}_{\\tt Ab}(U)$ preserves injective sheaves.\n\\end{lem}\n\\begin{proof}\nWe show that $\\nu_U$ has an exact  left adjoint $\\nu_\\mathbb{Z}^U:{\\tt Sh}_{\\tt Ab}(U)\\to {\\tt Sh}_{\\tt Ab}\\mathbf{X}$. \nWe first show that the restriction functor ${}^p\\nu_U:\\Pr_{\\tt Ab}\\mathbf{X}\\to \\Pr_{\\tt Ab}(U)$ fits into an adjoint pair\n$${}^p\\nu_\\mathbb{Z}^U:\\Pr_{\\tt Ab}(U)\\leftrightarrows \\Pr_{\\tt Ab}\\mathbf{X}:{}^p\\nu_U\\ .$$\nThe left-adjoint is given by\n $${}^p\\nu_\\mathbb{Z}^U(F)(A\\to X):= \\colim F(V)\\ ,$$\nwhere the colimit is taken over the category of diagrams\n$$\\xymatrix{V\\ar[d]&A\\ar@{.>}[dl]^\\phi\\ar[d]\\ar[l]\\\\U\\ar [r]&X}\\ ,$$\nwhere $V\\to U$ is the embedding of an open subset.\nAs explained in \\cite[II.3.18]{MR559531} we have a decomposition of this category into a union of  categories $S(\\phi)$ with  $\\phi\\in {\\tt Hom}_\\mathbf{X}((A\\to X),(U\\to X))$. The category $S(\\phi)$ is the category of open neighborhoods of $\\phi(A)$ and their inclusions.  It is cofiltered.\nTherefore  $F\\mapsto \\colim_{S(\\phi)} F(V)$ preserves finite limits and is in particular left exact.\n This implies that  \n${}^p\\nu_\\mathbb{Z}^U$ \ngiven by\n$${}^p\\nu_\\mathbb{Z}^U(F)(A\\to X)\\cong \\bigoplus_{\\phi} \\colim_{S(\\phi)} F(V)  $$\n is left-exact, too.\nWe now get $\\nu^U_\\mathbb{Z}:=i^\\sharp\\circ {}^p\\nu^U_\\mathbb{Z}\\circ i_U$.\nAs a left-adjoint it is right-exact. Since $i_U$ is left exact and $i^\\sharp$ is exact, this composition is also left-exact.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{}\n\n\n\\begin{lemma}\\label{lem:pullpush}\n  Consider the following Cartesian diagram in locally compact topological stacks\n  \\begin{equation*}\n    \\begin{CD}\n     H @>v>> G\\\\\n      @VVgV @VVfV\\\\\n      Y @>u>> X     \\end{CD}\n  \\end{equation*}\n  In this situation the two canonical ways to define a natural transformation\n$$u^*f_*\\to g_*v^*\\colon {\\tt Sh}_{\\tt Ab}(\\mathbf{G})\\to {\\tt Sh}_{\\tt Ab}(\\mathbf{Y})$$ \ngive the same result, i.e. the diagram\n\\begin{equation}\\label{hfiewufuwef}\n\\xymatrix{u^*f_*\\ar@{=}[d]\\ar[r]^{unit}&g_*g^*u^*f_*\\ar[r]^{u g=f v}&g_*v^*f^*f_*\\ar[r]^{\\hspace{0.1cm}counit}&g_*v^*\\ar@{=}[d]\\\\u^*f_*\\ar[r]^{unit}&u^*f_*v_*v^*\\ar[r]^{u  g=f v}&u^*u_*g_*v^*\\ar[r]^{\\hspace{0.2cm}counit}&g_*v^*}\n\\end{equation}\ncommutes. \nThis transformation is functorial with respect to composition of Cartesian diagrams.\n\n\nMoreover, \nif $u$ has local sections, then this transformation induces isomorphisms\n  \\begin{gather}\n    u^*f_*\\cong g_*v^*\\colon {\\tt Sh}_{\\tt Ab}(\\mathbf{G})\\to {\\tt Sh}_{\\tt Ab}(\\mathbf{Y}),\\label{fewfwfwefewf}\\\\\n    u^*Rf_* \\cong Rg_* v^* \\colon D^+{\\tt Sh}_{\\tt Ab}(\\mathbf{G})\\to D^+{\\tt Sh}_{\\tt Ab}(\\mathbf{Y}).\\label{wqdwqdqw}\n  \\end{gather}\nIf $u$ and $f$ have local sections, then we get \n  commutative diagrams\n $$\n\\xymatrix{&u_*\\ar[rd]^{unit}\\ar[dl]^{unit}&\\\\\nu_*g_*g^*\\ar[r]^\\sim&f_*v_*g^*&f_*f^*u_*\\ar[l]_\\sim}, \\xymatrix{&v_* &\\\\\nf^*f_*v_*\\ar[ur]^{counit}\\ar[r]^\\sim&f^*u_*g_*\\ar[r]^\\sim&v_*g^*g_*\\ar[ul]^{counit}}$$\n    \n$$\n\\xymatrix{&u^*\\ar[rd]^{unit}\\ar[dl]^{unit}&\\\\\nu^*f_*f^*\\ar[r]^\\sim&g_*v^*f^*&g_*g^*u^*\\ar[l]_\\sim}, \\xymatrix{&v^* &\\\\\nv^*f^*f_*\\ar[ur]^{counit}\\ar[r]^\\sim&g^*u^*f_*\\ar[r]^\\sim&g^*g_*v^*\\ar[ul]^{counit}}$$\n   and\ntheir derived versions, e.g.\n\n\\begin{equation}\\label{ddwqzwqidqwdww}\n \\xymatrix{&u^*\\ar[rd]^{unit}\\ar[dl]^{unit}&\\\\\nu^*Rf_*f^*\\ar[r]^\\sim&Rg_*v^*f^*&Rg_*g^*u^*\\ar[l]_\\sim}\\ ,\n  \\end{equation}\nand also\n   \\begin{equation}\\label{wdqidwqdwdwdewdwedwed}\n\\xymatrix{&&Ru_*u^*\\ar[drr]^{unit}\\ar[dll]_{unit}&&\\\\\nRu_*u^*Rf_*f^*\\ar[r]^\\sim&Ru_*Rg_*v^*f^*\\ar[r]^\\sim&Rf_*Rv_*v^*f^*\\ar[r]^\\sim&Rf_*Rv_*g^*u^*&Rf_*f^*Ru_*u^*\\ar[l]^\\sim}  \n   \\end{equation}\n\\end{lemma}\n\\begin{proof}\n\\textit{Most of the following arguments and the large diagrams  were supplied by \\textbf{A. Schneider}.}\nFor convenience we present a proof of (\\ref{hfiewufuwef}), see also \\cite[Expose XVII, Proposition 2.1.3]{deligne}. \nWe first observe that\n\\begin{equation}\\label{jdewkdwed}\n\\xymatrix{\nv^*f^*f_*v_*\\ar[r]^{counit}\\ar[d]^\\sim&v^*v_*\\ar[d]_{counit}\\\\\n(fv)^*(fv)_*\\ar[r]^{counit}&{\\tt id}\n}\n\\end{equation}\ncommutes. Using this in addition to standard functorial properties \nwe check that all squares in the following diagram commute:\n$$\\hspace{-0.8cm}\n \\xymatrix{\nu^*f_*\\ar[r]^{unit}\\ar@{=}[ddddd]&\ng_*g^* u^*f_*\\ar[r]^\\sim\\ar[d]^{unit}&g_*(ug)^*f_*\\ar[r]^=\\ar[d]^{unit}&\ng_*(fv)^*f_*\\ar[r]^\\sim\\ar[d]^{unit}&g_*v^* f^*f_*\\ar[r]^{counit}\\ar[d]^{unit}&\ng_*v^* \\ar[d]^{unit}\\ar@\/^1cm\/[dd]^{\\rm id}\\\\\n&\ng_*g^* u^*f_*v_*v^*\\ar[r]^\\sim\\ar@{=}[d]&g_*(ug)^*f_*v_*v^*\\ar[r]^=\\ar[d]^\\sim&\ng_*(fv)^*f_*v_*v^*\\ar[r]^\\sim\\ar[d]^\\sim&g_*v^* f^*f_*v_*v^*\\ar[r]^{counit}\\ar[d]^\\sim&\ng_*v^* v_*v^*\\ar[d]^{counit}\\\\\n&\ng_*g^* u^*f_*v_*v^*\\ar[r]^\\sim&g_*(ug)^*(fv)_*v^*\\ar[r]^=&\ng_*(fv)^*(fv)_*v^*\\ar@{=}[r]&g_* (fv)^*(fv)_*v^*\\ar[r]^{counit}&g_*v^*\\\\\n&\ng_*g^* u^*f_*v_*v^*\\ar[r]^\\sim\\ar@{=}[u]&g_*(ug)^*(fv)_*v^*\\ar[r]^=\\ar@{=}[u]&\ng_*(ug)^*(ug)_*v^*\\ar@{=}[r]\\ar[u]_=&g_* (ug)^*(ug)_*v^*\\ar[r]^{counit}\\ar[u]_=&g_*v^*\\ar@{=}[u]\\\\\n&\ng_*g^* u^*f_*v_*v^*\\ar[r]^\\sim\\ar@{=}[u]&g_*g^*u^*(fv)_*v^*\\ar[r]^=\\ar[u]_\\sim&\ng_*g^*u^*(ug)_*v^*\\ar[r]^\\sim\\ar[u]_\\sim&g_*g^* u^*u_*g^*v^*\\ar[r]^{counit}\\ar[u]_\\sim&\ng_*g^* g_*v^*\\ar[u]_{counit}\\\\\nu^*f_*\\ar[r]^{unit}&\nu^*f_*v_*v^*\\ar[r]^\\sim\\ar[u]_{unit}&u^*(fv)_*v^*\\ar[r]^=\\ar[u]_{unit}&\nu^*(ug)_*v^*\\ar[r]^\\sim\\ar[u]_{unit}&u^*u_*g^*v^*\\ar[r]^{counit}\\ar[u]_{unit}&\ng_*v^*\\ar[u]_{unit}\\ar@\/_1cm\/[uu]_{\\rm id}.\n}\n$$ \nThe two ways to go along the boundary from the upper left to lower right corner give the two maps $u^*f_*\\to g_*v^*$ in question.\n\n\nThe isomorphism (\\ref{fewfwfwefewf}) can be shown as in \\cite[Lemma 2.16]{bss}, where the\nassumption of smoothness of $u$ in  \\cite{bss} corresponds to the assumption of local sections in the present setting. The derived version (\\ref{wqdwqdqw}) can be shown using the simplicial models as in \\cite [Lemma 2.43]{bss}.\nAlternatively one can use the commutativity of the diagram asserted in Lemma \\ref{uiqehewqdqwdwqdqd}\nand the isomorphism (\\ref{uiidwqdwqdwqd}).\n\nWe now show the compatibility of the units and counits with Cartesian diagrams. The arguments are purely formal and only use\nthat the functors involved occur as parts of adjoint pairs. We will only give the details for the two triangles involving derived functors. \n If in addition to $u$ also  $f$ has local sections, then so has $g$. In this case we have the adjoint pairs $(f^*,Rf_*)$ and $(g^*,Rg_*)$.\nIn order to see (\\ref{ddwqzwqidqwdww}) we must show that  \n$$\n\\xymatrix{\nu^*\\ar[r]^{unit}\\ar@\/_1cm\/[rrrrr]_{unit}&\nu^*Rf_*f^*\\ar[r]^{\\Psi}&Rg_*v^*f^*\\ar[r]^\\sim&Rg_*(fv)^*\\ar[r]^=&\nRg_*(ug)^*\\ar[r]^\\sim&Rg_*g^*u^*,\n}\n$$\ncommutes, \nwhere $\\Psi:u^*Rf_*f^*\\to Rg_*v^*f^*$ is induced by (\\ref{wqdwqdqw}).\nThis is a consequence of the  commutativity of \n$$\n\\xymatrix{\nu^*\\ar[r]^{unit}&\nu^*Rf_*f^*\\ar[d]^{unit}\\ar[rrr]^{\\Psi}&&&\nRg_*v^*f^* \\ar@\/^2cm\/[dd]\\\\\n&\nRg_*g^* u^*f_*f^*\\ar[r]^\\sim&Rg_*(ug)^*Rf_*f^*\\ar[r]^=&\nRg_*(fv)^*Rf_*f^*\\ar[r]^\\sim&Rg_*v^* f^*Rf_*f^*\\ar[u]_{counit}\\\\\nu^*\\ar[r]^{unit}\\ar@{=}[uu]&\nRg_*g^* u^*\\ar[r]^\\sim\\ar[u]_{unit}&Rg_*(ug)^*\\ar[u]_{ unit}\\ar[r]^=\\ar@\/^0.5cm\/[l]&\nRg_*(fv)^*\\ar[u]_{unit}\\ar[r]^\\sim\\ar@\/^0.5cm\/[l]&Rg_*v^* f^*\\ar@\/^0.5cm\/[l]\\ar[u]_{ unit}\\ar@\/_1cm\/[uu]_{\\rm id}\n}\n$$\nwhich follows from standard functorial properties of units and counits.\n\nThe same properties are used in the proof of (\\ref{wdqidwqdwdwdewdwedwed}) which is\nrepresented by the boundary of the following big array of small commutative squares and triangles\n\\newpage\n\\pagestyle{empty\n\\hspace{13cm}\n{\\tiny \\begin{rotate}{270}\n$\n\\hspace{-2cm}\n\\xymatrix{\n&Rf_*f^*u_*u^*\\ar[r]^{unit}\\ar@\/^0.5cm\/[rrrrr]^\\Phi\\ar[d]^{unit}\t&Rf_*f^*u_*Rg_*g^*u^*\\ar[r]^\\sim\\ar[d]^{unit}\t\t&Rf_*f^*R(ug)_*g^*u^*\\ar[r]^=\t\\ar[d]^{unit}\t\t&Rf_*f^*R(fv)_*g^*u^*\\ar[r]^\\sim\\ar[d]^{unit}\t\t&Rf_*f^*Rf_*v_*g^*u^*\\ar[r]^{counit}\\ar[d]^{unit}\t\t&Rf_*v_*g^*u^*\\ar@{=}[rd]\\ar[d]^{unit}&\\\\\n&Rf_*f^*u_*u^*Rf_*f^*\\ar[r]^{unit}\\ar@{=}[d]&Rf_*f^*u_*Rg_*g^*u^*Rf_*f^*\\ar[r]^\\sim\\ar@{=}[d]\t&Rf_*f^*R(ug)_*g^*u^*Rf_*f^*\\ar[r]^=\t\\ar[d]^\\sim\t&Rf_*f^*R(fv)_*g^*u^*Rf_*f^*\\ar[r]^\\sim\\ar[d]^\\sim\t&Rf_*f^*Rf_*v_*g^*u^*Rf_*f^*\\ar[r]^{counit}\\ar[d]^\\sim\t&Rf_*v_*g^*u^*Rf_*f^*\\ar[d]^\\sim\n&Rf_*v_*g^*u^*\\ar[d]^\\sim\\ar[l]_{unit}\\\\\n&Rf_*f^*u_*u^*Rf_*f^*\\ar[r]^{unit}\\ar@{=}[d]&Rf_*f^*u_*Rg_*g^*u^*Rf_*f^*\\ar[r]^\\sim\\ar@{=}[d]\t&Rf_*f^*R(ug)_*(ug)^*Rf_*f^*\\ar[r]^=\t\\ar@{=}[d]\t\t&Rf_*f^*R(fv)_*(ug)^*Rf_*f^*\\ar[r]^\\sim\\ar[d]^=\t\t&Rf_*f^*Rf_*v_*(ug)^*Rf_*f^*\\ar[r]^{counit}\\ar[d]^=\t\t&Rf_*v_*(ug)^*Rf_*f^*\\ar[d]^=\n&Rf_*v_*(ug)^*\\ar[d]^=\\ar[l]_{unit}\\\\\n&Rf_*f^*u_*u^*Rf_*f^*\\ar[r]^{unit}\\ar@{=}[d]&Rf_*f^*u_*Rg_*g^*u^*Rf_*f^*\\ar[r]^\\sim\\ar@{=}[d]\t&Rf_*f^*R(ug)_*(ug)^*Rf_*f^*\\ar[r]^=\\ar@{=}[d]\t\t&Rf_*f^*R(fv)_*(fv)^*Rf_*f^*\\ar[r]^\\sim\\ar@{=}[d]\t&Rf_*f^*Rf_*v_*(fv)^*Rf_*f^*\\ar[r]^{counit}\\ar[d]^\\sim\t&Rf_*v_*(fv)^*Rf_*f^*\\ar[d]^\\sim\n&Rf_*v_*(fv)^*\\ar[l]_{unit}\\ar[d]^\\sim\\\\\n&Rf_*f^*u_*u^*Rf_*f^*\\ar[r]^{unit}\\ar@{=}[dd]&Rf_*f^*u_*Rg_*g^*u^*Rf_*f^*\\ar[r]^\\sim\\ar@{=}[dd]\t&Rf_*f^*R(ug)_*(ug)^*Rf_*f^*\\ar[r]^=\\ar@{=}[dd]\t\t&Rf_*f^*R(fv)_*(fv)^*Rf_*f^*\\ar[r]^\\sim\\ar@{=}[dd]\t&Rf_*f^*Rf_*v_*v^*f^*Rf_*f^*\\ar[r]^{counit}\\ar@{=}[dd]\t&Rf_*v_*v^*f^*Rf_*f^*\\ar[dr]^{counit}\n&Rf_*v_*v^*f^*\\ar[l]_{unit}\\ar[d]^{\\rm id}\\\\\nu_*u^*\\ar@\/^0.7cm\/[uuuuur]^{unit}\\ar@\/_0.7cm\/[dddddr]_{unit}&&&&&&&Rf_*v_*v^*f^*\\\\\n&Rf_*f^*u_*u^*Rf_*f^*\\ar[r]^{unit}\\ar@{=}[d]&Rf_*f^*u_*Rg_*g^*u^*Rf_*f^*\\ar[r]^\\sim\\ar@{=}[d]\t&Rf_*f^*R(ug)_*(ug)^*Rf_*f^*\\ar[r]^=\\ar@{=}[d]\t\t&Rf_*f^*R(fv)_*(fv)^*Rf_*f^*\\ar[r]^\\sim\\ar@{=}[d]\t&Rf_*f^*Rf_*v_*v^*f^*Rf_*f^*\\ar[r]^{counit}\t\t\t&Rf_*f^*Rf_*v_*v^*f^*\\ar[ur]^{counit}\n&Rf_*v_*v^*f^*\\ar[l]_{unit}\\ar[u]_{\\rm id}\\\\\n&Rf_*f^*u_*u^*Rf_*f^*\\ar[r]^{unit}\\ar@{=}[d]&Rf_*f^*u_*Rg_*g^*u^*Rf_*f^*\\ar[r]^\\sim\\ar@{=}[d]\t&Rf_*f^*R(ug)_*(ug)^*Rf_*f^*\\ar[r]^=\\ar@{=}[d]\t\t&Rf_*f^*R(fv)_*(fv)^*Rf_*f^*\\ar[r]^\\sim\t\t\t&Rf_*f^*R(fv)_*v^*f^*Rf_*f^*\\ar[r]^{counit}\\ar[u]_\\sim\t&Rf_*f^*R(fv)_*v^*f^*\\ar[u]_\\sim\n&R(fv)_*v^*f^*\\ar[u]_\\sim\\ar[l]_{unit}\\\\\n&Rf_*f^*u_*u^*Rf_*f^*\\ar[r]^{unit}\\ar@{=}[d]&Rf_*f^*u_*Rg_*g^*u^*Rf_*f^*\\ar[r]^\\sim\\ar@{=}[d]\t&Rf_*f^*R(ug)_*(ug)^*Rf_*f^*\\ar[r]^=\t\t\t\t&Rf_*f^*R(ug)_*(fv)^*Rf_*f^*\\ar[r]^\\sim\\ar[u]_=\t\t&Rf_*f^*R(ug)_*v^*f^*Rf_*f^*\\ar[r]^{counit}\\ar[u]_=\t\t&Rf_*f^*R(ug)_*v^*f^*\\ar[u]_=\n&R(ug)_*v^*f^*\\ar[u]_=\\ar[l]_{unit}\\\\\n&Rf_*f^*u_*u^*Rf_*f^*\\ar[r]^{unit}\t\t&Rf_*f^*u_*Rg_*g^*u^*Rf_*f^*\\ar[r]^\\sim\t\t\t&Rf_*f^*u_*Rg_*(ug)^*Rf_*f^*\\ar[r]^=\\ar[u]_\\sim\t&Rf_*f^*u_*Rg_*(fv)^*Rf_*f^*\\ar[r]^\\sim\\ar[u]_\\sim\t&Rf_*f^*u_*Rg_*v^*f^*Rf_*f^*\\ar[r]^{counit}\\ar[u]_\\sim\t&Rf_*f^*u_*Rg_*v^*f^*\\ar[u]_\\sim\n&u_*Rg_*v^*f^*\\ar[u]_\\sim\\ar[l]_{unit}\\\\\n&u_*u^*Rf_*f^*\\ar[r]^{unit}\\ar@\/_0.5cm\/[rrrrr]^\\Psi\\ar[u]_{unit}\t&u_*Rg_*g^*u^*Rf_*f^*\\ar[r]^\\sim\t\\ar[u]_{unit}\t&u_*Rg_*(ug)^*Rf_*f^*\\ar[r]^=\\ar[u]_{unit}\t\t&u_*Rg_*(fv)^*Rf_*f^*\\ar[r]^\\sim\\ar[u]_{unit}\t\t&u_*Rg_*v^*f^*Rf_*f^*\\ar[r]^{counit}\\ar[u]_{unit}\t\t\n&u_*Rg_*v^*f^*\\ar@{=}[ru]\\ar[u]_{unit}&\\\\\n}\n$\n\\end{rotate}}\n\\newpage\n\\pagestyle{plain}\n\n\n \n \n\n\n\n\n\n\n  \n\\end{proof}\n\n\n\n\n\n\n\n\n \n\\section{Tensor products and the projection formula}\n\n\\subsection{}\n \n\nWe consider a Grothendieck site $\\mathbf{X}$ and a commutative ring $R$. \nThe goal of the present Subsection is to discuss aspects of the closed monoidal structures on\nthe categories of presheaves $\\Pr_{R-{\\tt Mod}}\\mathbf{X}$ and sheaves ${\\tt Sh}_{R-{\\tt Mod}}\\mathbf{X}$ of $R$-modules on $\\mathbf{X}$. The material is standard, but we need to understand in detail the relation between the sheaf and presheaf versions in order to show the compatibility with the operations\ninduced by a morphism of stacks.\n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{}\n\n\nLet $F,G\\in \\Pr_{R-{\\tt Mod}}\\mathbf{X}$ be presheaves of $R$-modules.\nThe tensor product $F\\otimes^p G\\in \\Pr_{R-{\\tt Mod}}\\mathbf{X}$ is defined as the presheaf which associates to $(U\\to X)$ the $R$-module\n$F(U)\\otimes^p_R G(U)$. In this way $\\Pr_{R-{\\tt Mod}}\\mathbf{X}$ becomes a symmetric monoidal category.\n\nSince colimits of presheaves are defined objectwise\nwe have for a diagram of presheaves of $R$-modules $(F_i)_{i\\in I}$ that\n$$\\colim_{i\\in I} (F_i\\otimes^p_RG)\\cong (\\colim_{i\\in I}F_i)\\otimes^p_RG\\ .$$\n\n\n\n\n\n\\newcommand{\\underline{\\tt Hom}}{\\underline{\\tt Hom}}\n\\subsection{}\\label{system202}\n\nFor $U\\in \\mathbf{X}$ let $h_U\\in \\Pr\\mathbf{X}$ denote the presheaf represented by $U$ and\n$h_U^R\\in \\Pr_{R-{\\tt Mod}}\\mathbf{X}$ be the presheaf of $R$-modules generated by $h_U$.\nLet $F,G\\in \\Pr_{R-{\\tt Mod}}\\mathbf{X}$. We define the presheaf\n$$\\underline{\\tt Hom}^p(F,G)\\in {\\Pr}_{R-{\\tt Mod}}\\mathbf{X}$$ by\n$$\\underline{\\tt Hom}^p(F,G)(U):={\\tt Hom}_{\\Pr_{R-{\\tt Mod}}\\mathbf{X}}(h_U^R\\otimes^p F,G)\\ .$$\n\nThe topology of the site of a locally compact stack is sub-canonical. Hence, in this case $h_U$ is actually a sheaf.\nBut even in the case of a sub-canonical topology $h^R_U$ is only a presheaf, in general.\n\n\nIf $U\\to V$ is a morphism in $\\mathbf{X}$, then \n$\\underline{\\tt Hom}^p(F,G)(V)\\to \\underline{\\tt Hom}^p(F,G)(U)$ is induced by the morphism $h_U\\to h_V$.\nIf $H\\in  \\Pr_{R-{\\tt Mod}}\\mathbf{X}$, then we have\n\\begin{eqnarray*}\n{\\tt Hom}_{ \\Pr_{R-{\\tt Mod}}\\mathbf{X}}(H,\\underline{\\tt Hom}^p(F,G))&\\cong&{\\tt Hom}_{ \\Pr_{R-{\\tt Mod}}\\mathbf{X}}(\\colim_{h^R_V\\to H}h_V^R, \\underline{\\tt Hom}^p(F,G))\\\\\n&\\cong&\n\\lim_{h^R_V\\to H}{\\tt Hom}_{ \\Pr_{R-{\\tt Mod}}\\mathbf{X}}(h_V^R, \\underline{\\tt Hom}^p(F,G))\\\\\n&\\cong&\\lim_{h^R_V\\to H} \\underline{\\tt Hom}^p(F,G)(V)\\\\\n&=&\\lim_{h^R_V\\to H} {\\tt Hom}_{ \\Pr_{R-{\\tt Mod}}\\mathbf{X}}(h_V^R \\otimes^p F,G)\\\\\n&\\cong&{\\tt Hom}_{ \\Pr_{R-{\\tt Mod}}\\mathbf{X}}(\\colim_{h^R_V\\to H} (h_V^R \\otimes^p F),G)\\\\\n&\\cong&{\\tt Hom}_{ \\Pr_{R-{\\tt Mod}}\\mathbf{X}}((\\colim_{h^R_V\\to H} h_V^R) \\otimes^p F,G)\\\\\n&\\cong&{\\tt Hom}_{ \\Pr_{R-{\\tt Mod}}\\mathbf{X}}(H\\otimes^p F,G)\\end{eqnarray*}\nIn other words, the pair  $(\\otimes^p,\\underline{\\tt Hom}^p)$ together with this natural isomorphism defines a closed symmetric monoidal structure on $\\Pr_{R-{\\tt Mod}}\\mathbf{X}$.\nIn particular, if $(F_i)_{i\\in I}$ is a diagram of presheaves, then we have\n\\begin{equation}\\label{mb12}\\underline{\\tt Hom}^p(\\colim_{i\\in I} F_i,G)\\cong \\lim_{i\\in I}\\underline{\\tt Hom}^p(F_i,G)\\ .\\end{equation} \n\n\\subsection{}\n\nAn element of \n $$\\underline{\\tt Hom}(F,G)(U)={\\tt Hom}_{ \\Pr_{R-{\\tt Mod}}\\mathbf{X}}(h_U^R\\otimes^p F,G) $$\nis given by a collection of $R$-linear maps $(\\phi_{V\\to U}:F(V)\\to G(V))_{(V\\to U)\\in \\mathbf{X}\/U}$ such that for a morphism\n$(W\\to U)\\mapsto (V\\to U)$ in $\\mathbf{X}\/U$ the diagram\n$$\\xymatrix{F(V)\\ar[r]\\ar[d]^{\\phi_{V\\to U}}&F(W)\\ar[d]^{\\phi_{W\\to U}}\\\\G(V)\\ar[r]&G(W)}$$ commutes. Therefore\n$$\\underline{\\tt Hom}(F,G)(U)\\cong {\\tt Hom}_{ \\Pr_{R-{\\tt Mod}}\\mathbf{X}\/U}(F_{|U},G_{|U})\\ .$$\n\\begin{lem}\\label{vorh1}\nIf $G$ is a sheaf, then $\\underline{\\tt Hom}(F,G)$ is a sheaf.\n\\end{lem}\n\\begin{proof}\nLet $U\\in \\mathbf{X}$ and $(U_i\\to U)_{i\\in I}$ be a covering.\nIn order to simplify the notation we consider $V:=\\sqcup_{i\\in I} U_i$.\nWe must show that the sequence\n$$0\\to \\underline{\\tt Hom}(F,G)(U)\\to  \\underline{\\tt Hom}(F,G)(V)\\to  \\underline{\\tt Hom}(F,G)(V\\times_UV)$$\nis exact.\n\nLet $\\psi\\in  {\\tt Hom}_{\\Pr_{R-{\\tt Mod}}\\mathbf{X}\/U}(F_{|U},G_{|U})$ be such that\nits restriction to $V$ vanishes.\nIf $(W\\to U)\\in \\mathbf{X}\/U$, then $W\\times_UV\\to W$ is a covering of $W$,\nand ${\\tt pr}_W^*:G(W)\\to G(W\\times_UV)$ is injective since $G$ is a sheaf.\nIn view of the commutative diagram \n$$\\xymatrix{F(W)\\ar[r]^{{\\tt pr}_W^*}\\ar[d]^{\\psi_W}&F(W\\times_UV)\\ar[d]^{(\\psi_{|V})_{W\\times_UV}}\\\\G(W)\\ar[r]^{{\\tt pr}_W^*}&G(W\\times_UV)}$$\nwe see that $\\psi_W=0$.\n\nLet now $\\phi\\in  {\\tt Hom}_{\\Pr_{R-{\\tt Mod}}\\mathbf{X}\/V}(F_{|V},G_{|V})$\nbe such that the induced map \n$$\\Phi\\in {\\tt Hom}_{\\Pr_{R-{\\tt Mod}}\\mathbf{X}\/(V\\times_UV)}(F_{|V\\times_UV},G_{|V\\times_UV})$$\nvanishes. We will construct\n$\\psi\\in  {\\tt Hom}_{\\Pr_{R-{\\tt Mod}}\\mathbf{X}\/U}(F_{|U},G_{|U})$ such that $\\psi_{|V}=\\phi$.\nLet $(W\\to U)\\in \\mathbf{X}\/U$ and $f\\in F_{|U}(W\\to U)=F(W)$.\nThen $W\\times_UV\\to W$ is a covering of $W$ and ${\\tt pr}_W^*f\\in F_{|V}(W\\times_UV\\to V)=F(W\\times_UV)$. We get an element\n$$\\phi_{W\\times_UV\\to V} ({\\tt pr}_W^*(f))\\in G(W\\times_UV)= G_{|V}(W\\times_UV\\to V)\\ .$$\nNote that\n$(W\\times_UV)\\times_W(W\\times_UV)\\cong W\\times_U(V\\times_UV)$. The difference\nof the pull-backs of $\\phi_{W\\times_UV\\to V} ({\\tt pr}_W^*(f))$ with respect to the two projections to $W\\times_UV$ induces\n$$\\Phi_{W\\times_U(V\\times_UV)}({\\tt pr}_W^*(f))=0\\in G((W\\times_UV)\\times_W(W\\times_UV))\\ .$$\nAgain, since $G$ is a sheaf\nthere is a unique element $\\psi_W(f)\\in G(W)$ such that\n$$\\psi_W(f)_{|W\\times_UV}=\\phi_{W\\times_UV\\to V} ({\\tt pr}_W^*(f))\\ .$$\nThe morphism $\\psi$ is now given by the collection $(\\psi_W)_{(W\\to U)\\in \\mathbf{X}\/U}$.\n\\end{proof}\n\n\n\n\n\n\n\n\n\\subsection{}\\label{system101}\n\n \nIf $F,G\\in {\\tt Sh}_{R-{\\tt Mod}}\\mathbf{X}$, then we define\n$F\\otimes G\\in  {\\tt Sh}_{R-{\\tt Mod}}\\mathbf{X}$ to be\n$$F\\otimes G :=i^\\sharp(i(F)\\otimes^p i(G))\\ .$$ \nWe furthermore define\n$$\\underline{\\tt Hom}(F,G):=i^\\sharp\\underline{\\tt Hom}^p(i(F),i(G))\\ .$$\nUsing the fact \\ref{vorh1} that $\\underline{\\tt Hom}^p(i(F),i(G))$ is a sheaf at the isomorphism marked by $!$ we get for every $H\\in {\\tt Sh}_{R-{\\tt Mod}}\\mathbf{X}$ that \n\\begin{eqnarray*}\n{\\tt Hom}_{{\\tt Sh}_{R-{\\tt Mod}}\\mathbf{X}}(H\\otimes F,G)&\\cong&{\\tt Hom}_{{\\tt Sh}_{R-{\\tt Mod}}\\mathbf{X}}(i^\\sharp(i(H)\\otimes^p i(F),G)\\\\\n&\\cong&{\\tt Hom}_{\\Pr_{R-{\\tt Mod}}\\mathbf{X}}(i(H)\\otimes^p i(F),i(G))\\\\\n&\\cong&{\\tt Hom}_{\\Pr_{R-{\\tt Mod}}\\mathbf{X}}(i(H),\\underline{\\tt Hom}^p(i(F),i(G))\\\\\n&\\stackrel{!}{\\cong}&{\\tt Hom}_{\\Pr_{R-{\\tt Mod}}\\mathbf{X}}(i(H),i\\circ i^\\sharp (\\underline{\\tt Hom}^p(i(F),i(G))))\\\\\n&\\cong&{\\tt Hom}_{{\\tt Sh}_{R-{\\tt Mod}}\\mathbf{X}}(i^\\sharp\\circ i(H),\\underline{\\tt Hom}(F,G))\\\\\n&\\cong&{\\tt Hom}_{{\\tt Sh}_{R-{\\tt Mod}}\\mathbf{X}}(H,\\underline{\\tt Hom}(F,G))\\ .\\end{eqnarray*}\nIn other words, the pair \n$(\\otimes,\\underline{\\tt Hom})$ together with this natural isomorphism make ${\\tt Sh}_{R-{\\tt Mod}}\\mathbf{X}$ into a closed symmetric monoidal category. \n\n\n\n\n\n \n\n\n\n\n\n\n\n\n\\subsection{}\\label{flatdetect}\n\nLet $F,G\\in {\\tt Sh}_{R-{\\tt Mod}}\\mathbf{X}$ and $(U\\to X)\\in \\mathbf{X}$. Then we have\n$$(F\\otimes G)_U\\cong F_U\\otimes G_U\\ .$$\nIndeed, this follows from the fact that sheafification commutes with the restriction from the site $\\mathbf{X}$ to the site $(U)$, see \\ref{obstr1a}.\nSince the collection of functors $(\\nu_U)_{(U\\to X)\\in \\mathbf{X}}$ detects exact sequences\nit now follows that a sheaf $F\\in  {\\tt Sh}_{R-{\\tt Mod}}\\mathbf{X}$ is flat if and only if $F_U\\in {\\tt Sh}_{R-{\\tt Mod}}(U)$ is flat for all $(U\\to X)\\in \\mathbf{X}$. This fact was claimed in \\ref{obstr1a}.\n\n\n\n\n\\subsection{}\n\n\\begin{lem}\\label{kion}\nFor $F,G\\in \\Pr_{R-{\\tt Mod}}\\mathbf{X}$ we  have\n$i^\\sharp(F\\otimes^p G)\\cong i^\\sharp(F)\\otimes i^\\sharp(G)$.\n\\end{lem}\n\\begin{proof}\nThis follows from (we omit the functor $i$ at various places in order to simplify the formula)\n\\begin{eqnarray*}\n{\\tt Hom}_{{\\tt Sh}_{R-{\\tt Mod}}\\mathbf{X}}( i^\\sharp(F\\otimes^p G),H)&\\cong&\n{\\tt Hom}_{\\Pr_{R-{\\tt Mod}}\\mathbf{X}}( F\\otimes^p G,H)\\\\\n&\\cong&{\\tt Hom}_{\\Pr_{R-{\\tt Mod}}\\mathbf{X}}(F,\\underline{\\tt Hom}^p(G,H))\\\\\n&\\stackrel{!}{\\cong}&{\\tt Hom}_{\\Pr_{R-{\\tt Mod}}\\mathbf{X}}(i^\\sharp(F),\\underline{\\tt Hom}^p(G,H))\\\\\n&\\cong&{\\tt Hom}_{\\Pr_{R-{\\tt Mod}}\\mathbf{X}}(i^\\sharp(F)\\otimes^p G,H)\\\\\n&\\cong&{\\tt Hom}_{\\Pr_{R-{\\tt Mod}}\\mathbf{X}}(G,\\underline{\\tt Hom}^p(i^\\sharp F,H))\\\\\n&\\stackrel{!}{\\cong}&{\\tt Hom}_{\\Pr_{R-{\\tt Mod}}\\mathbf{X}}(i^\\sharp G,\\underline{\\tt Hom}^p(i^\\sharp F,H))\\\\\n&\\cong&{\\tt Hom}_{\\Pr_{R-{\\tt Mod}}\\mathbf{X}}(i^\\sharp G\\otimes^p i^\\sharp F,H)\\\\\n&\\cong&{\\tt Hom}_{{\\tt Sh}_{R-{\\tt Mod}}\\mathbf{X}}(i^\\sharp G\\otimes i^\\sharp F,H) \n \\end{eqnarray*}\nfor arbitrary $H\\in {\\tt Sh}_{R-{\\tt Mod}}\\mathbf{X}$, where we use Lemma \\ref{vorh1} at the isomorphisms marked by $!$.\n\\end{proof}\n\\subsection{}\n\nLet $f\\colon X\\to Y$ be a morphism of  locally compact stacks. \nLet $\\mathbf{X}$ and $\\mathbf{Y}$ be the sites associated to $X$ and $Y$.  Consider the adjoint pair of functors\n$${}^pf^*\\colon \\Pr_{R-{\\tt Mod}}\\mathbf{Y}\\leftrightarrows\\Pr_{R-{\\tt Mod}}\\mathbf{X}\\colon {}^pf_*\\ .$$\nThe proof of the following Lemma uses the product in $\\mathbf{Y}$ \ndescribed in \\cite[Lemma 3.1]{bss}\nin a specific way.\n\\begin{lem}\nFor $F,G\\in \\Pr_{R-{\\tt Mod}}\\mathbf{Y}$ we have a natural isomorphism\n$${}^pf^*(F\\otimes^pG)\\cong {}^pf^*F\\otimes^p{}^p f^*G\\ .$$\n\\end{lem}\n\\begin{proof}\nWe use the notation introduced in \\cite[2.1.4]{bss}.\nFor  $(U\\to X)\\in \\mathbf{X}$ we consider the category $U\/\\mathbf{Y}$ of diagrams $$\\xymatrix{U\\ar[r]\\ar[d]&X\\ar[d]\\\\V\\ar[r]&Y}\\ .$$\nThe functor ${}^pf^*$ is defined in \\cite[Definition 2.3]{bss}\nas a colimit over this category.\n\nWe consider  the diagonal functor\n$U\/\\mathbf{Y}\\to U\/\\mathbf{Y}\\times U\/\\mathbf{Y}$ given on objects by\n$$\\xymatrix{U\\ar[r]\\ar[d]&X\\ar[d]\\\\V\\ar[r]&Y}\\hspace{0.5cm}\\mapsto\\hspace{0.5cm} (\\xymatrix{U\\ar[r]\\ar[d]&X\\ar[d]\\\\V\\ar[r]&Y},\\xymatrix{U\\ar[r]\\ar[d]&X\\ar[d]\\\\V\\ar[r]&Y})\\ .$$\nIn view of the definition of ${}^pf^*$ by colimits\nit induces a map\n$${}^pf^*(F\\otimes^pG)\\to {}^pf^*F\\otimes^p {}^pf^*G\\ .$$\nIn the other direction we have the functor\n$U\/\\mathbf{Y}\\times U\/\\mathbf{Y}\\to U\/\\mathbf{Y}$ given by\n$$(\\xymatrix{U\\ar[r]\\ar[d]&X\\ar[d]\\\\V\\ar[r]&Y},\\xymatrix{U\\ar[r]\\ar[d]&X\\ar[d]\\\\V^\\prime\\ar[r]&Y})\\hspace{0.5cm}\\mapsto\\hspace{0.5cm} \\xymatrix{U\\ar[r]\\ar[d]&X\\ar[d]\\\\V\\times_Y V^\\prime\\ar[r]&Y}\\ .$$\nThis together with the projections\n$V\\times_Y V^\\prime\\to V$ and $V\\times_Y V^\\prime\\to V^\\prime$ it induces the inverse map\n$${}^pf^*F\\otimes^p {}^pf^*G \\to {}^pf^*(F\\otimes^pG)\\ .$$\n\\end{proof}\n\n\\subsection{}\n\n\nLet $f\\colon X\\to Y$ be a morphism of locally compact stacks.\n\\begin{lem}\\label{tens-pres}\nFor $F,G\\in {\\tt Sh}_{R-{\\tt Mod}}\\mathbf{Y}$ we have a natural isomorphism\n$$f^*(F\\otimes G)\\cong f^*F\\otimes f^*G\\ .$$\n\\end{lem}\n\\begin{proof}\nFor $H\\in {\\tt Sh}_{R-{\\tt Mod}} \\mathbf{X}$, using the fact that ${}^pf_*$ preserves sheaves (see \\ref{sec:morph_of_sheaves}) and Lemma \\ref{kion}, we have\n\\begin{eqnarray*}\n{\\tt Hom}_{{\\tt Sh}_{R-{\\tt Mod}}\\mathbf{X}}(f^*(F\\otimes G),H)&\\cong&{\\tt Hom}_{{\\tt Sh}_{R-{\\tt Mod}}\\mathbf{X}}(F\\otimes G,f_*(H))\\\\\n&\\cong&{\\tt Hom}_{{\\tt Sh}_{R-{\\tt Mod}}\\mathbf{Y}}(i^\\sharp(i(F)\\otimes^p i(G)),i^\\sharp \\circ f_*^p\\circ i(H))\\\\\n&\\cong&{\\tt Hom}_{\\Pr_{R-{\\tt Mod}}\\mathbf{Y}}((i(F)\\otimes^p i(G)), f_*^p\\circ i(H))\\\\\n&\\cong&{\\tt Hom}_{\\Pr_{R-{\\tt Mod}}\\mathbf{X}}({}^pf^*(i(F)\\otimes^p i(G)), i(H))\\\\\n&\\cong&{\\tt Hom}_{\\Pr_{R-{\\tt Mod}}\\mathbf{X}}({}^pf^*\\circ i(F)\\otimes^p {}^pf^*\\circ i(G), i(H))\\\\\n&\\cong&{\\tt Hom}_{{\\tt Sh}_{R-{\\tt Mod}}\\mathbf{X}}(i^\\sharp({}^pf^*\\circ i(F)\\otimes^p {}^pf^*\\circ i(G)), H)\\\\\n&\\cong&{\\tt Hom}_{{\\tt Sh}_{R-{\\tt Mod}}\\mathbf{X}}(f^*(F)\\otimes f^*(G), H)\n\\end{eqnarray*}\n\\end{proof}\n\n\\subsection{}\n\nFor a derived version of Lemma \\ref{tens-pres} we assume that the morphism  $f:X\\to Y$ of locally compact stacks has local sections. For simplicity we only consider the case\n$R=\\mathbb{Z}$, i.e. sheaves of abelian groups (finite cohomological dimension of $R$ would suffice).\nThen the exact functor $f^*=(f_\\sharp)^*$ preserves torsion-free sheaves of abelian groups. Since the derived tensor product can be calculated using torsion-free resolutions we get the corollary \n\\begin{kor} \\label{tens-pres1} If $f:X\\to Y$ has local sections, then\nfor $F,G\\in D^+({\\tt Sh}_{\\tt Ab}\\mathbf{Y})$ we have a natural isomorphism\n$$f^*(F\\otimes^L G)\\cong f^*F\\otimes^L f^*G\\ .$$ \n\\end{kor}\nof Lemma \\ref{tens-pres}.\n                                                                                                                                                                                                \n\n\n\\subsection{}\n\nLet $f\\colon X\\to Y$ be a morphism of locally compact stacks.\n\\begin{lem}\nFor $F\\in {\\tt Sh}_{R-{\\tt Mod}}\\mathbf{Y}$ and $G\\in {\\tt Sh}_{R-{\\tt Mod}}\\mathbf{X}$ we have a natural isomorphism\n$$\\underline{\\tt Hom}(F,f_*G)\\cong f_*\\underline{\\tt Hom}(f^*F,G)$$\nin  ${\\tt Sh}_{R-{\\tt Mod}}\\mathbf{Y}$\n\\end{lem}\n\\begin{proof}\nFor any $T\\in {\\tt Sh}_{R-{\\tt Mod}}\\mathbf{Y}$ we calculate\n\\begin{eqnarray*}\n{\\tt Hom}_{{\\tt Sh}_{R-{\\tt Mod}}\\mathbf{Y}}(T,f_*\\underline{\\tt Hom}(f^*F,G))&\\cong&{\\tt Hom}_{{\\tt Sh}_{R-{\\tt Mod}}\\mathbf{X}}(f^*T,\\underline{\\tt Hom}(f^*F,G))\\\\\n&\\cong& {\\tt Hom}_{{\\tt Sh}_{R-{\\tt Mod}}\\mathbf{X}}(f^*T\\otimes f^*F,G)\\\\\n&\\cong& {\\tt Hom}_{{\\tt Sh}_{R-{\\tt Mod}}\\mathbf{X}}(f^*(T\\otimes F),G)\\\\\n&\\cong&{\\tt Hom}_{{\\tt Sh}_{R-{\\tt Mod}}\\mathbf{Y}}(T\\otimes F,f_*G)\\\\\n&\\cong&{\\tt Hom}_{{\\tt Sh}_{R-{\\tt Mod}}\\mathbf{Y}}(T,\\underline{\\tt Hom}(F,f_*G))\n\\end{eqnarray*}\n\\end{proof}\n\n\n \n\n\\subsection{}\n\nLet $f\\colon X\\to Y$ be a morphism of locally compact stacks.\n\\begin{lem}\\label{system80}\nFor $F\\in {\\tt Sh}_{R-{\\tt Mod}}\\mathbf{Y}$ and $G\\in {\\tt Sh}_{R-{\\tt Mod}}\\mathbf{X}$ we have a natural morphism\n$$f_*G\\otimes F\\to f_*(G\\otimes f^* F)\\ .$$\n\\end{lem}\n\\begin{proof}\nThe transformation is the image of the identity under the following chain of maps, where the first is induced by the counit $f^*\\circ f_*\\to {\\tt id} $ of the adjoint pair $(f^*,f_*)$, and the second isomorphism is given by Lemma \\ref{tens-pres}.\n\\begin{eqnarray*}\n{\\tt Hom}_{{\\tt Sh}_{R-{\\tt Mod}}\\mathbf{X}}(G\\otimes f^*F,G\\otimes f^*F)&\\to&\n{\\tt Hom}_{{\\tt Sh}_{R-{\\tt Mod}}\\mathbf{X}}(f^* f_*G\\otimes f^*F,G\\otimes f^*F)\\\\\n&\\cong&{\\tt Hom}_{{\\tt Sh}_{R-{\\tt Mod}}\\mathbf{X}}(f^*(f_*G\\otimes F),G\\otimes f^*F)\\\\\n&\\cong&{\\tt Hom}_{{\\tt Sh}_{R-{\\tt Mod}}\\mathbf{Y}}(f_*G\\otimes F,f_*(G\\otimes f^*F))\\ .\n\\end{eqnarray*}\n\\end{proof}\n\n\\begin{lem}\nIf $f$ has local sections, then for $F\\in {\\tt Sh}_{{\\tt Ab}}\\mathbf{Y}$ and $G\\in {\\tt Sh}_{{\\tt Ab}}\\mathbf{X}$ we have a natural morphism\n$$f_*G\\otimes^L F\\to f_*(G\\otimes^L f^* F)\\ .$$\n \\end{lem}\n\\begin{proof}\nWe use the same argument as for Lemma \\ref{system80} based on the adjoint pair $(f^*,Rf_*)$ and Lemma \\ref{tens-pres1}. \\end{proof} \n\n\n\n\\subsection{}\n\nLet $f\\colon X\\to Y$ be a morphism of locally compact stacks.  \n\\begin{lem}\\label{gr213}\nLet $F\\in {\\tt Sh}_{R-{\\tt Mod}}\\mathbf{Y}$ be sheaf which is locally isomorphic to $\\underline{R}_{\\mathbf{Y}}$, i.e. there exist an atlas\n$a\\colon U\\to Y$ such that $a^*F\\cong  \\underline{R}_{\\mathbf{U}}$.\nIn this case we have the projection formula:\nFor all $G\\in {\\tt Sh}_{R-{\\tt Mod}}\\mathbf{X}$ or $H\\in D^+({\\tt Sh}_{\\tt Ab}\\mathbf{X})$ the natural morphism\n$$f_*G\\otimes F\\to f_*(G\\otimes f^* F)\\ , \\quad Rf_* H\\otimes^L F\\to Rf_*(H\\otimes^Lf^*F)$$ are isomorphisms.\n \\end{lem}\n\\begin{proof}\nThis can be checked locally on the atlas $U\\to Y$.\nWe consider the pull-back\n$$\\xymatrix{V\\ar[r]^b\\ar[d]^g&X\\ar[d]^f\\\\\nU\\ar@{:>}[ur]\\ar[r]^a&Y}\\ .$$\n We must check that\n$$a^*\\circ (f_*G\\otimes F)\\to a^*\\circ f_*(G\\otimes f^* F)$$\nis an isomorphism.\nThis map is equivalent to\n\n\\begin{eqnarray*}\n             a^*(f_*G\\otimes F) &\\cong& a^*f_*G\\otimes a^*F\\\\& \\cong& a^*f_*G\\otimes\n        \\underline{R}_U\\\\\n        &\\cong& a^*f_*G\\\\&\\cong& g_*b^*G\\\\\n        &\\cong& g_*b^*(G\\otimes\\underline{R}_X)\\\\& \\cong& g_*(b^*G\\otimes\n        b^*f^*\\underline{R}_Y) \\\\\n        &\\cong& g_*(b^*G\\otimes g^*a^*\\underline{R}_Y)\\\\& \\cong& g_*(b^*G\\otimes\n        g^*a^*F)\\\\\n        &\\cong& g_*b^*(G\\otimes f^*F)\\\\& \\cong &a^*f_*(G\\otimes f^*F)\\ .\n     \\end{eqnarray*}\nThe derived version is shown in  similar manner. \n\\end{proof} \n\n\\subsection{}\n\nWe will also need the projection formula with different assumptions.\nLet $f\\colon X\\to Y$ be a map of locally compact stacks.\nWe consider $F\\in {\\tt Sh}_{R-{\\tt Mod}}\\mathbf{Y}$ and $G\\in {\\tt Sh}_{R-{\\tt Mod}}\\mathbf{X}$.\n\n\\begin{lem}\\label{system81}\nAssume that $f$ is proper and representable, and that  $F$ is flat.\nThen the natural transformation\n$$f_* G\\otimes F\\to f_*(G\\otimes f^*F)$$\nof \\ref{system80}\nis an isomorphism.\n\\end{lem}\n\\begin{proof}\nUsing the observations \\ref{obstr1a} we see that\nit suffices to show that for all $(U\\to Y)\\in \\mathbf{Y}  $\nthe induced morphism\n\\begin{equation}\\label{kassp}\ng_*G_V\\otimes F_U\\to g_*(G_V\\otimes g^*F_U)\n\\end{equation}\nis an isomorphism. Here $g\\colon V\\to U$ is the proper map of locally compact spaces defined by the Cartesian diagram\n$$\\xymatrix{V\\ar[d]^g\\ar[r]&X\\ar[d]^f\\\\U\\ar[r]&Y}\\ .$$\nThe map (\\ref{kassp}) is an isomorphism by\n\\cite[Prop. 2.5.13]{MR1299726}. \\end{proof}\n \n\n\n\\subsection{}\n \nWe also have  a derived version of the projection formula in the case of sheaves of abelian groups.\nThe main point is that the ring $\\mathbb{Z}$ has a  finite cohomological dimension (in fact equal to $1$). \nLet $f\\colon X\\to Y$ be a morphism of locally compact stacks.\n\\begin{lem}\\label{projefoa}Assume that $f$ is proper and representable. \nIf $G\\in D^+({\\tt Sh}_{\\tt Ab}\\mathbf{Y}  )$ and $F\\in D^+({\\tt Sh}_{\\tt Ab} \\mathbf{X}  )$,\nthen we have\n$$Rf_*G\\otimes^LF\\stackrel{\\sim}{\\to} Rf_*(G\\otimes^L f^* F)$$\nin $D^{+ }({\\tt Sh}_{{\\tt Ab}} \\mathbf{Y}  )$.\n\\end{lem}\n\\begin{proof}\nAs in the proof of Lemma \\ref{system81} we can reduce to the small sites $(U)$ for all objects $(U\\to Y)\\in \\mathbf{Y}$. After this reduction we apply\n\\cite[Prop. 2.6.6]{MR1299726} and the fact that the cohomological dimension of $\\mathbb{Z}$ is $1$, hence finite. \n\\end{proof}\n\n\\subsection{}\n\nThe following derived adjunction again uses the finiteness of the cohomological dimension of $\\mathbb{Z}$.\n\n\n\\begin{lem}\\label{khkqdqwwqc}\nFor $F,G,H\\in D^+({\\tt Sh}_{\\tt Ab}\\mathbf{X})$ we have a natural isomorphism\n$$R{\\tt Hom}_{{\\tt Sh}_{\\tt Ab}\\mathbf{X}}(F\\otimes^LG,H)\\cong R{\\tt Hom}_{{\\tt Sh}_{\\tt Ab}\\mathbf{X}}(F,R\\underline{\\tt Hom}(G,H))\\ .$$\n\\end{lem}\n\\begin{proof}\nIf $G\\in {\\tt Sh}_{\\tt Ab}\\mathbf{X}$ is flat and $H\\in {\\tt Sh}_{\\tt Ab}\\mathbf{X}$ is injective,  then the functor $${\\tt Sh}_{\\tt Ab}\\mathbf{X}\\ni F\\mapsto {\\tt Hom}_{{\\tt Sh}_{\\tt Ab}\\mathbf{X}}(F,\\underline{\\tt Hom}(G,H))\\cong {\\tt Hom}_{{\\tt Sh}_{\\tt Ab}\\mathbf{X}}(F\\otimes G,H)\\in {\\tt Ab}$$ is, as a composition of exact functors, exact. It follows that $\\underline{\\tt Hom}(G,H)$ is again injective.\nWe now show the Lemma.\nWe can assume that $H$ is a complex of injectives. Furthermore, since the cohomological dimension of $\\mathbb{Z}$ is one, hence in particular finite, we can assume  that $G$  is a complex of flat sheaves.\nThen we have\n\\begin{eqnarray*}\nR{\\tt Hom}_{{\\tt Sh}_{\\tt Ab}\\mathbf{X}}(F\\otimes^LG,H)&\\cong&\n{\\tt Hom}_{{\\tt Sh}_{\\tt Ab}\\mathbf{X}}(F\\otimes G,H)\\\\\n&\\cong&{\\tt Hom}_{{\\tt Sh}_{\\tt Ab}\\mathbf{X}}(F,\\underline{\\tt Hom}(G,H))\\\\\n&\\cong&R{\\tt Hom}_{{\\tt Sh}_{\\tt Ab}\\mathbf{X}}(F,\\underline{\\tt Hom}(G,H))\\ .\n \\end{eqnarray*}\n\\end{proof}\n\n\n\\section{Verdier duality for locally compact stacks in detail}\n\n\n\n\n\\subsection{}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nLet $f\\colon X\\to Y$ be a map of locally compact stacks.\n\\begin{ddd}\nWe say that the cohomological dimension of $f_*$ is not greater  than $n\\in \\mathbb{N}$ if the derived functor $R^if_*:{\\tt Sh}_{\\tt Ab}\\mathbf{X}\\to {\\tt Sh}_{\\tt Ab}\\mathbf{Y}$ vanishes for all $i>n$. \n\\end{ddd}\n\n\n\nThe main theorem of the present subsection is\n\\begin{theorem}\\label{main123}\nAssume that $f:X\\to Y$ is a representable and proper map between locally compact stacks such that\n $f_*$  has finite cohomological dimension. Then the functor $Rf_*\\colon D^+({\\tt Sh}_{\\tt Ab}\\mathbf{X}  )\\to D^+({\\tt Sh}_{\\tt Ab}\\mathbf{Y}  )$ admits a right adjoint\n$f^!\\colon  D^+({\\tt Sh}_{\\tt Ab}\\mathbf{Y}  )\\to  D^+({\\tt Sh}_{\\tt Ab}\\mathbf{X}  )$.\n\\end{theorem}\n\nThe proof of Theorem \\ref{main123}  will be finished in \\ref{uidbhqwodqwdwqddqw}.\nThe main idea is to\ntransfer the construction of $f^!$ from \\cite[Section 3.1]{MR1299726} to the\npresent situation.\n\n\n\\subsection{}\\label{canokcioa}\n \n  We consider the functorial flabby resolution (see \\ref{system100}) of the\n  constant sheaf $\\underline{\\Z}_{\\mathbf{X}}\\to {\\mathcal Fl}(\\underline{\\Z}_{\\mathbf{X}})$ and form the truncated complex\n  $K:=\\tau^{\\le n} {\\mathcal Fl}(\\underline{\\Z}_{\\mathbf{X}})$ so that in particular\n  $K^n=\\ker({\\mathcal Fl}^n(\\underline{\\Z}_{\\mathbf{X}})\\to {\\mathcal Fl}^{n+1}(\\underline{\\Z}_{\\mathbf{X}}))$.  \n\\begin{lemma}\\label{shortrest12}\nAssume that $f$ is representable and that $f_*$ has cohomological dimension  not greater than $n$.\n Then the complex\n\\begin{equation}\\label{reconl}0\\to \\underline{\\Z}_{\\mathbf{X} }\\to K^0\\to K^1\\to\\dots\\to K^n \\to 0\\end{equation}\nis a flat and $f_*$-acyclic resolution of $\\underline{\\Z}_{\\mathbf{X}}$.\n\\end{lemma}\n\\begin{proof}\nThe sheaf $\\ker(K^n\\to K^{n+1})$ is a torsion-free subsheaf of a torsion-free sheaf\nand therefore flat (compare \\cite[Lemma 3.1.4]{MR1299726}). \nBy Lemma \\ref{hdqoiwdwqw} the flabby sheaves $K^i$ for $i=0,\\dots,n-1$ are $f_*$-acyclic.\nIn order to see that $K^n$ is $f_*$-acyclic, it\n suffices to show that\n$R^if_* (\\ker(K^n\\to K^{n+1}))\\cong 0$ for $i\\ge 1$.\nIn fact, we have \n$R^if_* (\\ker(K^n\\to K^{n+1}))\\cong R^{i+n}f_* \\underline{\\Z}_{\\mathbf{X} }\\cong 0$.\n\\end{proof}\n\n\n\n\n\\subsection{}\n\n The fibers of a representable and proper morphism of topological stacks are compact. This is explicitly used in the proof of the following Lemma.\n\n\n\\begin{lem}\\label{sumpre45}\nIf $f:X\\to Y$ is  a representable and proper morphism of locally compact stacks, then\nthe functor $f_*\\colon {\\tt Sh}_{\\tt Ab}\\mathbf{X}  \\to {\\tt Sh}_{\\tt Ab}\\mathbf{Y}  $ preserves direct sums.\n\\end{lem}\n\\begin{proof}\n Let $(G_i)_{i\\in I}$ be a family of sheaves in \n${\\tt Sh}_{\\tt Ab}\\mathbf{X}  $. Then we have a canonical map\n$$\\bigoplus_{i\\in I} \\circ f_*(G_i)\\to f_*\\circ \\bigoplus_{i\\in I}(G_i)\\ .$$\nIn order to show that this map is an isomorphism we show that the induced map\n$$(\\bigoplus_{i\\in I} \\circ f_*(G_i))_U\\to (f_*\\circ \\bigoplus_{i\\in I}(G_i))_U$$ is an isomorphism for all\n$(U\\to Y)\\in \\mathbf{Y}  $. Choose such  $(U\\to Y)$ and consider the Cartesian diagram\n$$\\xymatrix{V\\ar[d]^g\\ar[r]&X\\ar[d]^f\\\\U\\ar[r]&Y}\\ .$$ It suffices to show that the induced map\n$$\\bigoplus_{i\\in I} \\circ g_*(G_i)_U\\to g_*\\circ \\bigoplus_{i\\in I}(G_i)_U$$\nis an isomorphism. We consider the induced map on the stalk at $x\\in U$. Since the restriction to $g^{-1}(x)$ commutes with the sum and $g^{-1}(x)$ is compact it is given by\n$$\\bigoplus_{i\\in I} \\circ \\Gamma(g^{-1}(x),[(G_i)_U]_{|g^{-1}(x)})\\to\n\\Gamma(g^{-1}(x),\\bigoplus_{i\\in I} [(G_i)_U]_{|g^{-1}(x)})$$\n(see \\cite[Proposition 2.5.2]{MR1299726}). But this last map is an isomorphism since the\nglobal section functor on sheaves on a compact space commutes with sums.\n\\end{proof}\n\n\n\\subsection{}\\label{uihiewfe}\nFix $j\\in \\{0,1,2\\dots,n\\}$ and set $K:=K^j$, see \\ref{canokcioa}\n\\begin{lem}\\label{ggtre1}\nLet $f:X\\to Y$ be a representable, proper morphism of locally compact stacks such that $f_*$ has cohomological dimension not greater than $n$.\nThen the functor\n$G\\mapsto f_*(G\\otimes K)$ is an exact functor ${\\tt Sh}_{\\tt Ab} \\mathbf{X}  \\to {\\tt Sh}_{\\tt Ab} \\mathbf{Y}  $.\nFurthermore, $G\\otimes K$ is $f_*$-acyclic.\n\\end{lem}\n\\begin{proof}\nIn the following proof we freely use the facts listed in \\ref{desc_sheaves_on_U}.\nLet $G^\\cdot$ be an exact complex in ${\\tt Sh}_{\\tt Ab} \\mathbf{X}  $.\nFor $(U\\to Y)\\in \\mathbf{Y}  $  consider the Cartesian  diagram\n$$\\xymatrix{V\\ar[d]^g\\ar[r]&X\\ar[d]^f\\\\U\\ar[r]&Y}\\ .$$\nNote that $(V\\to X)\\in \\mathbf{X}  $.\nBy construction (see \\cite[Lemma 3.1.4]{MR1299726}) $K_V$ is flat and $g$-soft.\nThe complex $G_V^\\cdot$ is exact.\nBy \\cite[Lemma  3.1.2 (ii)]{MR1299726}\nthe complex $g_*(G_V^\\cdot\\otimes K_V)=(f_*(G^\\cdot\\otimes K))_U$ is exact.\nSince this is true for all $(U\\to Y)\\in \\mathbf{Y}  $ we conclude that\n$f_*(G^\\cdot\\otimes K)$ is exact.\n\nWe now show that $G\\otimes K$ is $f_*$-acyclic.\nWe must show that\n$R^if_*(G\\otimes K)\\cong 0$ for all $i\\ge 1$.\nFor $(U\\to Y)\\in \\mathbf{Y}  $ as above we have\n$(R^if_*(G\\otimes K))_U\\cong R^ig_*(G_U\\otimes K_U)\\cong 0$,\nsince $G_U\\otimes K_U$ is $g$-soft by \\cite[Lemma 3.1.2 (i)]{MR1299726} (note that $K_U$ is flasque and flat).\nSince $(U\\to Y)$ was arbitrary this implies that $R^if_*(G\\otimes K)\\cong0$\n\\end{proof}\n\n\n\\subsection{}\n\n\nFor $(V\\to X)\\in \\mathbf{X}$ let $\\hat h_V^\\mathbb{Z}$ denote the sheafification of the presheaf $h_V^\\mathbb{Z}$, the presheaf of free abelian groups generated by the sheaf $h_V$ represented by $V$.\nWe consider the functor $f^!_K\\colon {\\tt Sh}_{\\tt Ab}\\mathbf{Y}  \\to  \\Pr_{\\tt Ab} \\mathbf{X}  $ which associates to\na sheaf  $F\\in {\\tt Sh}_{\\tt Ab}\\mathbf{Y}  $ the presheaf\n$f^!_{K}( F)\\in \\Pr_{\\tt Ab} \\mathbf{X} $ given by\n$$\\mathbf{X}  \\ni (V\\to X)\\mapsto f^!_KF(V):={\\tt Hom}_{{\\tt Sh}_{\\tt Ab}\\mathbf{Y}}(f_*(\\hat h_V^\\mathbb{Z}\\otimes K),F)\\in {\\tt Ab}\\ .$$\nNote that $K\\to f^!_K(F)$ is also a functor in $K$ (for fixed $F$).\n\n\\begin{lem}\\label{injpw}\nLet $K$ be as in \\ref{uihiewfe} and $f:X\\to Y$ be a representable, proper morphism of locally compact stacks such that $f_*$ has cohomological dimension not greater than $n$.\nAssume that  $F\\in {\\tt Sh}_{\\tt Ab}\\mathbf{Y}  $ is an injective sheaf.   Then\n$f^!_K(F)$ is an injective  sheaf.  Furthermore, for $G\\in {\\tt Sh}_{\\tt Ab}\\mathbf{X}  $ there is a canonical isomorphism\n\\begin{equation}\\label{ggtre}{\\tt Hom}_{{\\tt Sh}_{\\tt Ab}\\mathbf{Y}}(f_*(G\\otimes K),F)\\cong {\\tt Hom}_{{\\tt Sh}_{\\tt Ab}\\mathbf{X}}(G,f^!_K(F))\\ .\\end{equation}\n\\end{lem}\n\\begin{proof}\nWe show that $f^!_KF$ is a sheaf by copying the corresponding argument in the proof of \\cite[Lemma 3.1.3]{MR1299726}. The functor\n$G\\mapsto {\\tt Hom}_{{\\tt Sh}_{\\tt Ab}\\mathbf{Y}}(f_*(G\\otimes K),F)$ is exact by Lemma \\ref{ggtre1} and injectivity of $F$. If we establish the isomorphism (\\ref{ggtre}), then we also have shown that $f^!_K(F)$ is injective.\n\n\n\n\n\nFor $(W\\to X)\\in \\mathbf{X}$ we have a canonical isomorphism\n\\begin{equation}\\label{sus55}{\\tt Hom}_{{\\tt Sh}_{\\tt Ab}\\mathbf{Y}}(f_*(\\hat h_W^\\mathbb{Z}\\otimes K),F)= f^!_K (F)(W)\\cong{\\tt Hom}_{{\\tt Sh}_{\\tt Ab}\\mathbf{X}}(\\hat h_W^\\mathbb{Z},f^!_K(F))\\ .\\end{equation}\nFor a system $(G_i)_{i\\in I}$ of sheaves we have a natural map $\\colim_{i\\in I}\\circ f_*(G_i)\\to f_*\\circ \\colim_{i\\in I} (G_i)$.\nFor $G\\in {\\tt Sh}_{\\tt Ab}\\mathbf{X}  $ we get\n\\begin{eqnarray*}\n {\\tt Hom}_{{\\tt Sh}_{\\tt Ab}\\mathbf{Y}}(f_*(G\\otimes K),F)&\\cong &{\\tt Hom}_{{\\tt Sh}_{\\tt Ab}\\mathbf{Y}}(f_*((\\colim_{\\hat h_W^\\mathbb{Z}\\to G} \\hat h_W^\\mathbb{Z})\\otimes K),F)\\\\\n&\\stackrel{!}{\\cong}&{\\tt Hom}_{{\\tt Sh}_{\\tt Ab}\\mathbf{Y}}(f_*\\circ \\colim_{\\hat h_W^\\mathbb{Z}\\to G} (\\hat h_W^\\mathbb{Z}\\otimes K),F)\\\\\n&\\to&{\\tt Hom}_{{\\tt Sh}_{\\tt Ab}\\mathbf{Y}}(\\colim_{\\hat h_W^\\mathbb{Z}\\to G}\\circ  f_*(\\hat h_W^\\mathbb{Z}\\otimes K),F)\\\\\n&\\cong&\\lim_{\\hat h_W^\\mathbb{Z}\\to G}  {\\tt Hom}_{{\\tt Sh}_{\\tt Ab}\\mathbf{Y}}(f_*(\\hat h_W^\\mathbb{Z}\\otimes K),F)\\\\\n&\\cong&\\lim_{\\hat h_W^\\mathbb{Z}\\to G}{\\tt Hom}_{{\\tt Sh}_{\\tt Ab}\\mathbf{X}}(\\hat h_W^\\mathbb{Z},f^!_K(F))\\\\\n&\\cong&{\\tt Hom}_{{\\tt Sh}_{\\tt Ab}\\mathbf{X}}(\\colim_{\\hat h_W^\\mathbb{Z}\\to G}\\hat h_W^\\mathbb{Z},f^!_K(F))\\\\\n&\\cong&{\\tt Hom}_{{\\tt Sh}_{\\tt Ab}\\mathbf{X}}(G,f^!_K(F))\\ .\n\\end{eqnarray*}\nThe marked isomorphism uses that the tensor product of sheaves commutes with colimits, a consequence of the fact  \\ref{system101} that it is part of a closed monoidal structure.\nIt remains to show that this composition is an isomorphism.\nIf we write out the definition of the colimit in\n$G\\cong \\colim_{\\hat h_W^\\mathbb{Z}\\to G}\\hat h_W^\\mathbb{Z}$ we obtain an exact sequence of the form\n\\begin{equation}\\label{sus54}\\bigoplus_{j\\in J}\\hat h^\\mathbb{Z}_{W_j}\\to \\bigoplus_{i\\in I}\\hat h^\\mathbb{Z}_{V_i}\\to G\\to 0\\ .\\end{equation}\nNow observe that for any collection $(G_i)_{i\\in I}$ of sheaves in ${\\tt Sh}_{\\tt Ab} \\mathbf{X}  $ we have\n$${\\tt Hom}_{{\\tt Sh}_{\\tt Ab}\\mathbf{Y}}(f_*((\\bigoplus_i G_i)\\otimes K),F)\\cong \\prod_{i\\in I} {\\tt Hom}_{{\\tt Sh}_{\\tt Ab}\\mathbf{Y}}(f_*(G_i\\otimes K),F)$$\nsince $f_*$ (Lemma \\ref{sumpre45})  and $\\dots\\otimes K$ commute with sums.\nClearly we also have\n$${\\tt Hom}_{{\\tt Sh}_{\\tt Ab}\\mathbf{X}}(\\bigoplus_i G_i,f^!_K(F))\\cong \\prod_{i\\in I} \n{\\tt Hom}_{{\\tt Sh}_{\\tt Ab}\\mathbf{X}}(G_i,f^!_K(F))\\ .$$ \n{}From (\\ref{sus54}) we thus get the diagram\n\\begin{equation*}\n  \\begin{CD}\n    0 && 0\\\\\n @VVV @VVV \\\\\n   {\\tt Hom}_{{\\tt Sh}_{\\tt Ab}\\mathbf{Y}}(f_*(G\\otimes K),F) @>>> {\\tt Hom}_{{\\tt Sh}_{\\tt Ab}\\mathbf{X}}(G,f^!_K(F)) \\\\\n   @VVV @VVV\\\\\n  \\prod_{i\\in I}{\\tt Hom}_{{\\tt Sh}_{\\tt Ab}\\mathbf{Y}}(f_*(\\hat h_{V_i}^\\mathbb{Z}\\otimes K),F) @>\\alpha>>\\prod_{i\\in I}{\\tt Hom}_{{\\tt Sh}_{\\tt Ab}\\mathbf{X}}(\\hat h_{V_i}^\\mathbb{Z},f^!_K(F)) \\\\\n  @VVV @VVV\\\\\n\\prod_{j\\in J}{\\tt Hom}_{{\\tt Sh}_{\\tt Ab}\\mathbf{Y}}(f_*(\\hat h_{W_j}^\\mathbb{Z}\\otimes K),F)\n@>\\beta>>\\prod_{j\\in J}{\\tt Hom}_{{\\tt Sh}_{\\tt Ab}\\mathbf{X}}(\\hat h_{W_j}^\\mathbb{Z},f^!_K(F)) .\\\\\n  \\end{CD}\n\\end{equation*}\nBecause of the isomorphism (\\ref{sus55}) the maps $\\alpha$ and $\\beta$ are isomorphisms.\nThe left vertical sequence is exact by Lemma \\ref{ggtre1}.\nThe right vertical sequence is exact by the left-exactness of the ${\\tt Hom}$-functor.\nIt follows from the five Lemma that (\\ref{ggtre}) is an isomorphism.\n\\end{proof}\n\n\n\\subsection{}\\label{uidbhqwodqwdwqddqw}\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n\n\n\n\n\n\n\n\n\nLet $I{\\tt Sh}_{{\\tt Ab}}\\mathbf{X}  \\subset {\\tt Sh}_{{\\tt Ab}}\\mathbf{X}  $ denote the full subcategory of injective objects and $K^+(I{\\tt Sh}_{\\tt Ab} \\mathbf{X}  )$ be the category of complexes in $I{\\tt Sh}_{\\tt Ab} \\mathbf{X} $ which are bounded below, and whose morphisms are\nhomotopy classes of chain maps.\nThen we have an equivalence of triangulated categories\n$$K^+(I{\\tt Sh}_{\\tt Ab} \\mathbf{X}  )\\cong D^+({\\tt Sh}_{\\tt Ab} \\mathbf{X}  )\\ .$$\n\nLet $f:X\\to Y$ be a representable, proper morphism of locally compact stacks such that $f_*$ has cohomological dimension not greater than $n$, and let $K^\\cdot$ be as in \\ref{canokcioa}.\nWe then define the functor\n$f^!\\colon K^+(I{\\tt Sh}_{\\tt Ab} \\mathbf{Y}  )\\to K^+(I{\\tt Sh}_{\\tt Ab} \\mathbf{X}  )$ by\n$$f^!(F^\\cdot)=(f^!_{K^\\cdot}(F^\\cdot))_{tot}\\ ,$$\nwhere $E^{\\cdot,\\cdot}_{tot}$ denotes the total complex of the double complex $E^{\\cdot,\\cdot}$.\nSince $f^!_K$ preserves injective sheaves by Lemma \\ref{injpw} this functor is well-defined.\nFurthermore, for $F\\in K^+(I{\\tt Sh}_{\\tt Ab} \\mathbf{Y}  )$ and $G\\in K^+(I{\\tt Sh}_{\\tt Ab} \\mathbf{X}  )$ we have by \n Lemma \\ref{injpw} a natural isomorphism between spaces of chain maps\n$${\\tt Hom}_{C^+({\\tt Sh}_{\\tt Ab}\\mathbf{Y}  )}(f_*(G^\\cdot\\otimes K^\\cdot)_{tot},F^\\cdot)\\cong  {\\tt Hom}_{C^+({\\tt Sh}_{\\tt Ab}\\mathbf{X}  )}(G^\\cdot, f^!(F^\\cdot))$$\nwhich descends to an isomorphism on the level of homotopy classes\n$${\\tt Hom}_{K^+(I{\\tt Sh}_{\\tt Ab}\\mathbf{Y}  )}(f_*(G^\\cdot\\otimes K^\\cdot)_{tot},F^\\cdot)\\cong  {\\tt Hom}_{K^+(I{\\tt Sh}_{\\tt Ab}\\mathbf{X}  )}(G^\\cdot, f^!(F^\\cdot))\\ .$$\nSince $f^!(F^\\cdot)$ is a complex of injective sheaves we have\n$${\\tt Hom}_{K^+(I{\\tt Sh}_{\\tt Ab}\\mathbf{X}  )}(G^\\cdot, f^!(F^\\cdot))\\cong {\\tt Hom}_{D^+({\\tt Sh}_{\\tt Ab}\\mathbf{X}  )}(G^\\cdot, f^!(F^\\cdot))\\ .$$\nNote that\n$G^\\cdot\\cong G^\\cdot\\otimes \\underline{\\Z}_{\\mathbf{X}  }\\to (G^\\cdot\\otimes K^\\cdot)_{tot}$\nis a quasi-isomorphism, and the complex $G^\\cdot\\otimes K^\\cdot$ consists of $f_*$-acyclic sheaves by Lemma \\ref{ggtre1}. Therefore $f_*(G^\\cdot\\otimes K^\\cdot)_{tot}\\cong Rf_*(G^\\cdot)$.\nSince $F^{\\cdot}$ is injective we have\n$${\\tt Hom}_{K^+({\\tt Sh}_{\\tt Ab}\\mathbf{Y}  )}(f_*(G^\\cdot\\otimes K^\\cdot)_{tot},F^\\cdot)\\cong {\\tt Hom}_{D^+({\\tt Sh}_{\\tt Ab}\\mathbf{Y}  )}(Rf_*(G^\\cdot),F^\\cdot)\\ .$$\nWe conclude that\n$${\\tt Hom}_{D^+({\\tt Sh}_{\\tt Ab}\\mathbf{Y}  )}(Rf_*(G^\\cdot),F^\\cdot)\\cong  {\\tt Hom}_{D^+({\\tt Sh}_{\\tt Ab}\\mathbf{X}  )}(G^\\cdot, f^!(F^\\cdot))\\ .$$\nThis finishes the proof of Theorem \\ref{main123}. \\hspace*{\\fill}$\\Box$ \\\\[0cm]\\noindent\n\n\n\n\n\n\n\n\n \\subsection{}\n\n We consider  morphisms $f\\colon X\\to Y$ and $u:U\\to Y$ of locally compact stacks and form the Cartesian diagram $$\\xymatrix{V\\ar[r]^v\\ar[d]^g&X\\ar[d]^f\\\\U\\ar[r]^u&Y}\\ .$$\n\n\n \n\\begin{lem}\\label{nattr121}\nAssume the  $f$ is representable, proper and that $f_*$ has finite cohomological dimension.\nAssume furthermore that $u$ has local sections. Then\nwe have a natural transformation\n $v^*\\circ f^!\\to g^!\\circ u^*$.\n\\end{lem}\n\\begin{proof}\nFirst note that $g$  is representable, proper and that $g_*$ has finite cohomological dimension.\nFurthermore, $v$ has local sections.\n We apply $f^!$ to the unit\n ${\\tt id}\\to Ru_*\\circ u^*$ and obtain a morphism\n \\begin{equation}\\label{plug1}f^!\\to f^!\\circ Ru_*\\circ u^*\\ .\\end{equation}\n Since $f$ is representable and $u$ has local sections we have the isomorphism (see Lemma \\ref{lem:pullpush} or \\cite[Lemma 2.43]{bss}) \n $$u^*\\circ Rf_*\\cong Rg_*\\circ v^*\\ .$$ Taking its right adjoint yields the isomorphism\n $$f^!\\circ Ru_*\\cong Rv_*\\circ g^!\\ .$$\n We plug this into (\\ref{plug1}) and obtain a transformation\n $$f^!\\to Rv_*\\circ g^!\\circ u^*\\ .$$\nIts adjoint is the desired transformation \\end{proof}\n\n\n \n\n\n\\subsection{}\n\nThe following adjunction is a consequence of the derived projection formula Lemma \\ref{projefoa}\nand the derived adjunction Lemma \\ref{khkqdqwwqc}\n\\begin{lem}\nIf  $f:X\\to Y$ is a representable proper morphism of locally compact stacks which has local sections and is such  that $f_*$ has finite cohomological dimension, then for $G\\in D^+({\\tt Sh}_{\\tt Ab}\\mathbf{X})$ and $F\\in D^+({\\tt Sh}_{\\tt Ab}\\mathbf{X})$\nwe have a natural isomorphism\n$$Rf_*R\\underline{\\tt Hom}(G,f^!F)\\cong R\\underline{\\tt Hom}(Rf_*G,F)\\ .$$\n\\end{lem}\n\\begin{proof}\nLet $H\\in D^+({\\tt Sh}_{\\tt Ab}\\mathbf{X})$ be arbitrary.\nThen we calculate using Lemma \\ref{derivedadj} and Lemma \\ref{projefoa} that\n\\begin{eqnarray*}\nR{\\tt Hom}_{{\\tt Sh}_{\\tt Ab}\\mathbf{Y}}(H,Rf_*R\\underline{\\tt Hom}(G,f^!F))&\\cong&R{\\tt Hom}_{{\\tt Sh}_{\\tt Ab}\\mathbf{X}}(f^*H,R\\underline{\\tt Hom}(G,f^!F))\\\\\n&\\cong&R{\\tt Hom}_{{\\tt Sh}_{\\tt Ab}\\mathbf{X}}(f^*H\\otimes^L G,f^!F)\\\\\n&\\cong&R{\\tt Hom}_{{\\tt Sh}_{\\tt Ab}\\mathbf{Y}}(Rf_*(f^*H\\otimes^L G),F)\\\\\n&\\cong&R{\\tt Hom}_{{\\tt Sh}_{\\tt Ab}\\mathbf{Y}}(H\\otimes^L Rf_*G,F)\\\\\n&\\cong&R{\\tt Hom}_{{\\tt Sh}_{\\tt Ab}\\mathbf{Y}}(H,R\\underline{\\tt Hom}(Rf_*G,F))\\ .\n\\end{eqnarray*}\n\\end{proof}\n\n\n\n\\subsection{}\n\\begin{ddd}\nIf  $f:X\\to Y$ is a proper morphism of locally compact stacks such that $f_*$ has finite cohomological dimension, then we define the relative dualizing complex by $$\\omega_{X\/Y}:=f^!(\\underline{\\Z}_\\mathbf{Y})\\ .$$\n\\end{ddd}\nIt would be interesting to know the structure of $\\omega_{X\/Y}$ for a topological submersion $f$ in the sense of \\cite[Def. 3.3.1]{MR1299726}. \n\n\\subsection{}\n\nIn a different setup  of Artin  stacks and  the lisse-\\'etale site in \\cite{laszlo-2005}\na six functor calculus was constructed. Starting with the observation that dualizing sheaves\non the small sites are sufficiently functorial the functors $Rf_!$ and $f^!$ are constructed on constructible sheaves by duality. In this approach one can relate the global $f^!$ \nwith the local versions without any difficulty.  \n\nA similar approach may work in the present topological context as well, but it is not clear how the\nresulting $f^!$ will relate to the construction in the present paper.\n\n\n\n\n\n\n\n\\section{The integration map}\\label{system103}\n\n\n\n \n\n\\subsection{}\n\nLet $M$ be a closed connected orientable $n$-dimensional topological manifold.\n\\begin{ddd}\nA map between locally compact stacks $f:X\\to Y$ is a locally trivial fiber bundle with fiber $M$ if\nfor every space $U\\to X$ the pull-back $U\\times_YX\\to U$ is a locally trivial\nfiber bundle of spaces with fiber $M$.\n\\end{ddd}\nNote that a locally trivial fiber bundle $f$ with fiber $M$ is representable, proper and has local sections, and $f_*$ has finite cohomological dimension.\n In order to see the last fact and to calculate $R^nf_*(\\underline{\\Z}_\\mathbf{X})$ we consider $(U\\to Y)\\in \\mathbf{Y}$ and let $V\\to U$ be surjective and locally an open embedding such that we have a diagram with Cartesian squares\n\\begin{equation}\\label{iudhqoiwdhqwidwqd}\n\\xymatrix{M\\ar[d]^q&V\\times_YX\\ar[l]\\ar[d]^h\\ar[r]&U\\times_YX\\ar[d]^g\\ar[r]&X\\ar[d]^f\\\\{*}&V\\ar[l]^p\\ar[r]&U\\ar[r]^u &Y}\\ .\n\\end{equation}\n\nThe map $g$ is a topological submersion in the sense of  \\cite[Def. 3.3.1]{MR1299726}.\nAs remarked in   \\cite[Sec.~3.3]{MR1299726}   the cohomological dimension of $g_*$  is  not greater than $n$. This implies  $(R^if_*F)_U\\cong R^ig_* (F_{U\\times_YX})=0$ for all $i>n$.\nSince this holds true for all $(U\\to Y)\\in \\mathbf{Y}$ we conclude that $R^if_*F=0$ for all $i>n$.\n\nWe use the left part of the diagram (\\ref{iudhqoiwdhqwidwqd})  in order to see that\n$R^nf_*(\\underline{\\Z}_\\mathbf{X})$ is locally isomorphic to $\\underline{\\Z}_\\mathbf{Y}$.\nIn fact, we have\n$$Rf_*(\\underline{\\Z}_\\mathbf{X})_V\\cong Rh_* \\underline{\\Z}_{(V\\times_YX)}\\cong p^* Rq_*\\underline{\\Z}_{(M)}\\ .$$ \nA choice of an orientation of $M$ gives an isomorphism $R^nq_*\\underline{\\Z}_{(M)}\\cong \\underline{\\Z}_{(*)}$ and therefore\n$ R^nf_*(\\underline{\\Z}_\\mathbf{X})_V\\cong p^*\\underline{\\Z}_{(*)}\\cong \\underline{\\Z}_{(V)}$.\n\n\n\\begin{ddd}\\label{iquhuiqwdddqw}\nA locally trivial fiber bundle $f\\colon X\\to Y$ with fiber $M$ is called orientable if\nthere exists an isomorphism $R^nf_*(\\underline{\\Z}_\\mathbf{X})\\cong \\underline{\\Z}_\\mathbf{Y}$. An orientation of $f$ is a choice of such an isomorphism.\n\\end{ddd}\n\n\n\n\n\n\n\n\\subsection{}\nLet $f:X\\to Y$ be a locally trivial fiber bundle with fiber $M$, where $M$ is  a compact closed $n$-dimensional topological manifold.\nWe consider the $f_*$-acyclic and flat resolution $K$ defined in (\\ref{reconl}).\n\nThe following was observed in \\ref{uidbhqwodqwdwqddqw} \n\\begin{corollary}\\label{corol:calculate_Rf}\n  The functor $Rf_*\\colon D^+({\\tt Sh}_{\\tt Ab}\\mathbf{X})\\to D^+({\\tt Sh}_{\\tt Ab}\\mathbf{Y})$ is represented by\n  $f_*\\circ T_K$, where $T_K$ is tensor product with the complex $K$.\n\\end{corollary}\n\nWe now define a natural transformation\n$$R\\underline{\\tt Hom}(R^nf_*( \\underline{\\Z}_{\\mathbf{X} }),F)\\to Rf_*\\circ f^!(F)\\ .$$\nLet $F\\in C^+(I{\\tt Sh}_{\\tt Ab}\\mathbf{Y} )$ be a complex of injectives. \nWe start from the observation that\n$$R^nf_*( \\underline{\\Z}_{\\mathbf{X} })\\cong f_*(K^n)\/{\\tt im}(f_*(K^{n-1})\\to f_*(K^n))\\ .$$\nFor $(U\\to Y)\\in \\mathbf{Y} $ we thus obtain a chain of maps of complexes\n\\begin{eqnarray*}\n\\underline{\\tt Hom}(R^nf_* \\underline{\\Z}_{\\mathbf{X} },F)(U)\n&\\cong&{\\tt Hom}_{{\\tt Sh}_{\\tt Ab}\\mathbf{Y}}(\\hat h_U^\\mathbb{Z},\\underline{\\tt Hom}(R^nf_* \\underline{\\Z}_{\\mathbf{X} },F))\\\\&\\cong&\n{\\tt Hom}_{{\\tt Sh}_{\\tt Ab}\\mathbf{Y}}(\\hat h^\\mathbb{Z}_U\\otimes R^nf_*( \\underline{\\Z}_{\\mathbf{X} }),F)\\\\&\\cong&{\\tt Hom}_{{\\tt Sh}_{\\tt Ab}\\mathbf{Y}}(\\hat  h_U^\\mathbb{Z}\\otimes f_*(K^n)\/{\\tt im}(f_*(K^{n-1})\\to f_*(K^n)),F)\\\\\n&\\stackrel{!}{\\to}&{\\tt Hom}_{{\\tt Sh}_{\\tt Ab}\\mathbf{Y}}(\\hat h_U^\\mathbb{Z}\\otimes f_*(K),F)\\\\\n&\\stackrel{\\ref{system81}}{\\cong}&{\\tt Hom}_{{\\tt Sh}_{\\tt Ab}\\mathbf{Y}}(f_*(f^*\\hat  h_U^\\mathbb{Z}\\otimes K),F)\\\\\n&\\stackrel{\\ref{injpw}}{\\cong}&{\\tt Hom}_{{\\tt Sh}_{\\tt Ab}\\mathbf{X}}(f^*\\hat   h_U^\\mathbb{Z} ,f^!_K(F))\\\\\n&\\cong&{\\tt Hom}_{{\\tt Sh}_{\\tt Ab}\\mathbf{X}}(\\hat   h_U^\\mathbb{Z},f_*\\circ f_K^!(F))\\\\\n&\\cong&f_*\\circ f_K^!(F)(U)\\ ,\n\\end{eqnarray*}\nwhere the map marked by $!$ has degree $n$.\nThe projection formula Lemma \\ref{system81} can be applied since $f^*\\hat h^\\mathbb{Z}_U$ is flat.\nThis transformation preserves homotopy classes of morphisms $F\\to F^\\prime$.\nSince $F$ is injective we have\n$$\\underline{\\tt Hom}(R^nf_* \\underline{\\Z}_{\\mathbf{X} },F)\\cong R\\underline{\\tt Hom}(R^nf_* \\underline{\\Z}_{\\mathbf{X} },F)\\ .$$\nFurther note that\n$f^!_K(F)$ is still a complex of injectives by Lemma \\ref{injpw}.\nTherefore\n$f_*\\circ f_K^!(F)\\cong Rf_*\\circ f^!(F)$.\nHence this chain of maps of complexes induces a transformation\n\\begin{equation}\\label{fetsch}R\\underline{\\tt Hom}(R^nf_* \\underline{\\Z}_{\\mathbf{X} },F)\\to Rf_*\\circ f^!(F)\\ .\\end{equation}\n\n\\subsection{}\n\nIts adjoint is a natural transformation\n$$Rf_* f^* R\\underline{\\tt Hom}(R^nf_* \\underline{\\Z}_{\\mathbf{X} },F)\\to F\\ .$$\nLet us now assume that $f:X\\to Y$ is in addition oriented by an isomorphism\n$R^nf_*\\underline{\\Z}_\\mathbf{X}\\cong \\underline{\\Z}_{\\mathbf{Y}}$.\nWe precompose with this isomorphism  and get the integration map. \n\\begin{ddd}\\label{rdphi}\nThe integration map\n$$\\int_f\\colon Rf_*\\circ f^*\\to {\\tt id}$$ is the natural transformation of functors $D^+({\\tt Sh}_{\\tt Ab}\\mathbf{Y})\\to D^+({\\tt Sh}_{\\tt Ab}\\mathbf{Y})$ of degree $-n$ defined as the composition\n$$Rf_* f^*(F)\\cong Rf_*f^*(\\underline{\\tt Hom}( \\underline{\\Z}_{\\mathbf{Y} },F))\\cong \nRf_* f^*(\\underline{\\tt Hom}(R^nf_*( \\underline{\\Z}_{\\mathbf{X} }),F))\\to F\\ .$$ \n\\end{ddd}\n\n\nIn Lemmas \\ref{ttt12} and \\ref{ttt13} we will verify in the more general case of unbounded derived categories that the integration map is functorial\nwith respect to compositions and compatible with pull-back diagrams.\n\n\n \n\n\n\n\n\n \n\n\n\\section{Operations with unbounded derived categories}\\label{system4001}\n\n\n\n\n\n\\subsection{}\n\nThe category of sheaves ${\\tt Sh}_{\\tt Ab}\\mathbf{X}$ on a locally compact stack is a Grothendieck abelian category (see \\ref{system104}). The category of complexes in a Grothendieck abelian category carries a model category structure (see \\ref{ztoou6}). The unbounded derived category is the associated homotopy category.\nThe goal of the present subsection is to extend the sheaf theory operations $(f^*,f_*)$ and the integration map\n$\\int_f$ to the unbounded derived category.\n\nMany results of the present subsection would continue to hold if one drops the assumption of local compactness in the definition of the site associated to stacks as well as for the stacks themselves.  \nBut the assumption of local compactness is important for the integration map\nsince it uses versions of the projection formula.\n\n\n\n\\subsection{}\n\n\nLet $f\\colon X\\to Y$ be a morphism between locally compact  stacks.\nThen we have an adjoint pair of functors\n$$f^*\\colon C({\\tt Sh}_{\\tt Ab}\\mathbf{Y})\\leftrightarrows C({\\tt Sh}_{\\tt Ab} \\mathbf{X}):f_*\\ .$$\nIn order to descend the functor $f_*$ to the bounded below derived category\nit was sufficient to know that $f_*$ is left exact. In this case the idea is to apply \n$f_*$ to injective resolutions. The descent of the other functor $f^*$ is usually only considered if it exact, but see e.g. \\cite{MR2312554} for more general constructions.  We know by  \\ref{prexact} that  the functor $f^*$ is exact \nif $f$ has local sections.\n\n\n\n\n\n\nIt is not possible to show using the left exactness that $f_*$ preserves quasi-isomorphisms between unbounded complexes of injectives. Even worse, it is not clear how to resolve an unbounded complex by an injective complex. The method to descend $f_*$ to the derived category  uses abstract homotopy theory and works under the additional assumption that $f$ has local sections.\n\n\nRecall that we use a model structure on the category $C({\\tt Sh}_{\\tt Ab}\\mathbf{X})$ of unbounded complexes of\nsheaves for which the equivalences are the quasi-isomorphisms,  and the cofibrations are the\nlevel-wise injections (see \\ref{ztoou6}).  The inclusion\n$C^+({\\tt Sh}_{\\tt Ab}\\mathbf{X})\\hookrightarrow C({\\tt Sh}_{\\tt Ab}\\mathbf{X})$ of the full subcategory of\nbounded below complexes induces an identification\n$D^+({\\tt Sh}_{\\tt Ab}\\mathbf{X})\\cong hC^+({\\tt Sh}_{\\tt Ab}\\mathbf{X})\\hookrightarrow\nhC({\\tt Sh}_{\\tt Ab}\\mathbf{X})=:D({\\tt Sh}_{\\tt Ab}\\mathbf{X})$ of the bounded below derived category as a full subcategory of the unbounded derived category.\n\nThe functor \n$Rf_*:D^+({\\tt Sh}_{\\tt Ab}\\mathbf{X})\\to D^+({\\tt Sh}_{\\tt Ab}\\mathbf{Y})$  is the adjoint of the restriction\nof $f^*$ to the bounded below derived categories, and it is therefore the restriction of $Rf_*:D({\\tt Sh}_{\\tt Ab}\\mathbf{X})\\to D({\\tt Sh}_{\\tt Ab}\\mathbf{Y})$ to be defined below. \n\n\n\\begin{lem}\nIf the morphism $f\\colon X\\to Y$  of locally compact stacks  has local sections, then\n$(f^*,f_*)$ is a Quillen adjoint pair.\n\\end{lem}\n\\begin{proof}\nWe use the  criterion \\cite[Def. 1.3.1 (2)]{MR1650134} in order to show that $f^*$ is a left Quillen functor. We must show that it preserves cofibrations and trivial cofibrations. In other words, we must show that $f^*$ preserves injections and injections which induce isomorphisms on cohomology. Both properties follow from the exactness of $f^*\\colon {\\tt Sh}_{\\tt Ab}\\mathbf{Y}\\to {\\tt Sh}_{\\tt Ab} \\mathbf{X}$. \n\\end{proof}\n\n\n\n\n\\subsection{}\n\nLet $f\\colon X\\to Y$ be a map between locally compact stacks which has local sections. Since $(f^*,f_*)$ is a Quillen adjoint pair\nit induces a derived adjoint pair\n$$Lf^*\\colon hC({\\tt Sh}_{\\tt Ab}\\mathbf{Y})\\leftrightarrows hC({\\tt Sh}_{\\tt Ab} \\mathbf{X}):Rf_*$$\n(see Lemma \\cite[Lemma 1.3.10]{MR1650134}).\nSince $f^*$ is exact it directly  descends to the homotopy category.  \n\n\n\n\n\n\n\n\\subsection{}\n\nLet $g\\colon Y\\to Z$ be a second map of locally compact stacks which admits local sections.\nThen we have adjoint  canonical isomorphisms\n\\begin{equation}\\label{wjehfdajewd}\n(g\\circ f)^*\\cong f^*\\circ g^*\\ ,\\quad (g\\circ f)_*\\cong g_*\\circ f_*\\ .\n\\end{equation}\n\\begin{lem}\\label{6.53}\nWe have a canonical isomorphism\n$$R(g\\circ f)_*\\cong Rg_*\\circ Rf_*\\ .$$\n\\end{lem} \n\\begin{proof}\nUsing \\cite[Thm. 1.3.7]{MR1650134} we have a natural transformation\n\\begin{equation}\\label{ajhwoirofg}\nR(g\\circ f)_*\\cong R(g_*\\circ f_*)\\to Rg_*\\circ Rf_*\n\\end{equation}\nwhich is adjoint to\n\\begin{equation}\\label{hjefdauzew}\nLf^*\\circ  Lg^*\\to L(f^*\\circ g^*)\\cong L(g\\circ f)^*\\ .\n\\end{equation}\nSince $Lf^*$, $Lg^*$, and $L(g\\circ f)^*$ are plain descents of\n$f^*$, $g^*$, and $(g\\circ f)^*$ to the homotopy category it follows from (\\ref{wjehfdajewd}) that (\\ref{hjefdauzew}) is an isomorphism. Therefore its adjoint\n(\\ref{ajhwoirofg}) is also an isomorphism. \n\\end{proof} \n\n\\subsection{}\n\nConsider a Cartesian diagram of locally compact stacks \n$$\\xymatrix{U\\ar[d]^g\\ar[r]^v&X\\ar[d]^f\\\\V\\ar[r]^u&Y}\\ ,$$\nwhere all maps have local sections.\nUsing the unit\n${\\tt id}\\to v_*\\circ v^*$, the counit $u^*\\circ u_*\\to {\\tt id}$,  and (\\ref{wjehfdajewd}) we define (see Lemma \\ref{lem:pullpush}) a transformation\n$$u^*\\circ f_*\\to u^*\\circ f_*\\circ v_*\\circ v^*\\cong u^*\\circ u_*\\circ g_*\\circ v^*\\to g_*\\circ v^*\\ .$$ It is functorial with respect to compositions of such Cartesian diagrams.\nBy the same method we obtain a transformation\n\\begin{equation}\\label{jfurfcscdde}\nLu^*\\circ Rf_*\\to Rg_*\\circ Lv^*\\ .\n\\end{equation}\n\n\n\n\n\n\n\n\n\n\n\\subsection{}\nBy Lemma \\ref{lem:pullpush} we know that the transformation\n$$u^*\\circ f_*\\to g_*\\circ v^*$$ is in fact\nan isomorphism. The derived version is more complicated\n and needs an additional assumption.\n \n\\begin{lem}\\label{eiuwh}\nAssume that $g$ is representable and  $g_*\\colon {\\tt Sh}_{\\tt Ab}\\mathbf{U}\\to {\\tt Sh}_{\\tt Ab} \\mathbf{V}$ has finite cohomological dimension. Then the transformation (\\ref{jfurfcscdde}) is an isomorphism.\n\\end{lem}\n\\begin{proof}\nWe choose  fibrant replacement functors $$I_X\\colon C({\\tt Sh}_{\\tt Ab}\\mathbf{X})\\to\nC({\\tt Sh}_{\\tt Ab}\\mathbf{X})\\ ,\\quad I_U: C({\\tt Sh}_{\\tt Ab}\\mathbf{U})\\to\nC({\\tt Sh}_{\\tt Ab}\\mathbf{U})\\ .$$ \nIn terms of these replacement functors we can write the compositions of derived functors as descents of quasi-isomorphism preserving functors on the level of chain complexes:\n$$Lu^*\\circ Rf_*\\cong u^*\\circ f_*\\circ I_X\\ ,\\quad Rg_* \\circ Lv^*\\cong g_*\\circ I_U\\circ v^*\\ .$$\nLet $F\\in  C({\\tt Sh}_{\\tt Ab}\\mathbf{X})$. We must show that the marked arrows  (induced by ${\\tt id}\\to I_U$ and ${\\tt id}\\to I_X$) in the following sequence are quasi-isomorphisms\n$$u^*f_*I_X(F)\\cong g_* v^*I_X(F)\\stackrel{(*)}{\\to} g_*I_Uv^*I_X(F)\\stackrel{(**)}{\\leftarrow} g_*I_Uv^*(F)\\ .$$\nThe arrow marked by $(**)$ is a quasi-isomorphism since the functors\n$g_*I_U$  and $v^*$ preserve quasi-isomorphisms, and  $F \\to I_X(F)$ is a quasi-isomorphism.\n\nThe morphism $(*)$ is more complicated, and it is here where we need the assumption.\nIt is a property of the injective model structure on the chain complexes of a Grothendieck  abelian category that a fibrant complex consists of  injective objects.\nAn injective sheaf is in particular flabby. Since $v$ has local sections $v^*$  preserves flabby sheaves (Lemma \\ref{flabbypres}). We conclude that $v^*I_X(F)$ is a complex of flabby sheaves.\n\nLet $G\\in C({\\tt Sh}_{\\tt Ab} \\mathbf{U})$ be a complex of flabby sheaves. We must show that\n$g_*(G)\\to g_*I_U(G)$ is a quasi-isomorphism. Since $g_*$ is an additive functor this assertion is equivalent to the assertion that $g_*(C)$ is exact, where $C$ is the mapping cone\nof $G\\to I_U(G)$. Note that $C$ is an exact complex of flabby sheaves. It decomposes into short exact sequences\n$$0\\to Z^n\\to C^n\\to Z^{n+1}\\to 0\\ ,$$\nwhere $Z^n:=\\ker(C^n\\to C^{n+1})$.\nSince $g$ is representable we know by Lemma \\ref{hdqoiwdwqw}  that flabby sheaves are $g_*$-acyclic.\nTherefore we obtain the exact sequence\n$$0\\to g_*(Z^n)\\to g_*(C^n)\\to g_*(Z^{n+1})\\to R^1g_*(Z^{n})\\to 0$$\nand the isomorphisms\n$$R^kg_*(Z^{n+1})\\cong R^{k+1}g_*(Z^{n})$$\nfor all $k\\ge 1$.\nBy induction we show that for $k\\ge 1$ and all $l\\in \\mathbb{N}$ we have\n$$R^kg_*(Z^{n})\\cong R^{k+l}g_*(Z^{n-l})\\ .$$\nSince we assume that $g_*$ has bounded cohomological dimension we conclude that\n$R^k(Z^n)\\cong 0$ for all $n\\in \\mathbb{Z}$ and $k\\ge 1$.\nIn particular the sequences \n$$0\\to g_*(Z^n)\\to g_*(C^n)\\to g_*(Z^{n+1})\\to 0$$\nare exact for all $n\\in \\mathbb{Z}$.  This shows the exactness of $g_*(C)$. \n\\end{proof}\n\n\n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n \n\n\\subsection{}\n\nLet now $f\\colon X\\to Y$ be a representable map between locally compact stacks which is an oriented locally trivial fiber bundle of closed  oriented manifolds of dimension $n$. In particular, $f$ has local sections and is proper, and $f_*$ has cohomological dimension $\\le n$. We consider the canonical flabby resolution (see \\ref{system100})\n$$0\\to \\underline{\\Z}_\\mathbf{X} \\to {\\mathcal Fl}^0(\\underline{\\Z}_\\mathbf{X})\\to {\\mathcal Fl}^1( \\underline{\\Z}_\\mathbf{X})\\to \\dots\n\\ .$$   \nThen we know that $f_*{\\mathcal Fl}( \\underline{\\Z}_\\mathbf{X})$ is exact above degree $n$.\nWe let\n$K $ denote the truncation (\\ref{reconl}) of this resolution at $n$. \nThen the orientation of the bundle (see \\ref{iquhuiqwdddqw}) gives the last isomorphism in the following composition\n$$f_*K^n\\to f_*K^n\/{\\tt im}(f_* K^{n-1}\\to f_*K^n)\\cong R^nf_*\\underline{\\Z}_\\mathbf{X}\\cong  \\underline{\\Z}_\\mathbf{Y}\\ .$$ We let \n$T_{K}\\colon C({\\tt Sh}_{\\tt Ab}\\mathbf{X})\\to C({\\tt Sh}_{\\tt Ab}\\mathbf{X})$ denote the functor which associates to the complex\n$F$ the total complex $T_{K}(F)$ of $F\\otimes K$.\nThe  projection formula Lemma \\ref{system81} for the proper representable map $f$ gives an isomorphism\n$$f_*\\circ T_{K}\\circ f^*(F)\\cong T_{f_*K}(F)$$\nfor complexes of flat sheaves $F\\in C({\\tt Sh}_{\\tt Ab}\\mathbf{Y})$.\nThe inclusion $ \\underline{\\Z}_\\mathbf{X}\\to K$ and the projection $f_*K\\to \\underline{\\Z}_\\mathbf{Y}[-n]$ induces transformations\n\\begin{equation}\\label{iofejoiwfwefwef}\n{\\tt id}\\to T_{K}\\ ,\\quad T_{f_*K^\\cdot}\\to {\\tt id}[-n]\\ .\n\\end{equation}\n\n\\subsection{}\n\nWe know by Lemma \\ref{ggtre1} that the functor $$f_*\\circ T_K\\colon {\\tt Sh}_{\\tt Ab} \\mathbf{X}\\to {\\tt Sh}_{\\tt Ab} \\mathbf{Y}$$\nis exact. We choose a functorial fibrant replacement functor ${\\tt id}\\to I$ on $C({\\tt Sh}_{\\tt Ab}\\mathbf{X})$. Let $R:C({\\tt Sh}_{\\tt Ab}\\mathbf{Y})\\to C({\\tt Sh}_{\\tt Ab}\\mathbf{Y})$ be the functorial flat\nresolution functor of \\ref{hdwidhwqdiwqdwdw}, extended to unbounded complexes.\nThen we consider sequence\n\\begin{equation}\\label{jhwjhdkqews}\nf_*\\circ I\\circ f^*\\to f_*\\circ T_K\\circ I\\circ f^*\\stackrel{!}{\\leftarrow} f_*\\circ T_K\\circ f^*\\stackrel{!}{\\leftarrow} \nf_*\\circ T_K\\circ f^*\\circ R \\cong T_{f_* K}\\circ R \\to R[-n]\\to  {\\tt id}[-n]\\ .\n\\end{equation} \nAll functors in this sequence preserve quasi-isomorphisms and therefore descend plainly to the homotopy category $hC({\\tt Sh}_{\\tt Ab} \\mathbf{X})$. Since $f_*\\circ T_K$ is exact the arrows marked by $!$  induce isomorphisms of functors on the homotopy category.\nNow observe that  the descent of $f_*\\circ I\\circ f^*$ to the homotopy category is isomorphic to $Rf_*\\circ Lf^*$.  Therefore (\\ref{jhwjhdkqews}) induces a transformation\n\\begin{equation}\\label{jkqwjhewuqieq3w}\n\\int_f\\colon Rf_*\\circ Lf^*\\to {\\tt id}[-n]\\ .\n\\end{equation}\n\n\\begin{ddd}\nThe transformation (\\ref{jkqwjhewuqieq3w}) is called the integration map.\n\\end{ddd}\nIt generalizes  Definition \\ref{rdphi} from the bounded below to the unbounded derived category.\n\n\n\\subsection{}\n\nIn order to have a simple definition we have defined the integration map using a canonical resolution of $\\underline{\\Z}_\\mathbf{X}$ of length $n$. In fact, we can use more general resolutions. This will turn out to be useful for the verification of functorial properties of the integration map.\n\n\\subsection{}\\label{system303}\n\nLet us first recall some notation.\nAn object  $(U\\to X)\\in \\mathbf{X}$ represents the presheaf $h_U\\in \\Pr\\mathbf{X}$ (see also \\ref{system202}).  We let $h_U^\\mathbb{Z}\\in \\Pr_{\\tt Ab} \\mathbf{X}$ be the free abelian presheaf generated by $h_U$ and form $\\hat h_U^\\mathbb{Z} := i^\\sharp h_U^\\mathbb{Z} \\in {\\tt Sh}_{\\tt Ab}\\mathbf{X}$. \n\\begin{ddd}\nLet $f:X\\to Y$ be a map of locally compact stacks.\nA sheaf $F\\in {\\tt Sh}_{\\tt Ab}\\mathbf{X}$ is called locally $f_*$-acyclic, if for every $(U\\to X)\\in \\mathbf{X}$ and $k\\ge 1$ we have $R^k f_*(\\hat h_U^\\mathbb{Z}\\otimes F)\\cong 0$.\n\\end{ddd}\n\n\\subsection{}\n\n Let  $f:X\\to Y$ be a map of locally compact stacks.\n\\begin{lem}\\label{9odclkmfwef}\n Assume that the cohomological dimension of $f_*$ is bounded by $n$.\nIf $$L^0\\to L^1\\to \\dots \\to L^{n-1}\\to L^n\\to0$$ is an exact complex such that the\n$L^i$ are $f_*$-acyclic (or locally $f_*$-acyclic) for $i=0,\\dots,n-1$, then $L^n$ is $f_*$-acyclic (or locally $f_*$-acyclic, respectively). \n\\end{lem}\nThis can be shown by a similar induction argument as used in the proof of Lemma \\ref{eiuwh}. \n\\hspace*{\\fill}$\\Box$ \\\\[0cm]\\noindent\n\n\\subsection{}\n\nLet $f:X\\to Y$ be a map of locally compact stacks.\n\\begin{lem}\\label{wgdqwededed}\nLet $(V\\to X)\\in \\mathbf{X}$ and $F$ be locally $f_*$-acyclic. Then $\\hat h_V^\\mathbb{Z}\\otimes F$ is locally $f_*$-acyclic.\n\\end{lem}\n\\begin{proof}\n Indeed, let $(U\\to  X)\\in \\mathbf{X}$.\nThen we have \n$$\\hat h_U^\\mathbb{Z}\\otimes(\\hat h_V^\\mathbb{Z}\\otimes F)\\cong (\\hat h_U^\\mathbb{Z}\\otimes \\hat h_V^\\mathbb{Z})\\otimes F\\ .$$\nFurthermore we have\n$$\\hat h_U^\\mathbb{Z}\\otimes \\hat h_V^\\mathbb{Z}\\stackrel{Lemma\\: \\ref{kion}}{\\cong} i^\\sharp (h^\\mathbb{Z}_U\\otimes^p h^\\mathbb{Z}_V)\\cong\ni^\\sharp (h_U\\times h_V)^\\mathbb{Z}\\cong i^\\sharp h_{U\\times_\\mathbf{X} V}^\\mathbb{Z}\\cong \n  \\hat h^\\mathbb{Z}_{U\\times_X V}\\ ,$$\nwhere we use the fact, that the absolute product in $\\mathbf{X}$ is given by the fiber product spaces over $X$ (\\cite[Lemma 2.3.3]{bss}).\nIt follows that\n$$R^kf_*(\\hat h_U^\\mathbb{Z}\\otimes(\\hat h_V^\\mathbb{Z}\\otimes F))\\cong R^kf_*(\\hat h_{U\\times_X V}^\\mathbb{Z}\\otimes F)\\cong 0$$\nfor all $k\\ge 1$. \n\\end{proof}\n\n\\subsection{}\n\nLet $f:X\\to Y$ be a map of locally compact stacks.\n\\begin{lem}\\label{wejjqklxskj}\nAssume that $f$ is proper, representable, and  that the cohomological dimension of $f_*$ is bounded.\nIf $F\\in {\\tt Sh}_{\\tt Ab}\\mathbf{X}$ is flat and locally $f_*$-acyclic, then for any sheaf\n$G\\in {\\tt Sh}_{\\tt Ab}\\mathbf{X}$ the tensor product $G\\otimes F$ is  $f_*$-acyclic and locally $f_*$-acyclic).\n\\end{lem}\n\\begin{proof}\nWe construct a resolution\n$\\dots\\to G_j\\to G_{j-1}\\to \\dots\\to G_0\\to G$, where all\n$G_j$ are coproducts of sheaves of the form $\\hat h^\\mathbb{Z}_U$.\nIn fact, we have a surjection\n$$\\bigoplus_{\\hat h_U^\\mathbb{Z}\\to G}\\hat h_U^\\mathbb{Z} \\to G\\ .$$\nIf we have already constructed $G_j\\to\\dots\\to G_0\\to G$, then we extend\nthis complex by\n$$\\bigoplus_{\\hat h_U^\\mathbb{Z}\\to \\ker(G_j\\to G_{j-1})}\\hat h^U_\\mathbb{Z} \\to G_j\\ .$$\nSince $F$ is flat, the complex\n$$\\dots \\to G_j\\otimes F\\to \\dots\\to G_0\\otimes F\\to G\\otimes F$$\nis exact. The tensor product commutes with direct sums.\nTherefore $G_j\\otimes F$ is a sum of  $f_*$-acyclic sheaves, and by Lemma \n\\ref{wgdqwededed} also of locally $f_*$-acyclic sheaves.  Since $f_*$ commutes with direct sums (Lemma \\ref{sumpre45}) the sheaves $G_j\\otimes F$ are themselves  $f_*$-acyclic and locally $f_*$-acyclic. With Lemma \\ref{9odclkmfwef} we conclude that\n$G\\otimes F$ is  $f_*$-acyclic and locally $f_*$-acyclic. \n\\end{proof}\n\n\\subsection{}\n\nLet $f:X\\to Y$ be a map of locally compact stack.\n\\begin{lem}\nIf $f$ is representable, then a flasque sheaf is locally $f_*$-acyclic.\n\\end{lem}\n\\begin{proof}\nLet $F\\in {\\tt Sh}_{\\tt Ab}\\mathbf{X}$ be flasque. We consider\n $(U\\to Y)\\in \\mathbf{Y}$ and form the Cartesian diagram\n$$\\xymatrix{V\\ar[r]\\ar[d]^g&X\\ar[d]^f\\\\U\\ar[r]&Y}\\ .$$\nThen $(V\\to X)\\in \\mathbf{X}$ and we have $Rf_*(F)_U\\cong Rg_*(F_V)$.\nThe restriction $F_V\\in {\\tt Sh}_{\\tt Ab}(V)$ is still flasque.\nA flasque sheaf on $(V)$ is $g$-soft (see  \\cite[Definition 3.1.1]{MR1299726}).\nBut this implies that\n$R^kg_*(F_V)=0$ for $k\\ge 1$.\nSince $U\\to Y$ was arbitrary we see that\n$R^kf_*(F)=0$ for $k\\ge 1$.\n\\end{proof}\n\n\n\n\n\n\n\\subsection{}\n\n\nLet us from now on until the end of this subsection assume that $f:X\\to Y$ is a proper representable map of locally compact stacks which is an oriented locally trivial fiber bundle with fiber a closed connected topological manifold of dimension $n$. \n\n\n\n\n\nSince a flat and flasque sheaf is locally $f_*$-acyclic and $K$ is a truncation of a flat and flasque resolution of $ \\underline{\\Z}_{\\mathbf{X}}$ we see by Lemma \\ref{9odclkmfwef} that $K$ is a complex of flat and locally $f_*$-acyclic sheaves.  These are the two properties which make the theory work.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nLet $L\\to M$ be a quasi-isomorphism between upper bounded complexes of locally $f_*$-acyclic and flat sheaves. \n\n\\begin{lem}\\label{uuusaaassq}\nFor every  complex $F\\in C({\\tt Sh}_{\\tt Ab}\\mathbf{X})$ the induced map\n$$f_*(F\\otimes L)\\to f_*(F\\otimes M)$$\nis a quasi-isomorphism.\n\\end{lem}\n\\begin{proof}\nWe form the mapping cone $C$ of $L\\to M$. It is an exact complex of locally $f_*$-acyclic and flat sheaves. Since the tensor product and $g_*$ commute with the formation of a mapping cone it suffices to show that $f_*(F\\otimes C)$ is exact.\n \nWe know by Lemma \\ref{wejjqklxskj} that $F\\otimes C$ is a complex of $f_*$-acyclic sheaves. We claim  that $F\\otimes C$ is exact. \n\nTo this end we first show that $H\\otimes C$ is exact for an arbitrary sheaf $H\\in {\\tt Sh}_{\\tt Ab}\\mathbf{X}$.\nWe decompose the exact complex $C$ into short exact sequences   \n$$S(k)\\colon 0\\to Z^k\\to C^k\\to Z^{k+1}\\to 0$$ where\n$Z^k:=\\ker(C^k\\to C^{k+1})$. \nUsing the fact that the sheaves $C^k$ are flat we obtain\n$$0\\to {\\tt Tor}_1(H,Z^{k+1})\\to H\\otimes Z^k\\to H\\otimes C^k \\to H\\otimes Z^{k+1}\\to 0$$ and the isomorphisms ${\\tt Tor}_{m+1}(H,Z^{k+1})\\cong {\\tt Tor}_{m}(H,Z^k)$ for all $m\\ge 1$.\nSince $\\mathbb{Z}$ is one-dimensional we know that ${\\tt Tor}_{m}\\cong 0$ for $m\\ge 2$. Inductively\nwe conclude that ${\\tt Tor}_1(H,Z^k)\\cong 0$ for all $k\\in \\mathbb{Z}$. \nIt follows that $H\\otimes S(k)$ is exact for all $k\\in \\mathbb{Z}$. \nThis implies that $H\\otimes C$ is exact.\n\n\nLet now $F$ be a complex. Using the previous result and a spectral sequence argument we conclude that  the total complex associated to the double complex $F\\otimes C$ is exact. \n\n\nLet now $C\\in C({\\tt Sh}_{\\tt Ab}\\mathbf{X})$ be an exact  complex of $f_*$-acyclic sheaves.\nWe show that this implies that $f_*(C)$ is exact.\n The complex $C$  decomposes into short exact sequences\n$$0\\to Z^n\\to C^n\\to Z^{n+1}\\to 0\\ ,$$\nwhere $Z^n:=\\ker(C^n\\to C^{n+1})$.\nUsing the fact that $C^n$ is $f_*$-acyclic we\nobtain the exact sequence\n$$0\\to f_*(Z^n)\\to f_*(C^n)\\to f_*(Z^{n+1})\\to R^1f_*(Z^{n})\\to 0$$\nand the isomorphisms\n$$R^kf_*(Z^{n+1})\\cong R^{k+1}f_*(Z^{n})$$\nfor all $k\\ge 1$.\nBy induction we show that for $k\\ge 1$ and all $l\\in \\mathbb{N}$ we have\n$$R^kf_*(Z^{n})\\cong R^{k+l}f_*(Z^{n-l})\\ .$$\nSince  $f_*$ has bounded cohomological dimension we conclude that\n$R^kf_*(Z^n)\\cong 0$ for all $n\\in \\mathbb{Z}$ and $k\\ge 1$.\nIn particular the sequences \n$$0\\to f_*(Z^n)\\to f_*(C^n)\\to f_*(Z^{n+1})\\to 0$$\nare exact for all $n\\in \\mathbb{Z}$.  This shows the exactness of $f_*(C)$. \n\\end{proof}\n\n\n\\subsection{}\n\n\\begin{lem}\\label{edwiuhed}\nThe integration map is independent of the choice of a flat locally $f_*$-acyclic resolution\n$K$ of $ \\underline{\\Z}_{\\mathbf{X}}$ of length $n$.\n\\end{lem}\n\\begin{proof}\nLet $K,L$ are two such resolutions. Assume that there exists a quasi-isomorphism\n$K\\to L$.\nThe identification\n$${\\tt coker} (f_*L^{n-1}\\to f_*L^n)\\cong {\\tt coker} (f_*K^{n-1}\\to f_*K^n)\\cong R^nf_*(\\underline{\\Z}_\\mathbf{X})\\cong \\underline{\\Z}_\\mathbf{Y}$$\ngives a map $f_*L\\to  \\underline{\\Z}_\\mathbf{Y}[-n]$  which induces\nthe transformation\n$T_{f_*L}\\to {\\tt id}$ of degree $-n$.\n\n\nIt induces a commutative diagram\n$$\\xymatrix{f_*If^*\\ar[r]\\ar@{=}[d]&f_*T_KIf^*\\ar[d]&f_*T_Kf^*\\ar[d]\\ar[l]_\\sim &f_*T_Kf^*R\\ar[l]_\\sim\\ar[r]^\\cong\\ar[d]&T_{f_*K}R\\ar[r]\\ar[d] &R\\ar[r]\\ar@{=}[d]&{\\tt id}\\ar@{=}[d]\\\\\nf_*If^*\\ar[r]&f_*T_LIf^*&f_*T_Lf^*\\ar[l]_\\sim&f_*T_Lf^*R\\ar[l]_\\sim\\ar[r]^\\cong&T_{f_*L}R\\ar[r]&R\\ar[r] &{\\tt id}}$$\n\nThe upper horizontal composition is the integration map defined using $K$ (see \\ref{jhwjhdkqews}), and the lower horizontal composition is the integration map defined using $L$. We see that both maps are equal.\n\nLet now $K,L$ again be flat and locally $f_*$-acyclic resolutions of $\\underline{\\Z}_\\mathbf{X}$ of length $n$.\nWe complete the proof of the Lemma by showing that there exists a third such resolution $M$\ntogether with quasi-isomorphisms $K\\stackrel{\\sim}{\\to}M\\stackrel{\\sim}{\\leftarrow} L$.\n\nThe maps $\\underline{\\Z}_{\\mathbf{X}}\\to K$ and $\\underline{\\Z}_{\\mathbf{X}}\\to L$, respectively, induce maps\n$K\\to K\\otimes L$ and $L\\to K\\otimes L$ which are quasi-isomorphisms.  We further get induced quasi-isomorphisms\n\\begin{equation}\\label{sswuebh}\nK\\to {\\mathcal Fl}(K\\otimes L)\\ ,\\quad L\\to {\\mathcal Fl}(K\\otimes L)\\ .\n\\end{equation}\nWe let $M:=\\tau^{\\le n} {\\mathcal Fl}(K\\otimes L)$. \nThe maps (\\ref{sswuebh}) factorize over $M$.\nNote that $K\\otimes L$ is flat. Since ${\\mathcal Fl}$ and truncation preserve flatness (see Lemma \\ref{flat-preserv}), we see that $M$ is flat. Since ${\\mathcal Fl}$ in fact produces flasque and hence locally $f_*$-acyclic resolutions, and the cohomological dimension of $f_*$ is bounded by $n$ we conclude by Lemma \\ref{9odclkmfwef} that $M$ is locally $f_*$-acyclic.\n\\end{proof}\n\n\n\n\n\n \n\\subsection{}\nIn this paragraph we show that the integration map is functorial. \nLet $g\\colon Y\\to Z$ be a second proper and representable map of locally compact stacks which is\nan oriented  locally trivial fiber bundle of closed   $m$-dimensional manifolds.\n\n\\begin{lem}\\label{ttt12}\nWe have a commutative diagram\n$$\\xymatrix{Rg_*\\circ Rf_*\\circ Lf^*\\circ Lg^*\\ar[d]^{Rg_*(\\int_f)}\\ar[r]^{\\cong}& R(g\\circ f)_*\\circ L(g\\circ f)^*\\ar[d]^{\\int_{g\\circ f}}\\\\\nRg_*\\circ Lg^*[-n]\\ar[r]^{\\int_g}&{\\tt id}[-n-m]}\\ .$$\n \\end{lem}\n\\begin{proof}\nThe following sequence of modifications transforms the down-right composition into the right-down composition. \n\\begin{multline}\n  g_*If_* If^*g^*\\to g_*I f_*T_K If^*g^*\\stackrel{\\sim}{\\leftarrow} g_*\n  If_*T_K f^*g^* R\n  \\to g_*Ig^*R\\\\ \n  \\to g_*T_L I g^*R\\stackrel{\\sim}{\\leftarrow}\n  g_*T_L g^*R \\to{\\tt id} \\label{riocsn1}\n\\end{multline}\n\\begin{multline}\n  g_*I f_* If^*g^*\\to g_* T_L I f_* If^*g^*\\to g_* T_L I f_*T_K\n  If^*g^*\n\\stackrel{\\sim}{\\leftarrow} g_* T_L f_* T_K I f^*g^*\\\\\n  \\stackrel{\\sim}{\\leftarrow} g_* T_L f_*T_K f^*g^*R \\to g_*T_L g^*R\n  \\to{\\tt id} \\label{riocsn2}\n\\end{multline}\n\\begin{multline}\n  g_*I f_* If^*g^* \\to g_* T_L I f_* If^*g^*\\stackrel{\\sim}{\\leftarrow} g_*\n  T_L f_* If^*g^*\n \\to g_* T_L f_*T_K If^*g^*\\\\\n\\stackrel{\\sim}{\\leftarrow} g_*\n  T_L f_* T_K f^*g^*R \\to g_*T_L g^* R\\to{\\tt id} \\label{riocsn3}\n\\end{multline}\n\\begin{multline}\n  g_*I f_* If^*g^* \\stackrel{\\sim}{\\leftarrow}g_* f_* If^*g^* \\to g_* T_L f_*\n  If^*g^*\n \\to g_* T_L f_*T_K If^*g^*\\\\\n\\stackrel{\\sim}{\\leftarrow} g_* T_L f_*\n  T_K f^*g^*R \\to g_*T_L g^*R \\to{\\tt id} \\label{riocsn4}\n\\end{multline}\n\\begin{multline}\n  g_* f_* If^*g^* \\to g_* T_L f_* If^*g^*\\to g_* T_L f_*T_K If^*g^*\n  \\stackrel{\\sim}{\\leftarrow} g_* T_L f_*T_KR  If^*g^*\\\\\n\\stackrel{\\sim}{\\leftarrow} g_* T_L f_* T_KR f^*g^*R \\to g_*T_L g^*R\n  \\to{\\tt id} \\label{riocsn44}\n\\end{multline}\n\\begin{multline}\n  g_* f_* If^*g^* \\to g_* T_L f_*T_K If^*g^*\\stackrel{\\sim}{\\leftarrow}g_*f_*\n  T_{f^*L\\otimes K} R If^*g^*\\\\\n\\stackrel{\\sim}{\\leftarrow} g_*f_* T_{f^*L\\otimes\n    K}R f^*g^* R \\to g_*T_L g^* R\\to{\\tt id} \\label{riocsn5}\n\\end{multline}\n\\begin{multline}\n  (g\\circ f)_* I (g\\circ f)^* \\to (g\\circ f)_* T_{M} I (g\\circ\n  f)^*\n\\stackrel{\\sim}{\\leftarrow} (g\\circ f)_* T_{M} (g\\circ f)^*R\n  \\to{\\tt id} \\label{riocsn6}\n\\end{multline}\nThe transition from (\\ref{riocsn1}) to (\\ref{riocsn2}) uses the fact  that tensoring  with $L$ and the map ${\\tt id}\\to T_L$ can be commuted with the intermediate operations.\nIn order to go from (\\ref{riocsn2}) to (\\ref{riocsn3}) we use the fact that $g_*T_L$ preserves quasi-isomorphisms. The same reason and the fact that $f_*$ preserves fibrant objects is behind the transition from (\\ref{riocsn3}) to (\\ref{riocsn4}). We use e.g. the isomorphism\n$g_*f_*If^*g^*\\stackrel{\\sim}{\\to} g_*I f_* I f^*g^*$. \nThere is a vertical quasi-isomorphism from (\\ref{riocsn44}) to (\\ref{riocsn4}).\nThe step from (\\ref{riocsn44}) to (\\ref{riocsn5}) uses the isomorphism $T_Lf_*T_KR\\stackrel{\\sim}{\\to}\n f_*T_{f^*L\\otimes K}R$ given by the  projection formula.\n The weak equivalence in (\\ref{riocsn5}) is not obvious (since $f^*L\\otimes K$ is not obviously $g_*f_*$-acyclic), but follows from the fact, that this line is isomorphic to the previous\n(\\ref{riocsn44}).\nIn the last step from (\\ref{riocsn5}) to (\\ref{riocsn6}) we use the map $f^*L\\otimes K\\to M$ given by a  truncated flabby resolution of $f^*L\\otimes K$ and the fact that the integration map is independent of the choice of the resolution.\n\\end{proof}\n\n\\subsection{}\n\n\nConsider a cartesian diagram of locally compact stacks \n\\begin{equation}\\label{udgqwuidqwdqwdqwdw}\\xymatrix{V\\ar[d]^g\\ar[r]^v&X\\ar[d]^f\\\\U\\ar[r]^u&Y}\\ .\\end{equation}\nWe assume that $f$ and $u$, and hence also $g$ and $v$ have local sections.\nFurthermore we assume that $f$ is representable and a locally trivial oriented fiber bundle\nwith a closed manifold as fiber. Then $g$ has these properties, too.\nThe orientation of $g$ is induced by\n$$R^ng_*\\underline{\\Z}_{\\mathbf{V}}\\cong R^ng_*v^*\\underline{\\Z}_{\\mathbf{X}}\\cong  u^*R^nf_*\\underline{\\Z}_{\\mathbf{X}}\\cong u^*\\underline{\\Z}_{\\mathbf{Y}}\\cong \\underline{\\Z}_\\mathbf{U}$$\nWe get   diagrams\n\\begin{equation}\\label{ieuwfhwefjkp}\\xymatrix{u^*Rf_*f^*\\ar[d]_{u^*\\int_f}\\ar[r]^{(\\ref{jfurfcscdde})}&Rg_*v^*f^*\\ar[d]^{(\\ref{hjefdauzew})}\\\\u^*&Rg_*g^*u^*\\ar[l]^{\\int_g}} \\end{equation}\\begin{equation}\\label{eq:comm_u_lower}\n    \\xymatrix{\n      Ru_* Rg_* g^* \\ar[d]^{Ru_*\\int_g}\\ar[r]&Rf_* Rv_* g^*\\\\\n       Ru_* & Rf_*f^* Ru_*\\ar[l]^{\\int_f Ru_*}\\ar[u]^\\sim\n    }\n\\end{equation}\n\n For the upper horizontal transformation in (\\ref{ieuwfhwefjkp}) we use \\ref{6.53}, and for the\n  right vertical one (\\ref{wqdwqdqw}) or \\ref{eiuwh}. Note that\n  only in the bounded below derived \n  category the right vertical morphism is an\n  equivalence for general $u$ (which is anyway the situation in which we will\n  apply the assertion). \n\n\\begin{lem}\\label{system300}\\label{ttt13}\\label{lem:funct_int} \\label{lem:funct_int1}\nThe diagrams (\\ref{ieuwfhwefjkp}) and  (\\ref{eq:comm_u_lower})  commutes.\n \\end{lem}\n\n\n\n\n\nTo prove Lemma \\ref{system300}, we start with the following two technical\nlemmas.\n \\begin{lem}       \\label{lem:two_projections}\n    Given a Cartesian diagram (\\ref{udgqwuidqwdqwdqwdw}) of locally compact stacks such that $f$ and $u$ have local sections, then  for \n     sheaves $K\\in {\\tt Sh}_{\\tt Ab}\\mathbf{X}$  and $F\\in {\\tt Sh}_{\\tt Ab}\\mathbf{U}$  the following diagram commutes:\n    \\begin{equation*}\n      \\begin{CD}\n        f_*K\\otimes u_* F @>=>> f_*K\\otimes u_*F\\\\\n        @VV{\\ref{system80}}V @VV{\\ref{system80}}V\\\\\n          f_*(K\\otimes f^*u_*F) &&  u_*(u^*f_*K\\otimes F)\\\\\n          @V{\\sim}V{\\ref{lem:pullpush}}V @V{\\sim}V{\\ref{lem:pullpush}}V\\\\\n          f_*(K\\otimes v_*g^*F) &&  u_*(g_*v^*K\\otimes F)\\\\\n          @VV{\\ref{system80}}V @VV{\\ref{system80}}V\\\\\n          f_*v_*(v^*K\\otimes g^*F)&&  u_*g_*(v^*K\\otimes g^*F)\\\\\n          @V{\\sim}V{\\ref{feriueewwwwzzz}}V  @V{\\sim}V{\\ref{feriueewwwwzzz}}V\\\\\n          h_*(v^*K\\otimes g^*F)@>=>> h_*(v^*K\\otimes g^*F)\n      \\end{CD}\\ ,\n    \\end{equation*}\n    where $h:=f\\circ v=u\\circ g$.\n  \\end{lem}\n  \\begin{proof}\n    By Definition \\ref{system80}, the left vertical morphism is the\n    image of the identity under the following sequence of maps\n    \\begin{multline*}\n      {\\tt Hom}(v^*K\\otimes g^*K,v^*K\\otimes g^*K) \\to {\\tt Hom}(v^*f^*f_*K\\otimes\n      v^*v_*g^*K,v^*K\\otimes g^*K)\\\\\n      \\to {\\tt Hom}(v^*(f^*f_*K\\otimes\n      f^*u_*K),v^*K\\otimes g^*K)\n      \\to {\\tt Hom}(f^*(f_*K\\otimes u_*K)\n      ,v_*(v^*K\\otimes g^*K))\\\\\n \\to {\\tt Hom}(f_*K\\otimes u_*K, f_*v_*(v^*K\\otimes\n      g^*K) ) \\to {\\tt Hom}(f_*K\\otimes u_*K, h_*(v^*K\\otimes g^*F)).\n    \\end{multline*}\n    The right vertical morphism, on the other hand, is given by\n    \\begin{multline*}\n       {\\tt Hom}(v^*K\\otimes g^*K,v^*K\\otimes g^*K) \\to {\\tt Hom}(g^*g_*v^*K\\otimes\n      g^*u^*u_*K,v^*K\\otimes g^*K)\\\\\n      \\to {\\tt Hom}(g^*(u^*f_*K\\otimes\n      u^*u_*K),v^*K\\otimes g^*K)\n      \\to {\\tt Hom}(u^*(f_*K\\otimes u_*K)\n      ,g_*(v^*K\\otimes g^*K))\\\\\n \\to {\\tt Hom}(f_*K\\otimes u_*K, u_*g_*(v^*K\\otimes\n      g^*K) ) \\to {\\tt Hom}(f_*K\\otimes u_*K, h_*(v^*K\\otimes g^*F)).\n    \\end{multline*}\n    In both cases, we first use the counit, then ``commute'' pushdown and\n    pullback using Lemma \\ref{lem:pullpush} and finally use adjunctions. By\n    Lemma \\ref{lem:pullpush}, the two ways to apply the counit and the push-pull isomorphism\n    commute. This implies commutativity of the diagram of homomorphism sets,\n    and therefore the commutativity of the original diagram.\n  \\end{proof}\n \n\n\n  \\begin{lemma}\\label{lem:projection_and_pull}\nIn the situation of Lemma \\ref{lem:two_projections}\nfor $K\\in {\\tt Sh}_{\\tt Ab}\\mathbf{X}$ and $F\\in {\\tt Sh}_{\\tt Ab} \\mathbf{Y}$ the following diagram commutes:\n    \\begin{equation*}\n      \\begin{CD}\n        u^*(f_*K\\otimes F) @>{\\ref{system80}}>> u^*f_*(K\\otimes f^*F)\\\\\n        @VV{\\ref{tens-pres}}V @VV{\\ref{lem:pullpush}}V\\\\\n        u^*f_*K\\otimes u^*F && g_*v^*(K\\otimes f^*F)\\\\\n        @VV{\\ref{lem:pullpush}}V @VV{\\ref{tens-pres}}V\\\\\n        g_*v^*K\\otimes u^*F && g_*(v^*K\\otimes v^*f^*F)\\\\\n        @VV=V @VV{\\ref{uefhewiufuwefzzz}}V\\\\\n         g_*v^*K\\otimes u^*F @>>{\\ref{system80}}> g_*(v^*K\\otimes\n        g^*u^*F) \n      \\end{CD}\\ .\n    \\end{equation*}\n\\end{lemma}\n\\begin{proof}\n  The left vertical and lower composition is by definition the image of the\n  identity under the sequence of maps\n  \\begin{multline*}\n    {\\tt Hom}(K\\otimes f^*F,K\\otimes f^*F) \\xrightarrow{unit} {\\tt Hom}(K\\otimes\n    f^*F,v_*v^* (K\\otimes f^*F))\\\\\n    \\xrightarrow{adj} {\\tt Hom}(v^*(K\\otimes f^*F),v^*(K\\otimes f^*F)) \\to {\\tt Hom}\n    (v^*K\\otimes g^*u^*F,v^*K\\otimes g^*u^*F)\\\\\n    \\xrightarrow{counit} {\\tt Hom}( g^*g_*v^*K \\otimes g^*u^*F,v^*K\\otimes g^*u^*F)\n    \\xrightarrow{adj} {\\tt Hom}(g_*v^*K\\otimes u^*F,g_*(v^*K\\otimes g^*u^*F))\\\\\n    \\to {\\tt Hom}(u^*(f_*K\\otimes F),g_*(v^*K\\otimes g^*u^*F)).\n  \\end{multline*}\n  The upper and right vertical composition is the image of the identity under\n  the sequence of maps\n  \\begin{multline*}\n    {\\tt Hom}(K\\otimes f^*F,K\\otimes f^*F) \\xrightarrow{counit} {\\tt Hom}(f^*f_*K\\otimes\n    f^*F,K\\otimes f^*F)\\\\\n    \\xrightarrow{adj} {\\tt Hom}(f_*K\\otimes F,f_*(K\\otimes f^*F))\n    \\xrightarrow{unit} {\\tt Hom}(f_*K\\otimes F,u_*u^*f_*(K\\otimes f^*F))\\\\\n    \\xrightarrow{adj} {\\tt Hom}(u^*(f_*K\\otimes F),u^*f_*(K\\otimes f^*F)) \\to\n    {\\tt Hom}(u^*(f_*K\\otimes F),g_*v^*(K\\otimes f^*F))\\\\\n    \\to {\\tt Hom}(u^*(f_*K\\otimes F),g_*(v^*K\\otimes v^*f^*F)) \\to\n    {\\tt Hom}(u^*(f_*K\\otimes F),g_*(v^*K\\otimes g^*u^*F)).\n  \\end{multline*}\n\n  These two maps coincide, as follows from the fact that units and counits\n  commute (in the appropriate sense) with $\\alpha_*$ and $\\beta^*$.\n\\end{proof}\n\\subsection{}\n\nWe now show that (\\ref{ieuwfhwefjkp}) commutes.\nWe  simplify the definition of the integration map which is represented by all horizontal compositions in the following diagram.\n$$\\xymatrix{f_*If^*\\ar[r]\\ar[d]^\\sim&f_*T_KIf^*\\ar[d]&f_*T_Kf^*R\\ar[l]_{\\sim}\\ar[r]\\ar[d]&{\\tt id}\\ar[d]^\\sim\\\\\nf_*If^*I\\ar[r]^\\sim&f_*T_KIf^*I&f_*T_Kf^*RI\\ar[l]_{\\sim}\\ar[r]&I\\ar@{=}[d]\\\\\nf_*f^*I\\ar[u]^\\sim\\ar[r]\\ar[d]^\\sim&f_*T_Kf^*I\\ar[d]^\\sim\\ar[u]^\\sim&f_*T_Kf^*RI\\ar[l]^\\sim\\ar[d]^\\sim\\ar@{=}[u]\\ar[r]&I\\ar[d]^\\sim\\\\\nf_*f^*I{\\mathcal Fl}\\ar[r]&f_*T_Kf^*I{\\mathcal Fl}&f_*T_Kf^*RI{\\mathcal Fl}\\ar[l]^\\sim\\ar[r]&I{\\mathcal Fl}\\\\\nf_*f^*{\\mathcal Fl}\\ar[u]^\\sim\\ar[r]&f_*T_Kf^*{\\mathcal Fl}\\ar[u]^\\sim&f_*T_Kf^*R\\ar[l]^\\sim{\\mathcal Fl}\\ar[u]^\\sim\\ar[r]&{\\mathcal Fl}\\ar[u]^\\sim\n}$$\nLet us comment about the isomorphisms in the first column. Let $F\\in C({\\tt Sh}_{\\tt Ab}\\mathbf{X})$.\nThen $f_*If^*(F)\\to f_*If^*I(F)$ is a quasi-isomorphism since $f_*If^*$ preserves quasi-isomorphisms and $F\\to I(F)$ is a quasi-isomorphism.\nThe map $f_*f^*I(F)\\to f_*If^*I(F)$ is a quasi-isomorphism since $I(F)$ is a complex of injective, hence flabby sheaves, the functor $f^*$ preserves flabby sheaves, and therefore the\nacyclic mapping cone of \n$C:=C(f^*I(F)\\to If^*I(F))$ is an exact complex of flabby sheaves. In particular it is \nan exact complex of $f_*$-acyclic sheaves. Since $f_*$ has bounded cohomological dimension this implies that $f_*(C)$ is exact (see the argument in the proof of Lemma \\ref{uuusaaassq}), and therefore $f_*f^*I(F)\\to f_*If^*I(F)$ is a quasi-isomorphism.\nThe map $f_*f^*I(F)\\to f_*f^*I{\\mathcal Fl}(F)$ is a quasi-isomorphism by a similar argument.\nIn fact, $f^*{\\mathcal Fl}(F)\\to f^*I{\\mathcal Fl}(F)$ is a quasi-isomorphism of $f_*$-acyclic sheaves.  This implies again by the  mapping cone argument, that\n$f_*f^*{\\mathcal Fl}(F)\\to f_*f^*I{\\mathcal Fl}(F)$ is a quasi-isomorphism.\n\n\nThe lower line of the diagram  (\\ref{ieuwfhwefjkp}) expresses the integration map in terms of \nthe flabby resolution functor ${\\mathcal Fl}$. Since we know that ${\\mathcal Fl}$ preserves flat sheaves (we do not know this for $I$) we can drop the flat resolution functor $R$ from the construction of the integration\nby adopting the convention that  the functors are applied  to complexes of \nflat sheaves.\n\nWe get the following commutative diagram\n\\begin{equation}\n  \\label{eq:6.78}\n  \\begin{CD}\n    u^* Rf_* f^* @>{\\sim}>> u^* Rf_*f^* @>{u^*\\int_f}>> u^*\\\\\n    @VV{\\sim}V @VV{\\sim}V @VV{\\sim}V\\\\\n    u^*f_*T_K f^*{\\mathcal Fl} @<{\\sim}<< u^* T_{f_*K} {\\mathcal Fl} @>>> u^*{\\mathcal Fl}\\\\\n    @VV{\\sim}V @VV{\\sim}V @VV{\\sim}V\\\\\n    g_*v^* T_K f^*{\\mathcal Fl} &&  T_{u^*f_*K} u^*{\\mathcal Fl} @>>> T_{u^*\\mathbb{Z}}u^*{\\mathcal Fl}\\\\\n    @VV{\\sim}V @VV{\\sim}V @VV{\\sim}V\\\\\n    g_* T_{v^*K} v^*f^*{\\mathcal Fl} && T_{g^*v_*K} u^*{\\mathcal Fl} @>>> u^*{\\mathcal Fl}\\\\\n    @VV{\\sim}V @VV=V @VV=V\\\\\n    g_* T_{v^*K}g^* u^*{\\mathcal Fl} @<{\\sim}<< T_{g_*v^*K} u^*{\\mathcal Fl} @>>> u^*{\\mathcal Fl}\\\\\n    @VV{\\sim}V @VV{\\sim}V @VV{=}V \\\\\n    g_*T_{v^*K}g^*{\\mathcal Fl} u^* @<{\\sim}<< T_{g_*v^*K} {\\mathcal Fl} u^* @>>> {\\mathcal Fl} u^*\\\\ \n    @VV{\\sim}V @VV{\\sim}V @VV{\\sim}V\\\\\n    Rg_* g^* u^* @>=>> Rg_*g^*u^*  @>{\\int_g u^*}>> u^*\\\\\n  \\end{CD}\n\\end{equation}\nThe commutativity of all the small squares is evident. The commutativity of\nthe large rectangle relies on the fact that the projection formula is\ncompatible with pullbacks, this is the statement of Lemma\n\\ref{lem:projection_and_pull}. The commutativity of the boundary of this diagram gives\n (\\ref{ieuwfhwefjkp}).\n\n \n\n\\subsection{}\n\nIn order to show that (\\ref{eq:comm_u_lower})  commutes we start with the following observation.\n\n  \\begin{lemma}\\label{lem:two_integrations}\n    Assume, in the situation of Lemma \\ref{lem:two_projections}, that $K$ is a\n    flat locally $f_*$-acyclic resolution of $\\underline{\\Z}_X$ of length $n$, and that $f$ is a\n    projection of a locally trivial orientable fiber bundle of $n$-dimensional\n    closed manifolds. Assume that $f_*K\\to \\underline{\\Z}_Y$ is an orientation. Let\n    $g_*v^*K\\to \\underline{\\Z}_U$ be the induced orientation of the pullback bundle\n    $g$. Then the following diagram commutes, where all the horizontal maps\n    are given by the orientations.\n    \\begin{equation*}\n      \\begin{CD}\n        f_*K\\otimes u_*F @>>> \\underline{\\Z}_Y\\otimes u_*F\\\\\n        @VVV      @VVV\\\\\n        u_*(u^*f_*K\\otimes F) @>>> u_*(u^*\\underline{\\Z}_Y\\otimes F)\\\\\n        @VV{\\sim}V   @VV{\\sim}V\\\\\n        u_*(g_*v^*K\\otimes F)  @>>> u_*(\\underline{\\Z}_U \\otimes F)\n      \\end{CD}\n    \\end{equation*}\n  \\end{lemma}\n  \\begin{proof}\n    The upper diagram commutes because of the naturality of the homomorphism\n    of the projection formula, the lower diagram commutes by the definition of\n    the induced orientation of $g$.\n  \\end{proof}\n\n\nTo understand the relation between derived pushdown along a non-representable\nmap and \nintegration we need to use an explicit model of the derived pushdown. If\n$u\\colon U\\to Y$ is a morphism between locally compact stacks which has local\nsections, then $Ru_*$ is given by $C_A\\circ {\\mathcal Fl}$, where ${\\mathcal Fl}$ is the\nfunctorial flabby resolution functor, and $C_A$ is defined in\nSection \\ref{iowefefwewqfqfefewf}, using an atlas $A\\to U$. Note that $C_A$ indeed can be decomposed as the\ncomposition of a functor $L_A$ on sheaves on $U$ and $u_*$. Here $L_A$ is the\nsheafification of the functor on presheaves given by $${}^pL^k_AF(W\\to \nU):= F(\\underbrace{A\\times_U\\dots\\times_U A}_{\\text{$k+1$ factors}}\\times_U\nW\\to U)\\ .$$ i.e.~${}^pL^k_A={p_k}_*p_k^*$, with $p_k\\colon\n\\underbrace{A\\times_U\\dots\\times_U A}_{\\text{$k+1$ factors}} \\to U$.\n\n\n  \\begin{lemma}\\label{lem:commute_pushdowns}\n    In the situation of Lemma \\ref{lem:two_integrations}, we obtain a\n    commutative diagram\n    \\begin{equation*}\n      \\begin{CD}\n        f_* T_K f^* u_* L_A{\\mathcal Fl} @>=>>   f_* T_K f^* u_* L_A{\\mathcal Fl}  @<{\\sim}<<  T_{f_*K} u_* L_A {\\mathcal Fl} @>>>\n        u_*         L_A{\\mathcal Fl}\\\\ \n     @VV{\\sim}V   @VV{\\sim}V   @VVV @VV=V\\\\\n  f_*T_Kv_*L_{g^*A}g^*{\\mathcal Fl} @>{\\ref{pulcomghjdf}}>{\\sim}>      f_*T_K v_*g^* L_A{\\mathcal Fl} && u_*T_{u^*f_*K} L_A{\\mathcal Fl} @>>> u_* L_A{\\mathcal Fl}\\\\\n&&        @VVV @VV{\\sim}V  @VV=V\\\\\n  &&      f_*v_*T_{v^*K} g^* L_A {\\mathcal Fl} && u_*T_{g_*v^*K} L_A{\\mathcal Fl} @>>> u_* L_A{\\mathcal Fl}\\\\\n    &&    @VV{\\sim}V @VV=V @VV=V\\\\\n    &&    u_*g_* T_{v^*K}g^* L_A{\\mathcal Fl} @<<< u_*T_{g_*v^*K} L_A{\\mathcal Fl} @>>> u_* L_A{\\mathcal Fl}.\n      \\end{CD}\n    \\end{equation*}\n    Here, the right horizontal maps are given by the orientations $f_*K\\to\\underline{\\Z}_Y$\n    and $g_*v^*K\\to\\underline{\\Z}_U$.\n  \\end{lemma}\n  \\begin{proof}\n    This is the direct translation of Lemma \\ref{lem:two_projections} and\n    Lemma \\ref{lem:two_integrations}. \n  \\end{proof}\n\nNote that the upper composition is a representation (when applied to flat\nsheaves) of\n\\begin{equation*}\n  Rf_*f^*Ru_* \\xrightarrow{\\int_f} Ru_*.\n\\end{equation*}\n\nThe leftmost vertical arrow represents the morphism\n\\begin{equation}\\label{eq:rightmap}\n  Rf_*f^*Ru_* \\to Rf_*Rv_*g^*,\n\\end{equation}\nsince $g^*$ preserves flabby sheaves, and $v_*L_{g^*A}$ indeed is a model for\n$C_{g^*A}$, which can be used to calculate $Rv_*$.\n\nTherefore the diagram in Lemma \\ref{lem:commute_pushdowns}\ncontains one part (lower right-up) of the diagram (\\ref{eq:comm_u_lower}).\n\n\\subsection{}\n\nTo represent the other composition of the diagram (\\ref{eq:comm_u_lower}), we have to  commute not only $u_*$\nbut also $L_A$ with the other operations. Recall that $L_A$ provides some kind of a resolution, i.e.~we have a\n  canonical map ${\\tt id}\\to L_A$, which is  used in the Lemma below.\n\n\\begin{lemma}\\label{lem:LA_and_int}\n  In the situation of Lemma \\ref{lem:two_integrations}, the following diagram\n  commutes, where the horizontal maps are induced by the orientation of\n  $g$. \n  \\begin{equation*}\n    \\begin{CD}\n      u_* T_{g_*v^*K}L_A{\\mathcal Fl} @>>>  u_* T_{\\mathbb{Z}} L_A {\\mathcal Fl}\\\\\n        @VVV   @VVV\\\\\n        u_* T_{L_A g_*v^*K} L_A {\\mathcal Fl} @>>> u_* T_{L_A\\mathbb{Z}} L_A{\\mathcal Fl}\\\\\n        @VVV @VVV\\\\\n        u_*L_A T_{g_*v^*K} {\\mathcal Fl} @>>> u_* L_A T_\\mathbb{Z} {\\mathcal Fl}\n    \\end{CD}\n  \\end{equation*}\n  The second vertical map in each column follows from a variant of the\n  projection \n  formula, using that $L_A$ is given by application of $(p_k)_*p_k^*$ (or by\n  directly inspecting the definitions).\n\\end{lemma}\n\\begin{proof}\nIf $G\\to H$ is a morphism of sheaves, then we get a natural transformation of functors $T_G\\to T_H$. This naturality \nimplies the commutativity of the first square.\n  The second square\n  is commutative by the naturality of the morphism in the projection formula.\n\\end{proof}\n\nObserve that we have a natural isomorphism $g^*L_A\\cong L_{g^*A}g^*$. \n\n\\begin{lemma}\\label{lem:move_LA}\n  In the situation of Lemma \\ref{lem:two_integrations}, we obtain the\n  following commutative diagram\n  \\begin{equation*}\n    \\begin{CD}\n      u_*g_* T_{v^*K}g^*L_A{\\mathcal Fl} @<<< u_*T_{g_*v^*K} L_A{\\mathcal Fl}\\\\\n      @VVV @VVV\\\\\n      u_*g_*T_{L_{g^*A}v^*K}g^*L_A{\\mathcal Fl} @<<< u_* T_{g_*L_{g^*A}v^*K} L_A {\\mathcal Fl}\\\\\n      @V{\\ref{pulcomghjdf}}V{\\sim}V @VV{\\sim}V\\\\\n      u_*g_* T_{L_{g^*A}v^*K} L_{g^*A}g^*{\\mathcal Fl} && u_*T_{L_Ag_*v^*K} L_A{\\mathcal Fl}\\\\\n      @VV{}V @VV{}V\\\\\n      u_*g_* L_{g^*A} T_{v^*K} g^*{\\mathcal Fl} && u_* L_A T_{g_*v^*K}{\\mathcal Fl}\\\\\n      @V{\\ref{pulcomghjdf}}V{\\sim}V   @VV=V\\\\\n      u_* L_A g_*T_{g_*v^*K}g^*{\\mathcal Fl} @<<< u_* L_A T_{g_*v^*K}{\\mathcal Fl}\n       \\end{CD}\n  \\end{equation*}\n\\end{lemma}\n\\begin{proof}\n  The upper square is commutative because of the naturality of the morphism in the projection\n  formula. The commutativity of the lower rectangle follows from Lemma\n  \\ref{lem:two_projections}, as we basically have to commute two different\n  applications of the projection formula.\n\\end{proof}\n\nWe now prove the commutativity of \\eqref{eq:comm_u_lower}. Using explicit\nrepresentatives of the maps in question, we obtain (applied to flat sheaves)\n\\begin{equation*}\n  \\begin{CD}\n    Rf_*f^* Ru_*   @>>=>  Rf_*f^*Ru_* @>{\\int_fRu_*}>> Ru_*\\\\\n    @VV{\\sim}V  @VV{\\sim}V @VV{\\sim}V\\\\\n        f_* T_K f^* u_* L_A{\\mathcal Fl}  @<{\\sim}<<  T_{f_*K} u_* L_A {\\mathcal Fl} @>>>\n        u_*         L_A{\\mathcal Fl}\\\\\n        @VVV @VVV @VV=V\\\\\n        u_*g_* T_{v^*K}g^* L_A{\\mathcal Fl} @<<< u_*T_{g_*v^*K} L_A{\\mathcal Fl} @>>> u_* L_A{\\mathcal Fl}\\\\\n        @VVV @VVV @VV=V\\\\\n      u_* L_A g_*T_{g_*v^*K}g^*{\\mathcal Fl} @<<< u_* L_A T_{g_*v^*K}{\\mathcal Fl} @>>> u_* L_A T_\\mathbb{Z} {\\mathcal Fl}\\\\\n      @VVV @VVV @VV=V\\\\\n      u_* L_A {\\mathcal Fl} g_*T_{g_*v^*K}g^*{\\mathcal Fl} @<<< u_* L_A {\\mathcal Fl} T_{g_*v^*K}{\\mathcal Fl} @>>>\n    u_* L_A  {\\mathcal Fl}\\\\\n    @VV{\\sim}V @VV{\\sim}V @VV{\\sim}V\\\\\n    Ru_*Rg_*g^* @>>=> Ru_*Rg_*g^* @>{Ru_*\\int_g}>> Ru_*\n    \\end{CD} \n\\end{equation*}\n\nHere, the first and the last rows are just added as illustration what the next\nor preceding line, respectively, computes in the derived category. The map\nfrom the third-last to the second-last row is induced by the inclusion into\nthe flabby resolution. This step is necessary because we don't know that the\nfunctors in question are $u_*$-acyclic, and explains why one can directly\ndefine only the map $f^*Ru_*\\to Rv_*g^*$, and why it is hard to show that this\nis an equivalence. The other vertical maps, and the commutativity of the\nremaining four squares, is given by Lemmas \\ref{lem:commute_pushdowns},\n\\ref{lem:LA_and_int}, \\ref{lem:move_LA}.\n\nNote that the left vertical composition is the composition \n\\begin{equation*}\nRf_*f^* Ru_* \\to Rf_* Rv_* g^* \\to Ru_*Rg_*g^*,\n\\end{equation*}\nas shown in the reasoning for \\eqref{eq:rightmap}. The assertion follows.\n\\hspace*{\\fill}$\\Box$ \\\\[0cm]\\noindent\n\n\\subsection{}\n\n\n\nCompared with the simplicity of its statement the proof of Lemma \\ref{system300} seems to be too long. But\nlet us mention that the proof of a similar result in the algebraic context is\nquite involved, too. The book \\cite{MR1804902} is devoted to this problem.\n\n\n\n\\section{Extended sites}\\label{system3000}\n\n\n\\subsection{}\n\nWe consider the lower right Cartesian square of the  diagram\n$$\\xymatrix{&U\\times_YB\\ar@{.>}[r]\\ar@{.>}[d]&B\\ar@{.>}[d]\\\\A\\times_YX\\ar@{.>}[d]\\ar@{.>}[r]&U\\times_YX\\ar[d]\\ar[r]&X\\ar[d]^f\\\\A\\ar@{.>}[r]&U\\ar[r]&Y}$$\nin stacks where $U,X,Y$ are locally compact.\n\\begin{lem}\\label{zqwduwdwdwdwqdwdzzz}\nIf $U$ is a space or $f$ is representable, then\n$U\\times_YX$ is a locally compact stack.\n \\end{lem}\n\\begin{proof}\nWe first assume that $U$ is a locally compact space.\nLet $B \\to X$ be a locally compact  atlas. Then\n$U\\times_YB\\to U\\times_YX$ is an atlas. Indeed, \nsurjectivity, representability, and local sections for this map are implied by the corresponding properties of the map $B\\to X$.  The stack $U\\times_YB$ is a space since $U\\to Y$ is representable by \nProposition \\ref{lem:representability}. By Lemma \\ref{qwuidiuwqdwqdwqd} the space $U\\times_YB$\nis locally compact.\nFurthermore, again by Lemma \\ref{qwuidiuwqdwqdwqd},\n$$(U\\times_YB)\\times_{(U\\times_YX)} (U\\times_YB)\\cong U\\times_Y(B\\times_XB)$$ is locally compact\nsince $B\\times_XB$ is locally compact.\nHence the atlas $U\\times_YB\\to U\\times_YX$ has the properties required in Definition \\ref{qwuidiuwqdwqdwqd1fwefw} so that  $U\\times_YX$ is a locally compact stack. \n\nWe now assume that $f$ is representable. \nLet $A\\to U$ be a locally compact atlas such that $A\\times_UA$ is locally compact.\nThen $A\\times_YX\\cong A\\times_U(U\\times_YX)\\to U\\times_YX$ is an atlas of $U\\times_YX$.\nWe again verify  the properties required in Definition \\ref{qwuidiuwqdwqdwqd1fwefw}.\nBy the special case of the Lemma already shown this atlas is locally compact. Moreover\n$[A\\times_U(U\\times_YX)]\\times_{U\\times_YX}[A\\times_U(U\\times_YX)]\\cong\n(A\\times_UA)\\times_YX$ is locally compact.\n\\end{proof}\n\n\\subsection{}\n\nIf $f\\colon X\\to Y$ is a representable map with local sections between   locally compact stacks, then\n\nfor $(U\\to Y)\\in \\mathbf{Y}$ we have ${}^pf^* h_U\\cong h_{U\\times_XY}$ (see the proof of Lemma \\ref{hhdf36434746zzz} below). If we drop the assumption that $f$ is representable, then in general ${}^pf^*h_U$ is not representable. In order to overcome this defect we enlarge the site $\\mathbf{X}$ to $\\tilde \\mathbf{X}$ so that it contains the stacks $U\\times_XY\\to X$ over $X$.\n\n \nWe consider the $2$-category ${\\tt Stacks}^{top, lc}\/_{ls,rep}X$ of locally compact stacks  $U\\to X$ over $X$, where the structure map is representable and has local sections. A morphism in this category is a diagram\n$$\\xymatrix{U\\ar[dr]\\ar[rr]&\\ar@{:>}[d]&V\\ar[dl]\\\\&X&}$$\nconsisting of a one-morphism  and a two-morphism. The composition is defined in the obvious way.\nIf there is a two-morphism  between two such one-morphisms, then it is unique by the representability of the structure maps. Therefore ${\\tt Stacks}^{top, lc}\/_{ls,rep}X$ is equivalent in two-categories to the one-category obtained by identifying all isomorphic one-morphisms.\n\n\n  \n\n\\subsection{}\n\nLet $f:X\\to Y$ be a map between locally compact stacks.\n\\begin{ddd}\nWe let $\\tilde \\mathbf{X}$  be the category obtained from   ${\\tt Stacks}^{top, lc}\/_{ls,rep}X$   by identifying all isomorphic one-morphisms. \n\\end{ddd}\n\n\n\n\n\n \n\n\n\n\n\n\n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n We now define the topology on $\\tilde \\mathbf{X}$.\n A covering family $(U_i\\to U)$ of $(U\\to X)\\in \\tilde \\mathbf{X}$  is a family of  locally compact  stacks over $U$ such that $U_i\\to U$ is representable, has local sections and  $\\sqcup_{i\\in  I} U_i\\to U$ is surjective\\footnote{These maps are actually equivalence classes, but in order to simplify the language we will not mention this explicitly in the following}. Using Lemma \\ref{zqwduwdwdwdwqdwdzzz}\none easily checks the axioms listed in \\cite[1.2.1]{MR1317816}.\n\nLet $\\hat \\mathbf{X}$ be  the site with the same underlying category as $\\tilde \\mathbf{X}$, but with the topology generated by the   covering families of $(U\\to X)$ given by families $(U_i\\to U)\\in {\\tt Stacks}^{top, lc}\/X$ such that $U_i\\to U$ is a map from a locally compact space with local sections and $\\sqcup_i U_i\\to U$ is surjective.\n \n\\begin{lem}\\label{wuiefhewffewfewfzzz}\nWe have a canonical isomorphism\n$${\\tt Sh} \\tilde \\mathbf{X}\\cong {\\tt Sh}\\hat \\mathbf{X} \\ .$$\n\\end{lem}\n\\begin{proof}\nThe covering families of $\\hat \\mathbf{X}$ are covering families in\n$\\tilde \\mathbf{X}$. Here we use Proposition \\ref{lem:representability} in order to see that the maps $U_i\\to U$ from spaces $U_i$ are representable.\nOn the other hand, every covering family  $(U_i\\to U)$ of $(U\\to X)$ in $\\tilde \\mathbf{X}$  can be refined\nto a covering family in $\\hat \\mathbf{X}$  by  choosing a locally compact   atlas $A_i\\to U_i$ for each $U_i$.  \nThis implies the lemma. \\end{proof}\n\n\\subsection{}\n\n\nThe natural functor ${\\tt Top}^{lc}\/X\\to {\\tt Stacks}^{top,lc}\/X$  from locally compact spaces over $X$ to locally compact stacks over $X$ induces a map of sites $j\\colon \\mathbf{X}\\to \\tilde \\mathbf{X}$.\n\\begin{lem}\\label{k7783nndnbdzzz}\nThe restriction functor\n$$j^*\\colon {\\tt Sh}\\tilde \\mathbf{X}\\to {\\tt Sh} \\mathbf{X} $$ is  an equivalence of\ncategories. \\end{lem}\n\\begin{proof}\nThe inverse of $j^*$ is the functor $j_*$ given by\n$$j_* F(U):=\\lim_{(V\\to U)\\in \\mathbf{X}\/\/U} F(V)$$\nfor all $(U\\to X)\\in \\tilde \\mathbf{X}$, where\n$\\mathbf{X}\/\/U$ is the category of all pairs $(V\\in \\mathbf{X}, j(V)\\to U\\in  \\mathrm{Mor}(\\tilde \\mathbf{X}))$ such that the map\n$j(V)\\to U$ has local sections.\n\n\n\nIf $U\\in j(\\mathbf{X})$, then $(U,{\\tt id}_{j(U)}\\colon j(U)\\to j(U))$ it is the final object of $\\mathbf{X}\/\/U$.\nThis gives a natural isomorphism $j^* j_*(F)(U)\\cong F(U)$.\n\nWe now define a natural isomorphism\n$j_*j^*(F)\\to F$ for all $F\\in {\\tt Sh}\\tilde \\mathbf{X}$.\nLet $(U\\to X)\\in \\tilde \\mathbf{X}$.\nThe family $(V\\to U)_{\\mathbf{X}\/\/U}$ is a covering family\nof $U\\to X$ in $\\hat \\mathbf{X}$. \nSince $F$ is also a sheaf on $\\hat \\mathbf{X}$  by Lemma \\ref{wuiefhewffewfewfzzz} we get an isomorphism\n$$j_*j^*(F)(U)\\cong \\lim_{(V\\to U)\\in \\mathbf{X}\/\/U} j^*(F)(V)\\cong F(U)\\ .$$\n\\end{proof}\n\n\\subsection{}\n\n\\begin{lem}\\label{wkuhwqeddqwzzz}\nA map $f:X\\to Y$  between locally compact  \nstacks induces a map of sites\n$$\\tilde f^\\sharp\\colon \\mathbf{Y}\\to \\tilde \\mathbf{X}$$ by\n$$\\tilde f^\\sharp(U\\to Y):=U\\times_YX\\to X\\ .$$\n\\end{lem}\n\\begin{proof}\nIndeed, if $U\\to Y$ is a map from a locally compact  space, then the stack\n$U\\times_YX$ is locally compact by Lemma \\ref{zqwduwdwdwdwqdwdzzz}.\nIf $(U_i\\to U)$ is a covering family of $(U\\to Y)\\in \\mathbf{Y}$ by open subspaces, then\n$(U_i\\times_YX\\to U\\times_YX)$ is a covering family in $\\tilde \\mathbf{X}$\nby open substacks.  \n \n\nFurthermore it is easy to see that $\\tilde f^\\sharp$  preserves fiber products, i.e.\nif $(U_i\\to U)$ is a covering family and $V\\to U$ is a morphism in $\\mathbf{Y}$, then\n$\\tilde f^\\sharp(U_i\\times_UV)\\cong \\tilde f^\\sharp(U_i)\\times_{\\tilde f^\\sharp(U)}\\tilde f^\\sharp(V)$. \n\\end{proof}\n\n\n\n\\subsection{}\nWe consider a map $f:X\\to Y$  between locally compact  \nstacks. Then we have  an adjoint pair of functors\n$$\\tilde f^\\sharp_* \\colon  {\\tt Sh} \\mathbf{Y}\\leftrightarrows {\\tt Sh}\\tilde \\mathbf{X} : (\\tilde f^\\sharp)^* \\ .$$\n\\begin{lem}\\label{hhdf36434746zzz}\nWe have an isomorphism of functors\n$j^*\\circ \\tilde f^\\sharp_* \\cong f^*\\colon {\\tt Sh}\\mathbf{Y}\\to {\\tt Sh}\\mathbf{X}$ \n\\end{lem}\n\\begin{proof}\n The map $j\\colon \\mathbf{X} \\to \\tilde \\mathbf{X} $ induces a map ${}^pj^*\\colon \\Pr\\tilde \\mathbf{X} \\to \\Pr\\mathbf{X} $.\nWe show the relation first on representable presheaves.\nLet $(U\\to Y)\\in \\mathbf{Y} $ and observe that $(U\\times_YX\\to X)\\in \\tilde \\mathbf{X} $\nby Lemma \\ref{zqwduwdwdwdwqdwdzzz}. \nThe following chain of natural isomorphisms (for arbitrary $F\\in \\Pr\\tilde \\mathbf{X}$) shows that\n$\\tilde f^\\sharp_* h_U\\cong h_{U\\times_YX}$:\n\\begin{eqnarray*}\n{\\tt Hom}_{\\Pr\\tilde \\mathbf{X}}(\\tilde f^\\sharp_* h_U,F)&\\cong&{\\tt Hom}_{\\Pr\\mathbf{Y}}(h_U,(\\tilde f^\\sharp)^*F)\\\\&\\cong&\n(\\tilde f^\\sharp)^*F(U)\\\\&\\cong&F(\\tilde f^\\sharp(U))\\\\&\\cong&F(U\\times_YX)\\\\&\\cong&\n{\\tt Hom}_{\\Pr\\tilde \\mathbf{X}}(h_{U\\times_YX},F)\\ .\n\\end{eqnarray*}\nFor $(U\\to Y)\\in \\mathbf{Y} $\nwe have $ {}^pf^* h_U\\cong {}^pj^* h_{U\\times_YX}$. Indeed,\nfor $(V\\to X)\\in \\mathbf{X} $ we have\n$${}^pj^* h_{U\\times_YX}(V)\\cong{\\tt Hom}_{\\tilde \\mathbf{X}}(j(V), U\\times_YX)\n\\stackrel{!}{\\cong} {}^pf^* h_U(V)\\ ,$$\nwhere the marked isomorphism can be seen by making the definition of ${}^pf^*$ explicit.\nSince ${}^pj^*\\circ {}^p\\tilde  f^\\sharp_*$ and  ${}^p f^*$ commute with colimits\nthe equation ${}^pj^*\\circ {}^p\\tilde f^\\sharp_*\\cong {}^pf^*$ holds on all presheaves.\nThe restriction to sheaves (note that all functors preserve sheaves) gives\n$j^*\\circ \\tilde f^\\sharp_*\\cong f^*$.\n\\end{proof}\nBy adjointness we get\n\\begin{equation}\\label{uqiwdiqwdqwdwqdwqdwdwqd}\n(\\tilde f^\\sharp)^*\\circ j_*\\cong f_*\\ .\n\\end{equation}\n\n \n\n\\subsection{}\n\n \nConsider two composeable  maps between locally compact stacks.\n$$X\\stackrel{f}{\\to} Y\\stackrel{g}{\\to} Z\\ . $$\nThe following lemma generalizes \\cite[Lemma 2.23]{bss}\nby dropping the unnecessary additional assumptions that $f$ has local sections or $g$ is representable.\n\\begin{lem}\\label{feriueewwwwzzz}\nWe have an isomorphism of functors\n$g_*\\circ f_*\\cong (g\\circ f)_*\\colon {\\tt Sh}\\mathbf{X}\\to {\\tt Sh} \\mathbf{Z}$.\n\\end{lem}\n\\begin{proof}\n\nWe consider the following diagram:\n$$\\xymatrix{{\\tt Sh}\\mathbf{X}\\ar@\/_2cm\/[dd]^{(g\\circ f)_*}\\ar[r]^{j^X_*}\\ar[d]^{f_*}&{\\tt Sh}\\tilde \\mathbf{X}\\ar@\/_-2cm\/[dd]^{(\\widetilde{(g\\circ f)}^\\sharp)^*}\\ar[d]^{(\\tilde f^\\sharp)^*}\\\\\n{\\tt Sh}\\mathbf{Y}\\ar[r]^{j^Y_*}\\ar[d]^{g_*}&{\\tt Sh}\\tilde \\mathbf{Y}\\ar[d]^{(\\tilde g^\\sharp)^*}\\\\\n{\\tt Sh}\\mathbf{Z}\\ar[r]^{j^Z_*}&{\\tt Sh}\\tilde \\mathbf{Z}}\\ .$$\nWe know that the squares commute (Equation (\\ref{uqiwdiqwdqwdwqdwqdwdwqd})), and that the horizontal arrows are isomorphisms\n(Lemma \\ref{k7783nndnbdzzz}).\nIt follows from the constructions that\n$$\\tilde f^\\sharp\\circ \\tilde g^\\sharp=\\widetilde{(g\\circ f)}^\\sharp$$\non the level of sites. Hence the right triangle commutes, too. This implies\ncommutativity of the left triangle. \n\\end{proof}\n\nTaking adjoints we get:\n\\begin{kor}\\label{uefhewiufuwefzzz}\nWe have an isomorphism $f^*\\circ g^*\\cong (g\\circ f)^*\\colon {\\tt Sh}\\mathbf{Z}\\to {\\tt Sh}\\mathbf{X}$.\n\\end{kor}\n\n\n\\subsection{}\n\nWe consider a topological stack $X$ and the inclusion $j\\colon \\mathbf{X}\\to \\tilde \\mathbf{X}$ which induces by Lemma \\ref{k7783nndnbdzzz}  an equivalence of categories of sheaves\n$$j^*\\colon {\\tt Sh}\\tilde\\mathbf{X}\\leftrightarrows {\\tt Sh} \\mathbf{X}\\colon j_*\\ .$$\nNote that the notion of flabbiness depends on the site. \n\\begin{ddd}\nWe call a sheaf $F\\in {\\tt Sh}_{\\tt Ab}\\mathbf{X}$ strongly flabby if $j_*(F)$ is flabby.\n\\end{ddd}\nSince flabbiness is a condition to be checked for all covering families\nand since all covering families in $\\mathbf{X}$ induce covering families in $\\tilde \\mathbf{X}$ it follows that a strongly flabby sheaf is flabby. Since injective sheaves are strongly flabby\neach sheaf admits a strongly flabby resolution.\n\n\\subsection{}\n\nLet $f\\colon X\\to Y$ be a morphism of locally compact stacks. \n\\begin{lem}\nStrongly flabby sheaves are $f_*$-acyclic.\n\\end{lem}\n\\begin{proof}\n In view of Lemma \\ref{hhdf36434746zzz}\nit suffices to show that flabby sheaves in ${\\tt Sh}_{\\tt Ab}\\tilde\\mathbf{X}$\nare $\\tilde f_*$-acyclic. We now can write\n$\\tilde f_*=\\tilde i^\\sharp\\circ {}^p\\tilde f_*\\circ \\tilde i$, where\n$\\tilde i^\\sharp$ and $\\tilde i$ are the sheafification\nfunctor and the inclusion of sheaves into presheaves for the tilded sites,\nand ${}^p\\tilde f_*={}^p(\\tilde f^\\sharp)^*\\colon \\Pr\\tilde \\mathbf{X}\\to \\Pr\\tilde \\mathbf{Y}$.\nSince\n${}^p\\tilde f_*(F)(V\\to Y)=F(V\\times_YX\\to X)$ we see that\n${}^p \\tilde f_*$ is exact. Since strongly flabby sheaves are $\\tilde i$-acyclic, and $\\tilde i^\\sharp$ is exact, it follows that strongly flabby sheaves are $\\tilde f_*$-acyclic.\n\\end{proof}\n\n\\begin{lem}\nThe functor $$f_*\\colon {\\tt Sh}_{\\tt Ab}\\mathbf{X}\\to {\\tt Sh}_{\\tt Ab}\\mathbf{Y}$$\n preserves strongly flabby sheaves. \n\\end{lem}\n\\begin{proof}\nWe must show that $\\tilde f_*$ preserves flabby sheaves.\nLet $F\\in {\\tt Sh}_{\\tt Ab} \\tilde \\mathbf{X}$ and $\\tau=(U_i\\to U)$ be a covering family of $(U\\to Y)$ in $\\mathbf{Y}$. We must show that the \\v{C}ech complex $C(\\tau,\\tilde f_*F)$ is acyclic. Note that\n$\\tilde f_*F(V)=F(V\\times_YX)$. The family $f^\\sharp(\\tau):=(U_i\\times_YX\\to U\\times_YX)$ is a covering family of $U\\times_YX$ in $\\tilde \\mathbf{X}$. We see that\n$C(\\tau,\\tilde  f_*F)\\cong C(f^\\sharp(\\tau),F)$. Since $F$ is strongly flabby, the complex $\nC(f^\\sharp\\tau,F)$ is acyclic. \n\\end{proof} \n\n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{}\n\n \n\n\n\n\n\n\n\n\n\n\n \nConsider again a sequence of composeable  maps between locally compact stacks.\n$$X\\stackrel{f}{\\to} Y\\stackrel{g}{\\to} Z\\ . $$\nThe following Lemma generalizes \\cite[Lemma 2.26]{bss}, again by dropping the unnecessary assumptions that $f$ has local sections or $g$ is representable.\n\\begin{lem}\\label{keykey}\nWe have an isomorphism of functors\n$$Rg_*\\circ Rf_*\\cong R(g\\circ f)_*\\colon D^+({\\tt Sh}_{\\tt Ab}\\mathbf{X})\\to D^+({\\tt Sh}_{\\tt Ab} \\mathbf{Z}).$$\n\\end{lem}\n\\begin{proof}\nThe isomorphism $(g\\circ f)_*\\to g_*\\circ f_*$ induces a transformation\n$R(g\\circ f)_*\\to Rg_*\\circ Rf_*$.\nSince injective sheaves are strongly flabby, $f_*$ preserves strongly flabby sheaves, and strongly flabby sheaves are $g_*$-acyclic, this transformation is indeed an isomorphism.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\\backmatter\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\nThe physics of current-driven Josephson (J) vortices  \\cite{BP,KL} and its manifestations in flux flow oscillators\\cite{ffo1,ffo2,ffo3}, THz radiation sources\\cite{thz1,thz2,thz3,thz4}, nanoscale superconducting structures for digital memory \\cite{qc,jm} current transport through grain boundaries\\cite{HM,D,physc} in superconducting polycrystals and radio-frequency superconducting cavities for particle accelerators \\cite{ag_srf}, have been areas of active experimental and theoretical investigations. Particularly, dynamics of J vortices in layered superconductors has attracted much attention since the discoveries of the cuprate and iron-based superconductors which exhibit an intrinsic Josephson effect between weakly coupled $ab$ planes \\citep{kliener92,csg1,klcsg,yugens}. Numerical simulations of stacks of  Josephson junctions (JJ) have revealed instabilities of sliding Josephson vortex lattices \\cite{kl-ch,jvlins,kosh} which affect the power of coherent THz radiation from single crystal BSCCO mesas  \\cite{kosh,machidarec,tachiki,radcal,recjvl,insrecjvl}. New imaging tools have probed vortices at nanometer scales and revealed hypersonic vortices moving much faster than the velocity of superfluid condensate \\cite{embon}.   \n\nIt has been usually assumed that a driven vortex preserves its identity as a topological defect no matter how fast it moves, because instability of a vortex would violate the fundamental conservation of the winding number $n=\\pm 1$ in the superconducting order parameter $\\Psi=\\Delta\\exp(in\\chi)$. One of the outstanding questions is whether this topologically protected stability of a moving vortex remains preserved at any current below the depairing limit or there is a terminal velocity above which a uniformly moving vortex cannot exist. As far as the Josephson vortices are concerned, numerical simulations of long underdamped junctions\\cite{screp}, planar JJ arrays \\cite{bob,nakajima,paco} and a few coupled JJs \\cite{a1,a2,a3,a4,a5}, and discrete sine-Gordon systems\\cite{sg1,sg2}, have shown that there is indeed a terminal velocity $v_c$ above which uniform motion of a vortex driven by a dc current breaks down due to Cherenkov radiation.  The Cherenkov radiation of a vortex moving with a constant velocity $v$ is characteristic of high-$J_c$ Josephson junctions (JJ) or arrays of coupled JJs  in which the phase velocity of electromagnetic waves $v_p(k)$ decreases as the wave number $k$ increases \\cite{sakai,csg-td,kleiner2,ngai,miccsg,lin-sust}, so that the Cherenkov condition $v>v_p(k)$ can be more easily satisfied at short wavelengths.  The resulting Cherenkov wake behind a moving J vortex causes a significant radiation drag in addition to the conventional quasiparticle viscous drag \\cite{a2}. It turns out that the steady-state motion of a J vortex in which the Lorentz force is balanced by the viscous and radiation drag forces can only be sustained at $v<v_c$. A vortex moving with a velocity $v>v_c$ starts producing a cascade of expanding vortex-antivortex (V-AV) pairs which form dynamic dissipative patterns \\cite{screp,paco}. Such resistive transition can occur at current densities $J>J_s$ which can be well below the critical current density of the interlayer junction $J_0$. Generation of V-AV pairs by a moving vortex pertains to a broader issue of stability of driven topological defects that can destroy global long range order in a way similar to the crack propagation resulting from the pileup of dislocations of opposite polarity \\cite{disl}. Such process was observed in simulations of vortices in long JJs and planar JJ arrays where driven vortices cause propagating phase cracks in superconducting long range order \\cite{screp,paco}. \n\nA question whether a fast Josephson vortex can initiate the V-AV pair production in layered superconductors is of interest to the theory of nonlinear flux flow of vortices along the $ab$ planes in high-$T_c$ cuprates and pnictides or artificial multilayer structures. For instance, revealing the materials parameters which control the values of $v_c$ and $J_s$ are essential for understanding the high-field electromagnetic response along the c-axis.  Another issue pertains to dynamic dissipative structures which appear due to the V-AV chain reaction triggered by a single moving vortex. The nonlinear dynamics of these structures and their effect on the radiation and other electromagnetic properties of layered superconductors are of particular interest.  The Cherenkov instability of vortices at high velocities is facilitated in underdamped interlayer junctions, as characteristic of highly anisotropic Bi-based cupraes, which can thus be testbeds for the experimental and theoretical investigations of these issues.       \n\nThe effects of Cherenkov radiation on a current-driven vortex in a few coupled junctions \\cite{a1,a2,a3,a4,a5} or structural instabilities of driven vortex lattices and their manifestations in the THz radiation sources  \\cite{kosh,kl-ch,jvlins} have been thoroughly investigated. Yet little is known about dynamics of macrovortex flux structures resulting from the V-AV pair production caused by a driven J vortex in multilayered superconductors. In this work we address this issue, including a nonlinear vortex viscosity controlled by the ohmic and radiation drag, and the factors determining the terminal velocity $v_c$ and the threshold critical current density $J_s$ at which the steady state flux flow breaks down. We investigate spontaneous generation of V-AV pairs by a moving vortex at $v>v_c$ and show that they result in macrovortex structures spreading both along and across the layers. It turns out that in a stack of underdamped JJs of finite length the V-AV pair production caused by a vortex shuttle excites  large-amplitude standing waves of magnetic flux, giving rise to oscillations in the total magnetic moment and magneto-dipole radiation from the sample.  In our simulation we used the well-established equations that describe J vortices in layered superconductors modeled as a stack of planar JJs coupled by inductive currents and charging effects \\cite{sakai,csg-td,kleiner2,ngai,miccsg,lin-sust}.\n        \nThe paper is organized as follows. Sec. \\ref{sec:elec} specifies the geometry of the problem and the equations used in numerical simulations. In Sec. \\ref{sec:ch} we discuss  Josephson plasmons and conditions of Cherenkov radiation in layered superconductors. Sec. \\ref{sec:sv} contains the results of our calculations of a nonlinear drag coefficient, terminal velocity and critical current density of the Cherenkov instability $J_s$ for a single vortex. It is shown that the production of V-AV pairs at $J>J_s$ results in branching dynamic patterns and macrovortex structures. In Sec. \\ref{sec:vc} and \\ref{sec:vl} we address the effects of vortex interaction on the Cherenkov instability of moving vortex chains and lattices in annular JJ stacks. In Sec \\ref{sec:finsize} we consider dynamics of bouncing macrovortices and self-sustained flux standing waves of large amplitude excited by a V-AV shuttle in a JJ stack of finite length. Contribution of this effect to the power $W$ radiated by the JJ stack, and a strong increase of $W$ with the number of layers are addressed. The conclusions and broader implications of our results are presented in Sec. \\ref{sec:disc}.\n\n\\section{Coupled sine-Gordon Equations}\n\\label{sec:elec}\n\n\\begin{figure}\n\\includegraphics[trim={1mm 0mm 0mm 0mm},clip, width=\\columnwidth ]{fig1}\n\\caption{Stack of intrinsic Josephson junctions (yellow) between superconducting layers (blue).}\n\\label{fig1}\n\\end{figure}\n\nConsider vortices in a stack of long JJs between superconducting layers shown in Fig. \\ref{fig1}. \nThe dynamics of the phase difference $\\theta_l(x,t)$ across the $l$-th junction, and the  \nmagnetic field $B_l(x,t)$ parallel to the layers can be described by the coupled sine-Gordon equations \\cite{thz1,tachiki,sakai,csg-td,kleiner2,ngai,miccsg,lin-sust}\n\\begin{gather}\n(1-\\alpha\\Delta_d)\\theta_l''=\\nonumber \\\\ (1-\\zeta\\Delta_d)[(1-\\alpha\\Delta_d)\\sin\\theta_l+\\beta+\\eta\\dot{\\theta_l}+\\ddot{\\theta_l}],\n\\label{csg} \\\\\nB_l=(1-\\zeta\\Delta_d)^{-1}\\theta_l'.\n\\label{B}\n\\end{gather}    \nHere $\\Delta_d f_l\\equiv f_{l+1}+f_{l-1}-2f_l$ is the lattice Laplacian,\nthe prime and overdot denote partial derivatives with respect to the dimensionless coordinate $x\/\\lambda_c$ and time $\\omega_J t$, respectively, $\\omega_J=c\/\\sqrt{\\epsilon_c} \\lambda_c$ is the Josephson plasma frequency, $c$ is the speed of light, $\\epsilon_c$ is the dielectric constant along the $z$ axis, $\\lambda_c$ is the magnetic field penetration depth along the layers ($\\textbf{B}$ parallel to the $ab$ planes in cuprates), and $B$ is measured in units of $\\phi_0\/2\\pi s\\lambda_c$ where $\\phi_0$ is the flux quantum. The viscous drag coefficient $\\eta$ and the dimensionless current $\\beta$ are defined by: \n\\begin{equation}\n\\eta=\\frac{\\sigma_c\\lambda_c}{\\epsilon_0\\sqrt{\\epsilon_c}c},\\qquad \\beta=\\frac{J}{J_0},\n\\label{par1}\n\\end{equation}\nwhere $J$ is the density of a uniform bias current flowing across the layers, $J_0$ is the critical current density of the junctions, $\\sigma_c$ is the interlayer quasiparticle conductivity, and $\\epsilon_0$ is the vacuum permittivity. The dimensionless damping parameter  $\\eta$ in BSCCO crystals is typically $\\simeq 0.005-0.05$\\cite{lin-sust,machida}. The parameters $\\alpha$ and $\\zeta$ in Eq. (\\ref{csg}) quantify charge and inductive coupling of the layers, respectively:\n\\begin{equation}\n\\alpha=\\epsilon_cl_{TF}^2\/s^2,\\qquad\\zeta=(\\lambda_{ab}\/s)^2.\n\\label{prmtr}\n\\end{equation}\nHere $l_{TF}$ is the Thomas-Fermi screening length along the layers, $\\lambda_{ab}$ is the magnetic field penetration depth for $\\textbf{B}$ parallel to the $c$ axis, and $s$ is the spacing between the superconducting layers. For a BSCCO crystal with the anisotropy parameter $\\Gamma\\equiv\\lambda_c\/\\lambda_{ab}\\sim 500$, $\\lambda_{ab}\\sim 400$ nm, $\\lambda_c\\sim 200$ $\\mu$m and $s=1.5$ nm, $\\zeta \\sim 10^5$ is much larger than the typical value of $\\alpha\\sim 1$. In this case the term $\\alpha\\Delta_d$ which describes deviations from charge neutrality in Eq. (\\ref{csg}) can be neglected \\cite{lin-sust}, so that Eq. (\\ref{csg}) reduces to:\n\\begin{equation}\n\\theta_l''=(1-\\zeta\\Delta_d)(\\sin\\theta_l+\\beta +\\eta\\dot{\\theta_l}+\\ddot{\\theta_l}).\n\\label{sge}\n\\end{equation}    \nIn this work we performed numerical simulations Eq. (\\ref{sge}) using the method of lines \\cite{mdln,mdabm}. Charging effects were neglected, unless specified otherwise. \n\n\\section{Cherenkov radiation and instability}\n\\label{sec:ch}\n \nJosephson vortices described by Eq. (\\ref{sge}) have two length scales along the $xy$ planes: the length of the Josephson core $\\lambda_J\\equiv\\Gamma s$ and the magnetic penetration depth $\\lambda_c$ determining the scale of circulating currents along the stack. Equation (\\ref{csg}) also describes small amplitude waves $\\delta\\theta \\propto e^{ik_x x+iqz-i\\omega t}$ ~ \\cite{sakai,lin-sust}. If the number of layers $N\\to\\infty$, linearization of Eq. (\\ref{csg}) with respect to $\\delta \\theta$ around the uniform current state $\\sin\\theta_0=-\\beta$ yields the following dispersion relation $\\omega(k_x,q)$ for the Josephson plasma waves (in the original units): \n\\begin{gather}\n\\omega(k_x,q)=\\Omega(k_x,q) - \\frac{i\\eta\\omega_J}{2},\n\\label{omk} \\\\\n\\!\\!\\Omega^2=\\bigl[(1+\\alpha_q)\\sqrt{1-\\beta^2}-\\frac{\\eta^2}{4}\\bigr]\\omega_J^2\n+\\biggl[\\frac{1+\\alpha_q}{1+\\zeta_q}\\biggr](k_x c_i)^2,\n\\label{Om} \\\\\n\\alpha_q=4\\alpha\\sin^2\\frac{qs}{2},\\qquad \\zeta_q=4\\zeta\\sin^2\\frac{qs}{2},\n\\label{alze}\n\\end{gather}\nwhere $c_i=\\lambda_c\\omega_J=c\/\\sqrt{\\epsilon_c}$ is the speed of light in the dielectric layers. At $\\eta\\to 0$ and $k_x=q=0$ \nEqs. (\\ref{omk})-(\\ref{alze}) yield $\\omega=\\omega_J(1-\\beta^2)^{1\/4}$ but at $\\lambda_c k_x \\gg 1$ the frequency of the Josephson plasmon $\\omega (k_x,q)=\\tilde{c}(q)k_x$ depends linearly on the in-plane wave number $k_x$. Here \nthe longitudinal phase velocity $\\omega\/k_x=\\tilde{c}(q)$ depends on the $z$-component $q$ of the wave vector:\n\\begin{equation}\n\\tilde{c}(q)=c_i\\!\\left[\\frac{1+4\\alpha\\sin^2(qs\/2)}{1+4\\zeta\\sin^2(qs\/2)}\\right]^{1\/2}.\n\\label{swih}\n\\end{equation} \nFor a stack of $N$ junctions, Eqs. (\\ref{omk})-(\\ref{alze}) with $q_n=\\pi n\/(N+1)s$ and $n=0,1, ... N$, describe \n$N+1$ branches of plasma waves\\cite{lin-sust}.\nIn the case of $\\zeta\\gg\\alpha$ characteristic of the layered cuprates, $\\tilde{c}$ decreases strongly as $q$ increases, from $\\tilde{c}=c_i$ at $q=0$ to \n$\\tilde{c}= c_i\/2\\sqrt{\\zeta}\\ll c_i$ at $q=\\pi\/s$. Thus, the plasma wave with alternating $\\theta_l$ in the $z$ direction has the minimum phase velocity $c_s=c_i\/2\\sqrt{\\zeta}=cs\/2\\lambda_{ab}\\sqrt{\\epsilon_c}$ corresponding to the Swihart velocity in a single junction \\cite{BP}. These features of $\\Omega(k_x,q)$ give rise to Cherenkov radiation produced by a moving vortex  \\cite{thz1,Kl,savelev,krasnov}. \n\nCherenkov radiation occurs if the velocity $v$ of a vortex exceeds the minimum phase velocity $\\Omega(k_x)\/k_x$ of the Josephson plasmons. As follows from Eq. (\\ref{swih}), the \ncondition $v>\\tilde{c}(q)$ at $(k_x\\lambda_c)^2\\gg 1$ and $\\zeta\\gg 1$ is first satisfied if $v>c_s$ at $q=\\pi\/s$.  For instance, Fig. \\ref{fig2} shows the Cherenkov radiation cone behind a moving vortex obtained by numerical simulations of Eq. (\\ref{csg}). \n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig2}\n\\caption{Colormap of Cherenkov radiation cone in the magnetic field $B_l(x)$ produced by a vortex moving uniformly in the middle layer in a stack of  $N=101$ junctions. Here $B_l(x)$ is obtained by simulations of Eqs. (\\ref{csg}) with $\\beta=0.25$, $\\zeta = 71111$, $\\alpha= 1$, $\\eta = 0.05$ and $B_0=\\phi_0\/2\\pi s\\lambda_c$. Only solutions for 15 neighboring junctions above and below the vortex are shown. Note that $L_z=Ns\\sim 10^{-3}\\lambda_c$ so the vortex is strongly elongated along the $x$ direction. }\n\\label{fig2}\n\\end{figure}\n \n\\section{Single vortex}\n\\label{sec:sv}\n\\subsection{Laterally infinite stack} \n\nIn this section we present results of simulations of Eq. (\\ref{sge}) describing vortices in a stack of $N=21$ junctions with $\\eta=0.05$.  Solution of Eq. (\\ref{sge}) for a  stationary vortex in the middle layer is shown in Fig. \\ref{fig3}. As the bias current $\\beta$ increases the vortex velocity $v(\\beta)$ controlled by the drag of quasiparticle currents and radiational forces increases. Here the viscous drag dominates at small  $\\beta$ for which the driving Lorentz force is balanced by the ohmic friction due to dissipative quasiparticle currents in the moving vortex \\cite{clem}.  At $\\beta\\simeq 0.075$ the velocity exceeds the threshold, $v>c_s$ at which the vortex starts radiating Cherenkov waves. As $\\beta$ further increases the amplitude and the wavelength of this Cherenkov wake increase and radiation spreads across the neighboring junctions. Figures \\ref{fig4} and \\ref{fig5} show the calculated phase and field profiles around the moving vortex at $\\beta=0.615$. \n\nUsing the solutions for $\\theta_l(x,t)$, we calculated the steady-state velocity of the vortex $v(\\beta)$ as a function of the driving current $\\beta$ at different values of $\\eta$. The so-obtained curves $v(\\beta)$ shown in Fig. \\ref{fig6} have two distinct parts corresponding to different mechanisms of vortex drag.  At small currents the vortex velocity is limited by the quasiparticle viscous drag $dv\/d\\beta\\propto \\eta^{-1}$ and $v(\\beta)$ increases sharply with $\\beta$ if $\\eta\\ll 1$. The kink in the $v(\\beta)$ curve at intermediate $\\beta$ occurs at the onset of Cherenkov radiation above which the slope of $v(\\beta)$ decreases as the radiation friction takes over \\cite{thz1,mints} and $v(\\beta)$ becomes weakly dependent on the dissipative term in Eq. (\\ref{sge}). At $\\eta\\ll 1$ the radiation friction dominates at practically all $\\beta$, significantly reducing $v(\\beta)$ which exceeds the Cherenkov threshold. As $\\eta$ increases the kink separating the ohmic and Cherenkov vortex drag regions of $v(\\beta)$ gets less pronounced. All $v(\\beta)$ curves have the endpoints at $\\beta=\\beta_s$ and $v=v_c$ beyond which Eq. (\\ref{sge}) no longer has solutions for uniformly moving vortices. Figure \\ref{fig7} shows the calculated critical current $\\beta_s$ and the corresponding terminal vortex velocity $v_c$ as functions of the damping parameter $\\eta$. For underdamped junctions $J_s(\\eta)$ is well below $J_0$ and increases monotonically with $\\eta$, approaching $J_0$ at $\\eta > 1$. In turn, the terminal velocity increases from $v_c\\approx 1.35c_s$ at $\\eta\\ll 1$ to $v_c \\approx 1.85c_s$ at $\\eta=1$. A similar behavior of $v(\\beta)$ and $v_c$ was obtained previously by Goldobin et al. ~\\cite{a2} in numerical simulations of two and three inductively coupled planar JJs.  \n\nAt $\\beta>\\beta_s$ in Eq. (\\ref{sge}), the moving vortex starts spontaneously generating V-AV pairs which spread both along and across the JJ stack. For instance, at $\\eta=0.05$ this process starts at $\\beta_s\\simeq 0.62$ and $v_c\\approx \\sqrt{2}c_s$. Such vortex splitting instability in a layered superconductor turned out to be similar to that of a driven vortex in a single JJ described by equations of nonlocal Josephson electrodynamics \\cite{screp}.  This mechanism is illustrated by Fig. \\ref{fig8} which shows that a critical nucleus being in the unstable $\\pi-$phase state with $5\\pi\/2 <\\theta<7\\pi\/2$ forms behind the vortex moving along the central layer where the maximum of Cherenkov radiation wake $\\theta_l(x,t)$ reaches the threshold value $\\theta_c\\approx 8.6$.  As $\\beta$ increases the amplitude and the width of this $\\pi-$phase domain grows and eventually it splits, triggering a cascade of V-AV pairs which expand along the middle junction. In turn, the V-AV pairs in the middle junction induce V-AV pairs on the neighboring junctions which then start splitting and propagating along the layers and across the stack. This process produces an expanding chain of macrovortices which spread across the entire stack, the macrovortices of positive polarity accumulating at one edge of the stack while macrovortices of negative polarity accumulating at the other edge, as shown in Figs.  \\ref{fig9}. A simulation video of this process is available in Ref. \\onlinecite{supp}. \n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig3}\n\\caption{Phase profile of a static vortex in the middle junction ($l=11$) and $\\theta_l(x)$ induced by the vortex on the layers with $l=10$ and $l =1$). Here $\\theta_l(x)$ are symmetric with respect to the central layer.}\n\\label{fig3}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig4}\n\\caption{Phase profiles of a single vortex propagating along the middle junction $(l=11)$ and the trailing tail of Cherenkov radiation produced on the neighboring junctions ($l = 1$ and $l=10$) calculated from Eq. (\\ref{sge}) at $\\beta=0.615$ and $\\eta= 0.05$. }\n\\label{fig4}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig5}\n\\caption{A color map of the magnetic field in the vortex moving along the central junction calculated from Eq. (\\ref{B}) at $\\beta=0.615$ and $\\eta = 0.05$. Here Cherenkov radiation behind the vortex manifests itself as color ripples. Since $L_z\\sim 10^{-4}\\lambda_c$, the vortex is strongly elongated along the $x$ direction.}\n\\label{fig5}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig6}\n\\caption{Stationary velocities of a vortex moving along the central JJ as a function of the bias current at different $\\eta$. The instability occurs at the endpoints of the curves. The sharp change in the slope of $v(\\beta)$ at $\\eta\\ll 1$ indicates the transition from the ohmic to radiation vortex drag.}\n\\label{fig6}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig7a}\n\\includegraphics[width=\\columnwidth]{fig7b}\n\\caption{The threshold instability current (a) and the terminal velocity (b) as functions of $\\eta$ calculated for $\\zeta=71111$. }\n\\label{fig7}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig8a}\\\\\n\\includegraphics[width=\\columnwidth]{fig8b}\n\\caption{Initial stages of generation of V-AV pairs by a vortex moving along the central junction (top panel), and snapshots of field distribution solutions showing the two dimensional growth of instability for junctions with $l=9,10$ and $11$ at three different times (bottom panel). The results are calculated at $\\eta=0.05$, $\\zeta=71111$ and $\\beta=0.62$.}\n\\label{fig8}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig9a}\\\\\n\\includegraphics[width=\\columnwidth]{fig9b}\n\\caption{Cross sectional view of the field distribution profiles in the stack after the instability (top panel, $t=125$) along with a close-up view of giant vortices moving to the left (bottom panel, $t=225$). Similar macro vortices with opposite polarity form at the other side of the stack (as shown in the top panel).}\n\\label{fig9}\n\\end{figure}\nThe dynamics of the V-AV pair production caused by a single moving vortex, and the subsequent formation of the expanding macrovortex structure does not change qualitatively as the number of layers increases above $N=21$ used in the simulations described above.  For instance, our simulations for a stack with $N=101$ have shown that the V-AV pair production starts at $\\beta=0.625$ which is very close to the instability current of a vortex in a stack with 21 junctions. Thus, the results obtained for $N=21$ can be representative of the BSCCO crystal mesas with $N\\sim 1000$, consistent with the conclusion of Ref. \\onlinecite{krasnov} that the behavior of vortices would become independent of the thickness of the stack if $N > \\lambda_{ab}\/s\\sim 200$.\n\n\\subsection{Annular stack}\n\nTo investigate how the vortex dynamics changes by imposing the periodic boundary conditions, we consider an annular stack in which \n\\begin{gather}\n\\theta_l(x=-L\/2)=\\theta_l(x=L\/2)+ 2n\\pi,\\nonumber \\\\\n\\theta'_l(x=-L\/2)=\\theta'_l(x=L\/2),\n\\label{bc}\n\\end{gather}\nwhere $n=n_f-n_a$ is the difference of the number of fluxons $(n_f)$ and antifluxons $(n_a)$ on the $l-$th layer, and $L$ is the circumference of the stack along the $x$ direction. In our simulations we choose $L=\\lambda_c\\gg\\lambda_J$ in which case the structure of a static vortex in the annular stack at $\\beta=0$ is nearly identical to the vortex in the infinite stack shown in Fig. \\ref{fig3}. If a transport current flows across the annular stack, a vortex moving along the central junction radiates Cherenkov waves in a way similar to that is shown in Fig. \\ref{fig4}. Likewise, the vortex starts producing V-AV pairs at a critical value $\\beta=\\beta_s$ that is very close to $\\beta_s$ for the laterally infinite stack considered above. The initial stages of the V-AV pair production spreading both along and across the junctions proceeds like it does in the infinite stack, resulting in expanding piles of vortices and antivortices.  However, in the annular JJ stack the propagating macrovortices of opposite polarity eventually collide and partly annihilate as they go through each other. The transient solution then evolves into a chaotically oscillating distribution of $\\theta_l(x,t)$ resulting in unidirectional traveling waves of magnetic field with nearly constant amplitudes in each junction, as shown in Fig. \\ref{fig10}. Eventually these traveling electromagnetic waves on different layers become more synchronized as shown in Fig. \\ref{fig11}. \n\nImposing the boundary condition $\\theta_1=\\theta_N$ models a periodic chain of vortices spaced by $N$ layers along the $z$ direction in an infinite annular JJ stack. Our simulations for this case show that, because of the  symmetry of this geometry, the solutions for $\\theta_l(x,t)$ and $B_l(x,t)$ are the same as in the above case of a finite annular stack. \n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig10}\n\\caption{Snapshots of representative solutions for $\\theta_l(x,t)$ (top) and $B_l(x,t)$ (bottom) along the middle JJ at the critical current $\\beta=\\beta_s=0.62$.}\n\\label{fig10}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig11}\n\\caption{Snapshots of the magnetic field (top) and electric field (bottom) in junctions 1-11 calculated at $\\beta=\\beta_s=0.62$, where $E_0=\\phi_0\\omega_J\/2\\pi c s$. Here the largest oscillation amplitude corresponds to the middle junction and the lowest amplitude corresponds to the top\/bottom junction.}\n\\label{fig11}\n\\end{figure}\n\n\\section{Vortex chain in an annular stack}\n\\label{sec:vc}\n\nThe above results show that the initial stage of the continuous V-AV pair production triggered by a single driven vortex is not very sensitive to the boundary conditions either across or along the stack. In this section we present the simulation results for a chain of $M$ vortices placed equidistantly in the middle junction of the 21 JJ stack. If vortices are far apart from each other, so that the spacing between vortices $d=L\/M\\gg\\lambda_J$, the initial stage of the V-AV pair production proceeds in the way similar to that of a single vortex. Namely, each vortex starts radiating Cherenkov wakes at $\\beta\\approx 0.075$ which matches that of a single vortex for up to $M=9$. The onset of the V-AV pair production at $M=9$ occurs at $\\beta=0.625$ close to $\\beta_s$ for a single vortex. In this case the intervortex spacing $d\\sim 30\\lambda_J$ is large so that no significant overlap between the Cherenkov wakes from neighboring vortices happens, as shown in Fig. \\ref{fig12}.  \n\nFor $M=9$, moving vortices start generating V-AV pairs at $\\beta=0.625$. The expanding pairs then overlap, resulting in the phase profile $\\theta_{11}(x,t)$ increasing nearly linearly with time while preserving the net winding number of the initial 9 vortices. In turn, the V-AV pair production in the central junction induces V-AV pairs in the neighboring junctions, causing propagation of the resistive state across the stack. Eventually $\\theta_l(x,t)$ evolves into a superposition of traveling waves propagating on the phase background increasing linearly with $t$. Our simulations of $M=14$ vortices in the middle layer have shown a similar dynamics of $\\theta_l(x,t)$ as for 9 vortices, except that the V-AV pair production starts at a lower value $\\beta\\approx0.59$. The latter may result from stronger overlap and the constructive interference of  the Cherenkov radiation tails which extend over the length $L_r \\sim \\lambda_J\/\\eta$ behind a moving vortex.\n\nThe dynamics of vortices changes as the intervortex spacing $d=L\/M$ becomes of the order of $\\lambda_J$. For instance,  at $M=50$ and $d\\simeq 5\\lambda_J$ the radiation tails of adjacent vortices overlap even at $\\beta\\ll \\beta_s$. As a result, vortices get trapped in the radiation wakes of neighboring vortices, and the unidirectional motion of the vortex chain at $J$ slightly below $J_s$ is accompanied by a low amplitude traveling wave in which the relative position of the adjacent vortices and their instantaneous velocities oscillate, as shown in Fig. \\ref{fig13}. The vortex chain starts producing V-AV pairs at $\\beta=0.445$ resulting in a quick  transition of the central junction into a resistive state in which $\\theta_{11}(x,t)$ becomes nearly a straight line in $x$ and increases linearly with $t$. Unlike the case of smaller $M$, the quick resistive transition of the central junction does not  spread across the stack and no V-AV pairs are generated on other junctions where only small amplitude plasma traveling waves appear. The electromagnetic oscillations in all layers are phase-locked, the amplitude of oscillations decreasing with the distance from the central layer. Snapshots of these solutions are shown in Fig. \\ref{fig14}.\n\nOur simulations have shown that the dynamics of 100 vortices with $d\\simeq 2.6\\lambda_J$ appears similar to that of 50 vortices. Yet because of stronger overlap of vortices and their Cherenkov radiation tails, the onset of the V-AV pair production $\\beta_s=0.455$ is slightly higher than for 50 vortices.  This trend becomes more apparent for 200 vortices for which $\\beta_s\\simeq 0.665$ not only exceeds $\\beta_s$ for 100 vortices but also $\\beta_s$ for a single vortex.  The increase of $\\beta_s$ with $M$ at large $M$ may result from the fact that, if vortices and their radiation tails overlap strongly, the spatial modulations of $\\theta(x,t)$ along the vortex chain get reduced, and the critical $\\pi$ phase nucleus which triggers the V-AV pair production can only appear at higher $\\beta$. For a very dense vortex chain with $d\\ll \\lambda_J$, the V-AV pair production does not occur before the central junction switches to the resistive state at $\\beta=1$.   \n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig12}\n\\caption{Comparison between $\\theta_l(x)$ in a single vortex and a chain of 9-vortices (only three are shown) moving along the central junction at $\\beta=0.6$.}\n\\label{fig12}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig13}\n\\caption{Snapshots of $\\theta_l(x,t)$ in a moving chain of 50 vortices at $\\beta=0.44$ near the instability threshold. The two profiles are superimposed for ease of comparison. Interaction of vortices with Cherenkov wakes causes temporal variations in the shape and velocity of moving vortices.}\n\\label{fig13}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig14}\n\\caption{Snapshots of the final form of the solution in electric field (top) and magnetic field (bottom) representations in junctions 1-11 for instability current $\\beta=0.445$. The oscillations are both in phase and periodic for all layers with amplitudes decaying from the middle junction across the stack.}\n\\label{fig14}\n\\end{figure}\n\n\\section{vortex lattice}\n\\label{sec:vl}\n\nIn this section we present the results of our simulations for the driven Josephson vortex lattice in an annular stack of planar junctions.\n\n\\subsection{Annular stack with finite $N$}\n\nConsider vortices initially placed along a line slightly tilted from being perpendicular to the layers with one vortex per layer in an annular stack with $N=21$. At zero current this structure then relaxes to that is shown in the top panel of Fig. \\ref{fig15}. The corresponding field distributions $B_l(x)$ are shown in the bottom panel of Fig. \\ref{fig15} for the top most, bottom most and middle layer. After a bias current is applied the vortices start moving uniformly and radiating Cherenkov waves with the amplitude and wavelengths increasing with $\\beta$. As shown in Fig. \\ref{fig16}, the average velocities of vortices in different layers are almost the same and their relative positions remain constant as the current is ramped up to the onset of the V-AV pair production, $\\beta=0.54$. At $\\beta_s=0.55$ the vortex moving with the velocity $v\\approx 1.34c_s$ along the $20$-th junction starts generating V-AV pairs which then spread to other junctions, driving the whole stack into a resistive state. As a result, the initial vortex structure evolves to $\\theta_l(x,t)$ which appears chaotic in both $x$ and $t$ on each junction, similar to that was obtained for a single vortex shown in Fig. \\ref{fig10}.\n\nIn our numerical simulations we observed that the symmetry of static vortex structures can depend strongly on the initial arrangement of vortices which can relax to many metastable states. This issue has been recognized in the literature as one of the main reasons why vortices do not necessarily form a triangular lattice in numerical simulations \\cite{jvlins,insrecjvl}. To produce a static vortex configuration with equidistant arrangement of vortices, we initially put chains of equidistant vortices in each layer with vortices on neighboring layers shifted with respect to each other. As a result, vortices relax to a periodic structure, as shown in Fig. \\ref{fig17} for ten vortices per layer. We found that, for a current-driven vortex lattice, the onset of the V-AV pair production is mostly determined by the vortex density within each layer and depends weakly on the symmetry of the vortex lattice.  For instance, for the structure shown in Fig. \\ref{fig17}, the V-AV pair production occurs at $\\beta\\approx 0.32$ irrespective of the arrangement of vortices as long as the linear density of vortices per junction is fixed.  From our calculations, it follows that the threshold current $J_s$ decreases monotonically with the increase of the linear density vortices per layer as shown in Fig. \\ref{fig18}. Hence, $J_s$ is reduced if a weak parallel magnetic field is applied to the stack.  \n\nAs the density of vortices is increased the vortex configuration becomes closer to a triangular lattice, as shown in Fig. \\ref{fig19} for a lattice of $50$ vortices per layer. If a bias current is applied, Cherenkov radiation occurs once the velocity of the lattice exceeds the threshold for the minimum plasma mode, but the radiation wakes are reduced due to strong overlap of vortices in both directions. Here the chain of vortices in the middle junction become unstable first at $\\beta_s=0.195$ producing only one V-AV pair after which the pair production stops. At a slightly larger current of $\\beta=0.2$ two more V-AV pairs are generated in the neighboring 10-th and 12-th junctions, while larger number of V-AV pairs are produced in the middle junction. As current is increased to $\\beta=0.205$ some vortices in the 9-th and 13-th junctions produce a few V-AV pairs. This stepwise process of limited V-AV pair production spreads across more and more junctions as the current further increases.  Finally, at $\\beta=0.2225$ the middle junction starts generating V-AV pairs, which triggers the V-AV pair production in all JJs. As a result, at $\\beta>0.2225$ the stack eventually switches into a dynamic resistive state comprised of propagating phase-locked waves which are synchronized for all junctions.   \n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig15}\n\\caption{Color map of the magnetic field across the stack for a stationary vortex lattice with one fluxon per layer (top) and $B_l(x)$ for the middle and surface JJs (bottom). }\n\\label{fig15}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig16}\n\\caption{Color map of the magnetic field across the stack for a uniformly moving vortex lattice with one fluxon per layer (top) and $B_l(x)$ for the middle and surface JJs (bottom) calculated at $\\beta=0.54$.}\n\\label{fig16}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig17}\n\\caption{Color map of the magnetic field across the stack for a stationary vortex lattice with ten fluxon per layer (top) and $B_l(x)$ for the middle and surface JJs (bottom). }\n\\label{fig17}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig18}\n\\caption{Calculated dependence of $J_s$ on the linear density of vortices per length $\\lambda_c$ along the layer in a vortex lattice. }\n\\label{fig18}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig19}\n\\caption{Color map of the magnetic field in a stationary vortex lattice composed of fifty fluxons per layer. The close up in the top left corner shows a triangle formed by three vortices in two adjacent layers. }\n\\label{fig19}\n\\end{figure}\n\n\\subsection{Annular stack with $\\theta_1=\\theta_N$}\n\nHere we impose the periodic boundary condition of $\\theta_1=\\theta_{N}$ which model periodic vortex structures in an annular stack infinite along $z$. Due to the symmetry of the problem, this boundary condition reduces the number of variables $\\theta_l(x,t)$ in Eqs. (\\ref{csg}) to $(N+1)\/2$ for odd $N$. Consider one fluxon per layer for which the situation is similar to that considered in the previous section. Bcause of the exact same position of vortices in 10-th and 12-th junctions, the magnitude of the image induced by these vortices on the middle junction $(l=11)$ doubles. As a result, the onset of the V-AV pair production on the central junction is reduced down to $\\beta_s=0.175$. At $\\beta=\\beta_s$ this image in the middle junction converts to a V-AV pair which then expand in such a way that two vortices move to the left and the antivortex moves to the right until it gets trapped between two vortices in the neighboring junctions 10 and 12. Shown in Fig. \\ref{fig20} are snapshots of magnetic field maps at $\\beta<\\beta_s$ and $\\beta>\\beta_s$ which illustrate the formation of transient V-AV-V triplets.  The antivortex trapped in the V-AV-V triplet slows it down relative to other vortices, so when the vortices from junction 9 and 13 reach the triplet, the antivortex escapes, producing a V-AV pair which then annihilates, as shown in the simulation movie \\cite{supp}. The process of creation and then annihilation of pairs during the disintegraion of the triplet occurs as $\\beta$ further increases. Finally, at $\\beta=0.3$ after the disintegration of the triplet, a cascade of V-AV pairs generated  continuously in the central junction spreads across the whole stack, resulting in a McCumber-type resistive state in which $\\theta_l(t)$ on each junction increases nearly linear with time \\cite{supp}. \n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig20}\n\\caption{Magnetic field color map in moving vortices in junctions $10$, $11$ and $12$ at $\\beta=0.1$ (top). Bottom panel illustrates how a transient triplet is formed out of the conversion of the image of vortices from 10th and 12th junction in the central JJ to a pair of V-AV at $\\beta=0.175$.}\n\\label{fig20}\n\\end{figure}\n\n\n\\section{Finite size effects and vortex bouncing}\n\\label{sec:finsize}\n\nProliferation of branching V-AV patterns or macrovortex (MV) structures caused by a single vortex is essentially a bulk effect which occurs in a sufficiently long sample or an annular JJ stack.  However, in a JJ stack of finite length $L_x$, the expanding MV chain eventually hits the edges of the JJs, where the boundary conditions of zero current $\\theta_l^\\prime=0$ are imposed.  In this section we consider peculiarities of vortex dynamics resulting from the finite size effects. It turns out that interaction of a MV with the edges of the stack occurs in a way similar to that of a moving J vortex in a single long JJ (see, e.g., Ref. \\onlinecite{physc}). This interaction proceeds as follows. As V approaches the edge of a JJ, it induces penetration of a counter-propagating AV which collides with the incoming vortex. The outcome of this collision depends on the damping parameter $\\eta$.  In an overdamped JJ  $(\\eta \\gtrsim 1)$, the colliding V and AV annihilate, fully extinguishing the fluxon of the initial vortex as it exits the junction. However in an underdamped JJ with $\\eta\\ll 1$, the colliding V and AV do not annihilate but go through each other, as characteristic of non-dissipative solitons described by the sine-Gordon equation \\cite{BP}. As a result, the incoming V exits while the AV moves into the JJ. This process can be regarded as a vortex analog of the Andreev reflection.  \n\nA current-driven V in an underdamped JJ stack gets periodically reflected from the edge where it transforms into a counter-propagating AV which in turn gets reflected as a vortex from the opposite edge. Such V-AV shuttle causes oscillations of the magnetic moment $M(t)$ with the flight frequency $\\nu=v\/2L_x$ depending on the JJ length. Here $M(t)=\\phi(t)L_y$ and the instantaneous magnetic flux threading the stack $\\phi(t)$ are calculated using\n\\begin{equation}\nM(t)=M_0\\sum_l\\int_0^{L_x}B_l(x)dx,\n\\label{M}\n\\end{equation} \nwhere $M_0=B_0 s\\lambda_c L_y\/\\mu_0=\\phi_0L_y\/2\\pi\\mu_0$, $L_y$ is the length of the stack along $y$, and the integral is expressed in terms of the dimensionless field $B_l$ and coordinates defined in Sec. \\ref{sec:elec}.\nShown in Fig. \\ref{fig21}a is $M(t)$ calculated for a vortex driven along the central layer at $\\beta <\\beta_s$ in a stack with $L_x=\\lambda_c$ and $N=21$. The magnitude of $|M(t)|\\simeq 0.0055 M_0$ in Fig. \\ref{fig21}a indicates that the vortex flux $\\phi\\simeq 9\\cdot 10^{-4}\\phi_0$ is much smaller than $\\phi_0$. This effect is similar to the well-known reduction of magnetic flux in a parallel Abrikosov vortex in a thin film \\cite{vf1,vf2,vf3}. Calculation of $\\phi$ of a vortex in a long JJ stack with $N\\gg 1$ and $L_x\\gg \\lambda_J$ given in Appendix \\ref{Ap} yields the same result as for the Abrikosov vortex \\cite{vf2}:   \n\\begin{equation}\n\\phi(u) = \\phi_0\\left[1-\\frac{\\cosh(u\/\\lambda_{ab})}{\\cosh(L_z\/2\\lambda_{ab})}\\right].\n\\label{ph}\n\\end{equation} \nHere $u$ is the position of the vortex relative to the center of the film. Notice that $\\phi(u)$ decreases as $u$ increases and vanishes at the surface $u=\\pm L_z\/2$ where the vortex flux is extinguished by AV images \\cite{vf1,vf2}. For the J vortex in the center of a thin JJ stack $(u=0,\\, L_z=sN\\ll 2\\lambda_{ab})$, Eq. (\\ref{ph}) gives:\n\\begin{equation}\n\\phi\\simeq \\frac{\\phi_0 N^2}{8}\\left(\\frac{s}{\\lambda_{ab}}\\right)^2,\\qquad N\\lesssim \\frac{2\\lambda_{ab}}{s}.\n\\label{phi}\n\\end{equation}\nTaking here $N=21$, $s=1.5$ nm and $\\lambda_{ab}=400$ nm for BSCCO, we obtain $\\phi\\simeq 8\\cdot 10^{-4}\\phi_0$ in agreement with the simulation results presented in Fig. \\ref{fig21}a.\n\nShown in Fig. \\ref{fig21}b is $M(t)$ calculated for a dynamic flux state with one vortex per layer below the Cherenkov instability threshold at $\\beta <\\beta_s$.  Here the magnitude of $M(t)$ for 21 vortices is about 12 times larger than for a single vortex. The fact that $M(t)$ for one vortex per layer is not 21 times larger than $M(t)$ for a single vortex is consistent with Eq. (\\ref{ph}) according to which the flux of vortices on outer layers is smaller than $\\phi$ for the vortex on the central layer. The shape of $M(t)$ changes from rectangular pulses for a single vortex to triangular pulses for many vortices. This happens because the repelling vortices tend to arrange themselves to maximize the intervortex spacing so the reflections of vortices from the edges on different layers occur at different times.        \n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig21}\n\\caption{Temporal oscillations of a magnetic moment $M(t)$ due to periodic reflections of driven vortices and antivortices from the sample edges at $\\eta=0.1$. (a) $M(t)$ caused by a vortex shuttle in which a single vortex gets reflected from the edges as antivortex at $\\beta=0.585<\\beta_s$. The features marked by the arrows result from Cherenkov and bremstrahlung radiation after reflection of a V or AV. (b) $M(t)$ caused by a bouncing flux structure with one vortex per layer at $\\beta=0.53<\\beta_s$. }\n\\label{fig21}\n\\end{figure}\n\nAbove the Cherenkov instability threshold $\\beta>\\beta_s$ a single V-AV shuttle excites counter-propagating MVs and anti-macrovortices (AMV) which then get reflected from the edges in the same way as single Vs and AVs. For instance, the collision of MVs with the edge of an underdamped stack with $\\eta=0.1$ is shown in Fig. \\ref{fig22}. As the MV exits the stack it induces penetration of a counterpropagating AMV, the structure of this AMV remains preserved as it goes through the incoming MV without fragmentation into single vortices. Such bouncing MVs and AMVs generated by a V-AV shuttle give rise to temporal oscillations of the magnetic moment $M(t)=L_y\\phi(t)\/\\mu_0$, where $\\phi(t)$ is the net magnetic flux produced by all Vs and AVs. As shown in Fig. \\ref{fig23}, the magnitude of $M(t)$ is of the order of that of a stable flux structure with one vortex per layer (see Fig. \\ref{fig21}b). Notice that $M(t)$ for bouncing MVs contains multiple harmonics with frequencies much higher than those for the stable flux structures shown in Fig. \\ref{fig21}. \n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig22a}\n\\includegraphics[width=\\columnwidth]{fig22b}\n\\caption{Magnetic field color map in moving macrovortices colliding with the edge of the stack at $x\/\\lambda_c=-0.5$.  Top: A chain of macrovortices reaching the edge just before the collision. Bottom: The same chain after the leading macrovortex collided with the edge and got transformed into a counter-propagating anti-macrovortex. }\n\\label{fig22}\n\\end{figure}\n\nA big transient spike in $M(t)$ at the onset of the MV formation can be understood as follows. At $\\beta>\\beta_s$ the initial vortex placed near the right edge of the stack accelerates and starts producing V-AV pairs which form the MV structures spreading both along and across the JJ stack. Here MVs move to the left along with the initial vortex while AMVs move to the right and get reflected as MVs from the right edge before the leading MV reaches the left edge. As a result, the number of vortices in the stack keeps growing until the leading MV reaches the left edge, after which the process reverses as the number of AMVs increases and exceeds the number of MVs. After a few bouncing of MVs and AMVs back and forth, generation of new V-AV pairs stops and a standing wave, resulting in self-sustained oscillations of $M(t)$ forms, as shown in Fig. \\ref{fig23}. A snapshot of this standing wave in Fig. \\ref{fig24} indicates nonlinear interference and multiplication of harmonics with frequencies ranging from $\\omega\\sim\\omega_J$ to much lower frequencies $\\omega \\sim v\/d$ determined by the velocity $v(\\beta)$ and the spacing $d(\\beta)$ between MVs. Simulation movies of this process are available in Ref. \\onlinecite{supp}.\n \n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig23}\n\\caption{Temporal magnetic moment $M(t)$ due to bouncing macrovortices excited by a single V-AV shuttle. Inset shows $M(t)$ caused by self-sustained MV standing waves  superimposed onto $M(t)$ due to stable oscillations of the flux structure with one vortex per layer taken from Fig. \\ref{fig21}.}\n\\label{fig23}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig24}\n\\caption{A snapshot of beating standing waves of $B_l(x,t)$ on different layers in a finite stack with $N=41$ calculated for self-sustained oscillations of $M(t)$ shown in Fig. \\ref{fig23}.}\n\\label{fig24}\n\\end{figure}\n\nSelf-sustained MV standing waves excited by a V-AV shuttle at $J>J_s$ increase the power of electromagnetic radiation $W$ caused by temporal oscillations of $M(t)$ and a charge density at the surface of the stack. We do not consider here all essential contributions to $W$ which depend on the geometry of the stack and details of its electromagnetic coupling with surrounding structures (see, e.g., reviews \\onlinecite{thz1,thz2,lin-sust} and the references therein) but only estimate a magneto-dipole part of  $W$ which has not been addressed in the literature.  As follows from the inset in Fig. \\ref{fig23}, each MV at $N=21$ has $\\sim N\\phi_0$ bunched vortices lined perpendicular to the layers. Such bouncing multi-quanta MVs greatly increase the magneto-dipole radiation power $W\\propto \\ddot{M}^2$ as compared to the V-AV shuttle at $\\beta<\\beta_s$. Indeed, once $J$ exceeds $J_s$, both the magnitude and the frequency of $M(t)$ shown in Figs. \\ref{fig21} and \\ref{fig23} increases by more than an order of magnitude, which translates to $\\sim 10^7$ fold increase in $W$. \n\nBoth the magnitudes and the frequencies of different harmonics in $M(t)$ change significantly as the number of layers increases. Shown in Fig. \\ref{fig25} are $M(t)=\\phi(t)L_y$ calculated at $N=21$, $N=41$, and $N=81$ after the transient spikes in $M(t)$ decayed completely. Parts of these $M(t)$ curves calculated with much finer time steps $\\Delta t=0.01\\omega_J^{-1}$ shown in Fig. \\ref{fig26} clearly exhibit multiple harmonics with high frequencies $\\omega\\sim \\omega_J$ and low beating frequencies $\\omega\\ll \\omega_J$ which increase nearly linearly with $N$. As was mentioned above, the low-frequency part of $M(t)$ is related to traveling times of MVs. Characteristic magnitudes $M_N$ of $M(t)$ also increase as $N$ increases: $M_{81}\\simeq 4M_{41}$ and $M_{41}\\simeq (4-5) M_{21}$. This trend is qualitatively consistent with the quadratic increase of the  magnetic flux per vortex $M_N \\propto \\phi \\propto N^2$ in $JJ$ stacks with $L_z\\ll 2\\lambda_{ab}$ given by Eq. (\\ref{phi}).    \n\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig25}\n\\caption{Self-sustained oscillations of $M(t)$ calculated for $N=21$, $N=41$ and $N=81$ at $\\beta=0.6$ and $\\eta=0.1$ after complete decay of initial transient spikes in $M(t)$.}\n\\label{fig25}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig26a}\n\\includegraphics[width=\\columnwidth]{fig26b}\n\\caption{Parts of $M(t)$ at $N=41$ and $N=81$ shown in Fig. \\ref{fig25} but calculated with the finer time steps $\\Delta t =0.01\\omega_J^{-1}$ \nto reveal high-frequency harmonics in $M(t)$.}\n\\label{fig26}\n\\end{figure}\n\nThe mean radiation power $W=\\mu_0\\langle \\ddot{M}^2\\rangle\/6\\pi c^3$ for JJ stacks smaller than the radiated wavelength \\cite{griffiths} can be estimated using $M(t)$ from Eq. (\\ref{M}), where   $M_0=\\phi_0L_y\/2\\pi\\mu_0$ and $\\omega_J=c\/\\sqrt{\\epsilon_c}\\lambda_c$. Hence, $W$ can be presented in the form  \n\\begin{equation}\nW\\simeq \\frac{c(\\phi_0L_y)^2G_N}{24\\pi^3\\mu_0\\epsilon_c^2\\lambda_c^4},\n\\qquad \nG_N=\\int_{t_0}^{t_0+T}\\ddot{m}^2\\frac{dt}{T},\n\\label{ww}\n\\end{equation}\nwhere $m(t)=M(t)\/M_0$. The dimensionless factor $G_N$ takes into account the effect of the number of layers on the amplitudes and frequencies of different harmonics in $M$ which contribute to $W$, where $t_0\\simeq 800$. We evaluated $G_N$ by averaging numerical derivatives in $\\ddot{m}^2$ for the calculated $M(t)$ over the time interval $T=200$. Calculations of $G_N$ for different $N$ using the results shown in Fig. \\ref{fig26} give $G_{21}=0.0336$, $G_{41}=2.05$ and $G_{81}=154.1$. Such strong increase of $G_N$ with $N$ is much faster than $W\\propto N^4$ resulting from only the quadratic increase of the magnetic flux of the vortex with $N$. Another part of this rapid growth of $G_N$  comes from the enhancement of higher-frequency harmonics at larger $N$ evident from Figs. \\ref{fig25} and \\ref{fig26}.  All in all, the calculated $G_N$ roughly follows the $N^6$ dependence at $N\\lesssim 10^2$. \n \nTaking $\\lambda_c=200\\, \\mu$m,  $L_y=1$ mm, $\\epsilon_c=10$, and $G_{81}=154$ in Eq. (\\ref{ww}), we obtain $W\\simeq 1.32$ nW of the order of the lower end of radiated power observed on BSCCO mesas \\cite{thz2,thz3} with a much larger number of layers $N\\sim 10^3$. Yet given the very rapid increase of $W_N\\propto N^6$ revealed in our simulations at $N\\lesssim 10^2$, a much greater $W$ at $N\\sim 10^3$ may occur. Direct calculation of $W$ for $N\\sim 10^3$ is beyond our current computational capabilities. Yet if the trend $W\\propto N^6$ would continue up to $N\\simeq 2\\lambda_{ab}\/s\\simeq 500$ at which the flux per vortex reaches $\\phi_0$ (see Eqs. (\\ref{ph}) and (\\ref{phi})), one might expect $W_{500}\\sim W_{81}(500\/81)^6 \\sim1$ mW (for an ideal cooling of the sample and no Joule heating caused by the motion of MVs).    \n\n\\section{Discussion}\n\\label{sec:disc}\n\nIn this paper we show that uniform motion of a Josephson vortex driven by a dc current in layered superconductors breaks down as the velocity of the vortex exceeds the terminal velocity $v_c$ at current densities $J>J_s$. If  $v>v_c$ the moving vortex starts emitting V-AV pairs, causing a dendritic flux branching in which vortices and antivortices become spatially separated and form dissipative structures which depend on the sample geometry. For instance, a single vortex in a long stack can produce a chain of dissipative macrovortices that extend across the entire stack as shown in Fig. \\ref{fig9}. The breakdown of the dc flux flow state caused by V-AV pair production can occur at current densities $J_s$ well below the Josephson critical currents $J_0$ across the stack. \n\nIn an underdamped JJ stack of finite length $L_x$ a vortex driven by a dc current at $J<J_s$ turns into a V-AV shuttle in which the vortex periodically changes its polarity and direction of motion after each reflection from the sample edge.  This process results in oscillations of the magnetic moment $M(t)$ with the flight frequency $v\/2L_x$ depending on the length of the stack. At $J>J_s$ the V-AV shuttle produces propagating macrovortices consisting of bunched vortices aligned perpendicular to the layers. These macrovortices periodically change both the polarity and the direction of motion without fragmentation into single vortices after each reflection from the edges of the JJ stack. Such bouncing macrovortices eventually form large-amplitude flux standing waves, giving rise to oscillations of $M(t)$. Here $M(t)$ contains multiple harmonics the amplitudes and frequencies of which increase as the number of layers increases.      \n\nProliferation of V-AV pairs at $J>J_s$ can manifest itself in hysteretic jumps on the V-I curves. These jumps appear similar to those produced by heating effects\\cite{KL,thz2} yet the initial stage of the Cherenkov vortex instability is affected\nby neither cooling conditions nor the nonequilibrium kinetics of quasiparticles. Moreover, heating is most pronounced in overdamped junctions with $\\eta>1$ in which radiation is suppressed, whereas the Cherenkov instability is most pronounced in weakly-dissipative underdamped interlayer junctions characteristic of the BSCCO cuprates.  The V-AV pair production can be facilitated by interaction of vortices with edges or materials defects, resulting in vortex bremsstrahlung and further reduction of the terminal velocity $v_c$ and the threshold of instability current density $J_s$. These effects are similar to those revealed in our previous simulations of current-driven vortices in a single Josephson junction of finite length \\cite{screp}.   \n  \nThe V-AV pair production and bouncing macrovortices caused by a single vortex at $J>J_s$ can contribute to the power of radiation $W$ from a JJ stack.  As was shown in Sec. \\ref{sec:finsize}, the V-AV shuttle generates self-sustained MV standing waves and oscillations of the total magnetic moment. In turn, oscillations of $M(t)$ gives a contribution to the radiation power which increases greatly as the number of layers increases. For the parameters of BSCCO and $N\\leq 81$ our calculations gave $W\\sim 1$ nW, so one might expect $W\\sim 1$ mW at $N\\sim 10^3$ characteristic of the BSCCO mesas. Hence,  bouncing macrovortices could contribute to the radiation power observed in the BSCCO mesas, although specifying the fraction of this contribution in the total $W$ requires more elaborate calculations taking into account the sample geometry and cooling conditions.  The nonlinear MV standing wave at $J>J_s$ eventually give rise to strong dissipation which can produce hotspots in the sample \\cite{hs1,hs2}, even though heating is not the underlying cause for the V-AV pair production but rather its consequence. Our results thus suggest a mechanism by which the formation of hotspots may be linked to peaks in the radiation power, as was indeed observed on the BSCCO mesas \\cite{ths1,ths2,ths3,ths4,ths5}. \n\n\\section*{Acknowledgments}\n\nThis work was supported by the US Department of Energy under Grant No. DE-SC0010081-020. We thank A.E. Koshelev for a useful discussion.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\subsection{The Reduction}\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Preliminaries, continued}\n\\subsection{Proof of Claim~\\ref{clm: budget_loss}}\n\\label{app:budget_loss}\n\nWe will prove both results at once, noting that they both directly follow from showing that there exists matrices $M_1^\\varepsilon,\\ldots,M_k^\\varepsilon$ that are feasible for the discretized problems, such that for all $j \\in [w]$,\n\\[\nR^\\top M_{k-1}^\\varepsilon \\ldots M_{1}^\\varepsilon e_j \\geq R^\\top M^*_{k-1} \\ldots M^*_{1} e_j - (k-1) \\varepsilon,\n\\] \nwhere $M^*_1,\\ldots,M^*_{k-1}$ is an optimal solution to Program~\\eqref{SW_program}, respectively~\\eqref{maxmin_program}. To do so, we first note that it is feasible for Program~\\eqref{dscrt_SW_program} to pick a split of the budget $B_1^\\varepsilon,\\ldots,B_{k-1}^\\varepsilon$ such that for all $t$, $B_t^\\varepsilon \\geq \\max(c(M_t^*,M_t^0) - \\varepsilon,0)$, by construction of $\\mathcal{B}(\\varepsilon)$. We are going to construct a matrix $M^\\varepsilon_t$ that is close to $M_t^*$ and requires budget at most $B_t^\\varepsilon$. \n\nWhen $M_t^* = M_t^0$, one can just let $M^\\varepsilon_t = M_t^0$. Now, suppose $c(M_t^*,M_t^0) > 0$. For every pair of nodes $u \\in L_{t}$, we let $S_u^+$ the set of vertices $v \\in L_{t+1}$ such that $M_t^*(v,u) > M_t^0(v,u)$ (i.e. the transition from $u$ to $v$ has higher probability in $M_t^*$ than in $M_0$), and $S_u^-$ the set of vertices $v \\in L_{t+1}$ such that $M_t^*(v,u) < M_t^0(v,u)$. We note immediately that \n\\[\nc(M_t^*,M_t^0) = \\sum_{u \\in L_t} \\left(\n\\sum_{v \\in S_u^+} \\left(M_t^*(v,u) - M_t^0(v,u)\\right) + \\sum_{v \\in S_u^-} \\left(M_t^0(v,u) - M_t^*(v,u)\\right),\n\\right)\n\\]\nNow, let us construct $M^\\varepsilon_t \\in \\mathcal{M}$ such that for every $u$, \n\\[\nM^\\varepsilon_t(v,u) = M_t^*(v,u) - \\alpha(v,u)~\\forall v \\in S_u^+\n\\]\nand\n\\[\nM^\\varepsilon_t(v,u) = M_t^*(v,u) + \\alpha(v,u)~\\forall v \\in S_u^-,\n\\]\nwhere $\\alpha(v,u) \\geq 0$ for all $u,v$, $\\sum_{u,v} \\alpha(v,u) = \\min(\\varepsilon,c(M_t^*,M_t^0))$, $\\sum_{v \\in S_u^+} \\alpha(v,u) = \\sum_{v \\in S_u^-} \\alpha(v,u)$, and $M^\\varepsilon_t(u,v) \\geq M_t^0(u,v)$ for $v \\in S_u^+$ and $M_t^\\varepsilon (u,v) \\leq M_t^0(u,v)$ for $v \\in S_u^-$. Note that such $\\alpha$'s exist by virtue of $\\min(\\varepsilon,c(M_t^*,M_t^0)) \\leq c(M_t^*,M_t^0)$, which is the absolute value amount by which $M_t^*$ differs from $M_t^0$ coordinate-by-coordinate. Second, note that only malleable edges $(u,v)$ have $M_t^\\varepsilon(v,u) \\neq M_t^0(v,u)$, since we only modify malleable edges where $M_t^*(v,u) \\neq M_t^0(v,u)$. Further, $M_t \\in \\mathcal{M}$ since all the coefficients of $M_t$ remain between $0$ and $1$, and for all $u$,\n\\[\n\\sum_v M_t(v,u) = \\sum_v M_t^*(v,u) + \\sum_{v \\in S_u^-} \\alpha(v,u) - \\sum_{v \\in S_u^+} \\alpha(v,u) = \\sum_v M_t^*(v,u) = 1.\n\\]\nFurther, the cost of moving from $M_t^0$ to $M_t^\\varepsilon$ is given by\n\\begin{align*}\nc(M_t^\\varepsilon,M_t^0) \n&= \\sum_{u \\in L_t} \\left(\n\\sum_{v \\in S_u^+} \\left(M_t^\\varepsilon(v,u) - M_t^0(v,u)\\right) + \\sum_{v \\in S_u^-} \\left(M_t^0(v,u) - M_t^\\varepsilon(v,u)\\right)\n\\right)\n\\\\&= \\sum_{u \\in L_t} \\left(\n\\sum_{v \\in S_u^+} \\left(M_t^*(v,u) - M_t^0(v,u) -\\alpha(v,u) \\right) + \\sum_{v \\in S_u^-} \\left(M_t^0(v,u) - M_t^*(v,u) - \\alpha(v,u)\\right)\n\\right)\n\\\\&= c(M_t^*,M_t^0) - \\sum_{u,v} \\alpha(v,u)\n\\\\&= \\max\\left(0,c(M_t^*,M_t^0) - \\varepsilon\\right),\n\\end{align*}\nnoting that if $v$ was in $S_u^+$ (resp $S_u^-$), it is still the case that $M_t^\\varepsilon(v,u) \\geq M_t^0(v,u)$ (resp. $M_t^\\varepsilon(v,u) \\leq M_t^0(v,u)$). In turn, $M_t^\\varepsilon$ requires at most budget $B_t^\\varepsilon$, and $M_1^\\varepsilon,\\ldots,M_{k-1}^\\varepsilon$ is a feasible solution for the discretized programs. Finally, for any transition matrices $M_1,\\ldots,M_{k-1}$, letting $R_{t+1}^\\top = R^\\top M_{k-1} \\ldots, M_{t+1}$ (trivially, $0 \\leq R_{t+1} \\leq \\left\\Vert R \\right\\Vert_{\\infty}$) and $D_{t,j} = M_{t-1} \\ldots M_1 e_j$ (trivially, $D_{t,j} \\in \\mathcal{D}$), we note that (letting $\\alpha(v,u) = 0$ where not defined) \n\\begin{align*}\nR_{t+1}^\\top M_t D_{t,j}\n&=\\sum_{u \\in L_t,~v \\in L_{t+1}} M_t^\\varepsilon(v,u) R_{t+1}(v) D_{t,j}(u)\n\\\\&\\geq \\sum_{u,v} M_t^*(v,u) R_{t+1}(v) D_{t,j}(u)\n- \\sum_{u,v} \\alpha(v,u) R_{t+1}(v) D_{t,j}(u)\n\\\\&\\geq \\sum_{u,v} M_t^*(v,u) R_{t+1}(v) D_{t,j}(u)\n-\\left\\Vert R \\right\\Vert_{\\infty} \\sum_{u,v} \\alpha(v,u)\n\\\\&\\geq \\sum_{u,v} M_t^*(v,u) R_{t+1}(v) D_{t,j}(u)\n-\\left\\Vert R \\right\\Vert_{\\infty} \\varepsilon,\n\\end{align*}\nwhere the first inequality uses that for all $u,v$, $M_t^\\varepsilon(v,u) \\geq M_t^*(u,v) - \\alpha(v,u)$ by construction, the second inequality that $0 \\leq \\alpha(v,u)$, $0 \\leq D_{t,j}(u) \\leq 1$ and $0 \\leq R_{t+1}(v) \\leq \\left\\Vert R \\right\\Vert_{\\infty}$, and the last inequality from the fact that $\\sum_{u,v} \\alpha(v,u) = \\min(\\varepsilon,c(M_t^*,M_t^0)) \\leq \\varepsilon$. The proof can be concluded noting that for all $t$, the above inequality implies\n\\begin{align*}\nR^\\top M_{k-1}^* \\ldots M_{t+1}^* M_t^\\varepsilon \\ldots M_1^\\varepsilon e_j\n\\geq R^\\top M_{k-1}^* \\ldots M_{t+2}^* M_{t+1}^\\varepsilon M_t^\\varepsilon \\ldots M_1^\\varepsilon e_j -\\left\\Vert R \\right\\Vert_{\\infty} \\varepsilon\n\\end{align*}\nand applying this new inequality recursively. \n\n\\subsection{Proof of Claim~\\ref{clm:eps_net}}\\label{app:eps_net}\n\nThis is a well-known result; we provide a proof for completeness. The first observation is that such a net can be constructed recursively, as follows. Start with an empty set $S$. Initialize by picking any point $D$ in $\\mathcal{D}$, and let $S = \\{D\\}$. Then, recursively keep finding points $D' \\in \\mathcal{D}$ such that for all $D \\in S$, $\\left\\Vert D - D' \\right\\Vert_1 > \\varepsilon$, and augment $S := S \\cup D'$. Finally, stop the algorithm when no such point $D' \\in \\mathcal{D}$ exists. By construction, it must be that when the algorithm terminates, for all $D' \\in \\mathcal{D}$, there exists $D \\in S$ with $\\left\\Vert D - D' \\right\\Vert_1 \\leq \\varepsilon$. As such, $S$ constitutes an $\\varepsilon$-net in $\\ell_1$ distance for $\\mathcal{D}$.\n\n Second, we bound the number of steps needed. To do so, we remark that by construction, for all $D_1,D_2 \\in S$, it must be the case that $\\left\\Vert D_1 - D_2 \\right\\Vert_1 > \\varepsilon$; in turn, the $\\ell_1$-balls of radius $\\varepsilon\/2$ around each of the elements of $S$ must be disjoint, and the sum of their volumes is less than the volume of $\\mathcal{D}$. Since the volume of the probability simplex is given by $\\frac{1}{w!}$, and the volume of an $\\ell_1$-ball of radius $r$ is given by $\\frac{1}{w!} \\left(2r\\right)^w$, this yields that $|S| \\times \\frac{\\varepsilon^w}{w!} \\leq \\frac{1}{w!}$, or equivalently $|S| \\leq \\left(\\frac{1}{\\varepsilon}\\right)^w$.\n\n\n\n\\section{A More General Cost Model}\\label{app:Lipschitz}\n\\input{more_general_cost}\n\n\n\\section{A Separation between Ex-ante and Ex-post Maximin Values}\\label{app:separation}\n\\input{separation}\n\n\n\n\\section{Omitted Proofs}\n\\subsection{Proof of Theorem~\\ref{thm:MMW_approximation}: Algorithmic Guarantees for Ex-Post-Fairness}\\label{app:maximin_program}\n\nRecall that $OPT_{MM}^\\varepsilon$ is the optimum maximin value under discretized splits of the budget. Let $M_1^\\varepsilon, \\ldots, M_{k-1}^\\varepsilon$ be a set of transition matrices with expected reward for each starting position $i$ lower-bounded by by $R^\\top M_{k-1}^\\varepsilon \\ldots M_1^\\varepsilon e_i \\geq OPT_{MM}^\\varepsilon \\triangleq OPT_{MM} - (k-1) \\varepsilon \\left\\Vert R \\right\\Vert_{\\infty}$ that is feasible with respect to budget split $B_1^\\varepsilon,\\ldots, B_{k-1}^\\varepsilon$. Note that such matrices exist by Claim~\\ref{clm: budget_loss}. Let $E_t \\in \\mathcal{D}^w$ denote the population-wise probability distribution that is induced by these transition matrices on layer $t$, i.e.\n\\[\nE_t^j = M^\\varepsilon_{t-1} \\ldots M^\\varepsilon_1 e_j~\\forall j \\in [w].\n\\]\n\nTo prove the result, we will show by induction that for all $B_{\\geq t} \\geq B^\\varepsilon_{\\geq t}$, for $A_t \\in \\mathcal{A}(\\varepsilon)$ such that $\\Vert A^j_t - E^j_t \\Vert_1 \\leq \\varepsilon~~\\forall j \\in [w]$, we have\n\\[\nR^\\top M(B_{\\geq t},A_t) A^j_t \\geq OPT_{MM}^\\varepsilon - 2(k - t) \\varepsilon \\Vert  R \\Vert_{\\infty}, ~~\\forall j \\in [w],\n\\]\ni.e., a population-wise welfare approximation guarantee starting at any layer $t$. Since we can take $B_{\\geq 1} = B^\\varepsilon$, this directly implies\n\\[\nR^\\top M(B^\\varepsilon,A_1) e_j \\geq OPT_{MM}^\\varepsilon - 2(k-1) \\varepsilon \\left\\Vert R \\right\\Vert_{\\infty}~\\forall j \\in [w].\n\\]\nCombined with Claim~\\ref{clm: budget_loss}, which states that $OPT_{MM}^\\varepsilon \\geq OPT_{MM} - (k-1) \\varepsilon \\left\\Vert R \\right\\Vert_{\\infty}$, we obtain the result. \n\nLet us now provide our inductive proof. First, consider the transition from layer $L_{k-1}$ to layer $L_k$. Note that \n\\[\nOPT_{MM}^\\varepsilon \\le  R^\\top M_{k-1}^\\varepsilon\\ldots M_1^\\varepsilon e_j = R^\\top M_{k-1}^\\varepsilon E_{k-1}^j~~\\forall j \\in [w]\n\\]\nusing the fact that $E_{t+1}^j = M_t^\\varepsilon E_t^j$. Let $A_{k-1} \\in \\mathcal{A}(\\varepsilon)$ be such that $\\Vert A^j_{k-1} - E^j_{k-1} \\Vert \\leq \\varepsilon~~\\forall j \\in [w]$. Note that such a $A_{k-1}$ always exists (by definition of $\\mathcal{A}(\\varepsilon)$), and is considered by Algorithm~\\ref{alg: max_MM}. By Corollary~\\ref{cor:approx_loss}, \n\\[\nR^\\top M_{k-1}^\\varepsilon A^j_{k-1} \\geq R^\\top M_{k-1}^\\varepsilon E^j_{k-1} - \\varepsilon \\Vert  R \\Vert_{\\infty} \\geq OPT_{MM}^\\varepsilon  - \\varepsilon \\Vert  R \\Vert_{\\infty}~~\\forall j \\in [w].\n\\]\nFurther, $M_{k-1}^\\varepsilon$ is feasible for Program~\\eqref{MMW_DP} with respect to $B_{\\geq k-1}, B_{\\geq k} = 0$, given $B_{\\geq k-1} \\geq B_{\\geq k-1}^\\varepsilon$. As such, for $B_{\\geq k-1} \\geq B_{\\geq k-1}^\\varepsilon$, by optimality of $M(B_{\\geq k-1},A_{k-1})$, we have that\n\\[\n\\min_{j \\in [w]} R^\\top M(B_{\\geq k-1},A_{k-1}) A^j_{k-1}  \\geq \\min_{j \\in [w]} R^\\top M_{k-1}^\\varepsilon A^j_{k-1},\n\\]\nand in turn\n\\[\nR^\\top M(B_{\\geq k-1},A_{k-1}) A^j_{k-1} \n\\geq OPT_{MM}^\\varepsilon - \\varepsilon \\Vert  R \\Vert_{\\infty} ~~\\forall j \\in [w].\n\\]\n\nNow, suppose the induction hypothesis holds at layer $t+1$. I.e., for all $B_{\\geq t+1} \\geq B^\\varepsilon_{\\geq t+1}$, for $A_{t+1} \\in \\mathcal{A}(\\varepsilon)$ such that $\\Vert A^j_{t+1} - E_{t+1}^j \\Vert_1 \\leq \\varepsilon~~\\forall j \\in [w]$,\n\\[\nR^\\top M(B_{\\geq t+1},A_{t+1}) A^j_{t+1} \\geq OPT_{MM}^\\varepsilon - 2(k - t - 1) \\varepsilon \\Vert  R \\Vert_{\\infty}~~\\forall j \\in [w].\n\\]\nFor any given $B_{\\geq t} \\geq B_{\\geq t}^\\varepsilon$, note that one can set $B_{\\geq t+1} = B_{\\geq t+1}^\\varepsilon$ and have $B_t \\geq B_t^\\varepsilon$; hence, $M_t^\\varepsilon$ is feasible for Program~\\eqref{MMW_DP} with respect to $B_t \\geq B_t^\\varepsilon,B^\\varepsilon_{\\geq t+1}$. Consider $A_t \\in \\mathcal{A}(\\varepsilon)$ such that $\\Vert A^j_{t} - E_{t}^j \\Vert_1 \\leq \\varepsilon~~\\forall j \\in [w]$. Note that such a $A_{t}$ always exists (by definition of $\\mathcal{A}(\\varepsilon)$, and is considered by Algorithm~\\ref{alg: max_MM}. Since we have $\\Vert A^j_t -E_t^J \\Vert_1 \\leq \\varepsilon$ and $\\Vert A^j_{t+1} - M_t^\\varepsilon E_t^j \\Vert_1 \\leq \\varepsilon$ $\\forall j \\in [w]$, applying Corollary~\\ref{cor:approx_loss} yields that $\\forall j \\in [w]$,  \n\\begin{align*}\nR^\\top M(B^\\varepsilon_{\\geq t+1}, A_{t+1}) M^\\varepsilon_t A^j_t \n&\\geq R^\\top M(B^\\varepsilon_{\\geq t+1}, A_{t+1}) M^\\varepsilon_t E_t^j - \\varepsilon \\Vert  R \\Vert_{\\infty}\n\\\\ &\\geq R^\\top M(B^\\varepsilon_{\\geq t+1}, A_{t+1}) A^j_{t+1} - 2 \\varepsilon \\Vert  R \\Vert_{\\infty}.\n\\end{align*}\nUsing the induction hypothesis, we obtain that\n\\[\nR^\\top M(B^\\varepsilon_{\\geq t+1}, D_{t+1}) M^\\varepsilon_t A^j_t \\geq OPT_{MM}^\\varepsilon - 2(k - t) \\varepsilon \\Vert  R \\Vert_{\\infty}~~\\forall j \\in [w],\n\\]\nwhich can be rewritten as \n\\[\n\\min_{j \\in [w]} R^\\top M(B^\\varepsilon_{\\geq t+1}, D_{t+1}) M^\\varepsilon_t A^j_t \\geq OPT_{MM}^\\varepsilon - 2(k - t) \\varepsilon \\Vert  R \\Vert_{\\infty}.\n\\]\nIn particular, by optimality of $M(B_{\\geq t},A_t)$, it must be the case that\n\\[\n\\min_{j \\in [w]} R^\\top M(B_{\\geq t}, A_{t}) A^j_t \\geq OPT_{MM}^\\varepsilon - 2(k - t) \\varepsilon \\Vert  R \\Vert_{\\infty},\n\\]\nwhich concludes the proof of the accuracy guarantee. The running time is obtained noting that at each time step $t$, we solve one Program~\\ref{MMW_DP} for each of the (at most) $\\frac{B}{\\varepsilon}$ possible budget splits of $B_{\\geq t}$ and for each of the $\\left(\\left(\\frac{1}{\\varepsilon}\\right)^w\\right)^w =\n\\left(\\frac{1}{\\varepsilon}\\right)^{w^2}$ population-wise probability distributions in $\\mathcal{A}(\\varepsilon)$ on both layer $L_t$ and layer $L_{t+1}$; i.e., in a given time step, the algorithm solves $O\\left( \\frac{B}{\\varepsilon} \\left(\\frac{1}{\\varepsilon}\\right)^{w^4}\\right)$ optimization programs. Then, the algorithm finds the solution of all of these programs with the best objective value, which can be done in time linear in the number of such solutions, i.e. $O \\left(\\frac{B}{\\varepsilon} \\left(\\frac{k}{\\varepsilon}\\right)^{w^4}\\right)$. The algorithm does so over $k-1$ time steps.\n\n\n\n\\subsection{Proof of Lemma~\\ref{lem:exante_accuracy}: Algorithmic Guarantees for Ex-Ante Fairness}\\label{app:exante_accuracy}\n\nThe proof follows that of Theorem 1 of \\citet{FS96}. Note that we can rewrite the objective in the normal form given in Corollary 2 of \\citet{FS96}, by letting the payoff matrix $G$ be such that $G\\left(\\left(M_1,\\ldots,M_{k-1}\\right),q\\right) = R^\\top M_{k-1} \\ldots M_1 e_q$ when the designer plays $\\left(M_1,\\ldots,M_{k-1}\\right) \\in \\mathcal{F}$ and the learner plays $q \\in [w]$. Noting that $G$ has entries bounded between $0$ and $\\Vert R \\Vert_{\\infty}$, we can apply Corollary 2 of \\citet{FS96} to loss $R^\\top M^t_{k-1} \\ldots M^t_1 D^t$ with an appropriate renormalization to show the following low-regret statement:\n\\[\n\\frac{1}{T} \\sum_{t=1}^T R^\\top M^t_{k-1} \\ldots M^t_1 D^t \\leq \\min_{D \\in \\mathcal{D}} \\frac{1}{T} \\sum_{t=1}^T R^\\top M^t_{k-1} \\ldots M^t_1 D \n+\\left( \\sqrt{2 \\frac{\\ln w}{T}} + \\frac{\\ln w}{T} \\right) \\Vert R \\Vert_{\\infty},\n\\]\nSince $R^\\top M^t_{k-1} \\ldots M^t_1 D$ is linear in $D$, we have\n\\[\n\\min_{D \\in \\mathcal{D}} \\frac{1}{T} \\sum_{t=1}^T R^\\top M^t_{k-1} \\ldots M^t_1 D = \\min_{q \\in [w]} \\frac{1}{T} \\sum_{t=1}^T R^\\top M^t_{k-1} \\ldots M^t_1 e_q,\n\\]\nand the following low-regret statement\n\\begin{align}\\label{eq: regret_MW}\n\\frac{1}{T} \\sum_{t=1}^T R^\\top M^t_{k-1} \\ldots M^t_1 D^t \n\\leq \n\\min_{q \\in [w]} \\frac{1}{T} \\sum_{t=1}^T R^\\top M^t_{k-1} \\ldots M^t_1 e_q \n+ \\left( \\sqrt{2 \\frac{\\ln w}{T}} + \\frac{\\ln w}{T} \\right) \\Vert R \\Vert_{\\infty}.\n\\end{align}\n\nWe can now show the result, using a similar argument to that of Theorem 1 of \\citet{FS96}. To do so, we let $\\bar{D} \\in \\mathcal{D}$ be the probability distribution given by $\\bar{D} \\triangleq \\frac{1}{T} \\sum_{t=1}^T D^t$. We have that\n\\begin{align*}\n    &\\min_{q \\in [w]} R^\\top \\mathbb{E}_{M \\sim \\overline{\\Delta M}} \\mathbb{E} \\left[M_{k-1} \\ldots M_1\\right] e_q\n    \\\\&= \\min_{q \\in [w]} \\frac{1}{T} \\sum_{t=1}^T R^\\top M^t_{k-1} \\ldots M^t_1 e_q \\tag{by definition of $\\overline{\\Delta M}$}\n    \\\\&\\geq \\frac{1}{T} \\sum_{t=1}^T R^\\top M^t_{k-1} \\ldots M^t_1 D^t \n    - \\left( \\sqrt{2 \\frac{\\ln w}{T}} + \\frac{\\ln w}{T} \\right) \\Vert R \\Vert_{\\infty}\n    \\tag{by Equation~\\eqref{eq: regret_MW}}\n    \\\\&\\geq  \\frac{1}{T} \\sum_{t=1}^T \\max_{M \\in \\mathcal{F}} R^\\top M_{k-1} \\ldots M_1 D^t\n    -\\varepsilon\n    -\\left( \\sqrt{2 \\frac{\\ln w}{T}} + \\frac{\\ln w}{T} \\right) \\Vert R \\Vert_{\\infty} \\tag{$M^t$ $\\varepsilon$-approx. best response to $D^t$}\n    \\\\&\\geq \\max_{M \\in \\mathcal{F}} \\frac{1}{T} \\sum_{t=1}^T R^\\top M_{k-1} \\ldots M_1 D^t\n    -\\varepsilon\n    -\\left( \\sqrt{2 \\frac{\\ln w}{T}} + \\frac{\\ln w}{T} \\right) \\Vert R \\Vert_{\\infty}\n    \\tag{Max of sum less than sum of max}\n    \\\\& \\geq \\max_{\\Delta M \\in \\Delta \\mathcal{F}} \\frac{1}{T} \\sum_{t=1}^T  \\mathbb{E}_{M \\sim \\Delta M} \\left[R^\\top M_{k-1} \\ldots M_1 D^t\\right] \n    -\\varepsilon \\tag{Expectation over $\\Delta M$ less than best realization of $\\Delta M$, and the realization is in $\\mathcal{F}$}\n    -\\left( \\sqrt{2 \\frac{\\ln w}{T}} + \\frac{\\ln w}{T} \\right) \\Vert R \\Vert_{\\infty}\n    \\\\& = \\max_{\\Delta M \\in \\Delta \\mathcal{F}} R^\\top \\mathbb{E}_{M \\sim \\Delta M} \\left[M_{k-1} \\ldots M_1\\right] \\bar{D}\n    -\\varepsilon\n    -\\left( \\sqrt{2 \\frac{\\ln w}{T}} + \\frac{\\ln w}{T} \\right) \\Vert R \\Vert_{\\infty}\n    \\\\&\\geq \\min_{D \\in \\mathcal{D}}  \\max_{M \\in \\Delta\\mathcal{F}} R^\\top \\mathbb{E}_{M \\sim \\Delta M} \\left[M_{k-1} \\ldots M_1\\right] D\n    -\\varepsilon\n    -\\left( \\sqrt{2 \\frac{\\ln w}{T}} + \\frac{\\ln w}{T} \\right) \\Vert R \\Vert_{\\infty}\n    \\\\&\\geq \\max_{\\Delta M \\in \\Delta \\mathcal{F}} \\min_{D \\in \\mathcal{D}} R^\\top \\mathbb{E}_{M \\sim \\Delta M} \\left[M_{k-1} \\ldots M_1\\right] D\n    -\\varepsilon\n    -\\left( \\sqrt{2 \\frac{\\ln w}{T}} + \\frac{\\ln w}{T} \\right) \\Vert R \\Vert_{\\infty} \\tag{Max-min inequality}\n    \\\\&= \\max_{\\Delta M \\in \\Delta \\mathcal{F}} \\min_{q \\in [w]} R^\\top \\mathbb{E}_{M \\sim \\Delta M} \\left[M_{k-1} \\ldots M_1\\right] e_q\n    -\\varepsilon\n    -\\left( \\sqrt{2 \\frac{\\ln w}{T}} + \\frac{\\ln w}{T} \\right) \\Vert R \\Vert_{\\infty}.\n\\end{align*}\nThis concludes the proof.\n\n\n\n\\subsection{Omitted Proofs for Price of Fairness}\\label{app:price_fairness}\n\n\\subsubsection{Proof of the Lower Bound of Theorem~\\ref{thm: fairness_lb}}\\label{app:fairness_lb}\n\nNote that $P_f \\geq 1$ is always true, by definition. The proof of the other two cases when $B \\leq 2w$ is based on Example~\\ref{ex: lb_construction}. We divide the analysis of the construction into the following cases:\n\\begin{enumerate}\n\\item $B \\leq 2$. It is easy to see that $OPT_{SW} = \\frac{B}{2} (1-(w-1) \\varepsilon)$, and is achieved by the following transition matrix:\n\\begin{align*}\nM^*_1 = \\left(\n\\begin{matrix}\nB\/2 & 0 & \\ldots & 0\n\\\\1 - B\/2 & 1 & \\ldots & 1\n\\end{matrix}\n\\right)\n\\end{align*}\nNow note that the maximin solution is unique (and in particular, is the maximim solution with the highest social welfare), and this unique maximin solution splits the budget evenly among the starting nodes and yields social welfare $\\frac{B}{2w}$, via transition matrix\n\\begin{align*}\nM_1^{f} = \\left(\n\\begin{matrix}\n\\frac{B}{2w} & \\ldots & \\frac{B}{2w} \n\\\\1- \\frac{B}{2w} & \\ldots & 1- \\frac{B}{2w}\n\\end{matrix}\n\\right)\n\\end{align*}\nTherefore, we have that \n\\[\nP_f^+(\\varepsilon) = \\frac{B(1- w\\varepsilon)\/2}{\\frac{B}{2w}} = w(1 - w \\varepsilon),\n\\]\nand\n\\[\n\\lim_{\\varepsilon \\to 0} P_f(\\varepsilon) = w.\n\\]\n\n\\item Now, consider the case when $2 \\leq B \\leq 2w$. On the one hand, note that $OPT_{SW} \\geq 1 - (w-1)\\varepsilon$, as setting\n\\begin{align*}\nM^*_1 = \\left(\n\\begin{matrix}\n1 & 0 & \\ldots & 0\n\\\\ 0 & 1 & \\ldots & 1\n\\end{matrix}\n\\right)\n\\end{align*}\nonly requires a budget of $2$ hence is feasible for Program~\\eqref{SW_program}. The unique maximin solution is still given by $M_1^f$ and has welfare $\\frac{B}{2w}$. As such, we have \n\\[\nP_f(\\varepsilon) \\geq \\frac{1- w \\varepsilon}{\\frac{B}{2w}} = 2w \\frac{1 - (w-1)\\varepsilon}{B}.\n\\]\nIn particular, taking $\\varepsilon \\to 0$, we get that a lower bound on the price of fairness is given by $\\frac{2w}{B}$.\n\\end{enumerate}\nThe proof for $B \\geq 2w$ is immediate, noting that \n\\begin{align*}\nM^*_1 = \\left(\n\\begin{matrix}\n1 & 1 & \\ldots & 1\n\\\\ 0 & 0 & \\ldots & 0\n\\end{matrix}\n\\right)\n\\end{align*}\nis feasible. As such $OPT_{SW} = 1$, and $M^*_1$ is a maximin solution with welfare $1$.\n\n\n\\subsubsection{Proof of Lemma~\\ref{lem: ub_OPT}}\\label{app:ub_OPT}\n\n\\begin{proof}\nThe first part of the claim is immediate from noting that given an optimal solution $M_1^*,\\ldots,M_{k-1}^*$ to Program~\\eqref{SW_program}, \n\\[\nOPT_{SW} = R^\\top M_{k-1}^* \\ldots M_1^* D_1^0\n\\leq \\left\\Vert R \\right\\Vert_{\\infty} \\left\\Vert M_{k-1}^* \\ldots M_1^* D_1^0 \\right\\Vert\n= \\left\\Vert R \\right\\Vert_{\\infty},\n\\]\nwhere the last equality follows from $M_{k-1}^* \\ldots M_1^* D_1^0$ being a probability distribution.\n\nFor the second part of the claim, consider any feasible solution $M_1,\\ldots,M_{k-1}$ with corresponding split $B_1, \\ldots, B_{k-1}$ of the budget. I.e., $B = \\sum_{t = 1}^{k-1} B_t$, and $\\sum_i \\sum_j \\left\\vert M_t(i,j) - M_t^0(i,j) \\right\\vert \\leq B$ for all $t$. Note that at layer $L_{t}$, for any input distribution $D_{t}$, and vector $R_{t+1}$ with non-negative coordinates at layer $t+1$, we have that \n\\begin{align*}\nR_{t+1}^\\top \\left(M_{t} - M_{t}^0\\right) D_{t}\n&= \\sum_{i=1}^w R_{t+1}(i) \\sum_{j=1}^w \\left(M_t(i,j) - M_t^0(i,j)\\right)  D_t(j)\n\\\\&= \\sum_{j=1}^w D_t(j) \\sum_{i=1}^w R_{t+1}(i) \\left(M_t(i,j) - M_t^0(i,j)\\right)  \n\\\\&\\leq \\sum_{j=1}^w D_t(j) \n\\sum_{i \\in S_j} R_{t+1}(i) \\left(M_t(i,j) - M_t^0(i,j)\\right)  \n\\\\&\\leq \\left\\Vert R_{t+1} \\right\\Vert_{\\infty }\n\\sum_{j=1}^w D_t(j) \\sum_{i \\in S_j} M_t(i,j) - M_t^0(i,j)\n\\end{align*}\nwhere $S_j = \\{i:~ M_t(i,j) - M_t^0(i,j) \\geq 0\\}$ and where the second-to-last inequality follows from the fact that $R_{t+1}(i) \\geq 0$. As $M_t,M_t^0 \\in \\mathcal{M}$, we have that\n\\[\n\\sum_{i=1}^w  M_t(i,j) - M_t^0(i,j) = 1 - 1 = 0,\n\\]\nwhich implies that \n\\begin{align*}\n    \\sum_{i \\in S_j}  M_t(i,j) - M_t^0(i,j)\n    = \\sum_{i = 1}^w  M_t(i,j) - M_t^0(i,j) - \\sum_{i \\notin S_j} M_t(i,j) - M_t^0(i,j)\n    = - \\sum_{i \\notin S_j} M_t(i,j) - M_t^0(i,j).\n\\end{align*}\nIn turn, we have that \n\\begin{align*}\n    \\frac{B_t}{2} \n    \\geq \\frac{c(M_t,M_t^0)}{2}\n    &=\\frac{1}{2}\\sum_j \\sum_i \\left\\vert M_t(i,j) - M_t^0(i,j) \\right\\vert \n    \\\\&= \\frac{1}{2} \\sum_j \\left(\\sum_{i \\in S_j} M_t(i,j) - M_t^0(i,j) -  \\sum_{i \\notin S_j} M_t(i,j) - M_t^0(i,j) \\right)\n    \\\\&= \\sum_j \\sum_{i \\in S_j} M_t(i,j) - M_t^0(i,j).\n\\end{align*}\nThis can also be seen noting that increasing edges of $M_t^0$ by a total amount of $\\delta B_t$ requires decreasing other edges by a total amount of $\\delta B_t$ for $M_t$ to be a stochastic matrix and so requires a total budget of $2\\delta B_t$, which in turn implies that $\\delta \\leq \\frac{1}{2}$ necessarily. This implies that\n\\begin{align*}\nR_{t+1}^\\top \\left(M_{t} - M_{t}^0\\right) D_{t}\n&\\leq \\left\\Vert R_{t+1} \\right\\Vert_{\\infty }\n\\sum_{j=1}^w D_t(j) \\sum_{i \\in S_j} M_t(i,j) - M_t^0(i,j)\n\\\\&\\leq \\left\\Vert R_{t+1} \\right\\Vert_{\\infty }\n\\sum_{j=1}^w \\sum_{i \\in S_j} M_t(i,j) - M_t^0(i,j)\n\\\\&\\leq \\frac{B_t}{2} \\left\\Vert R_{t+1} \\right\\Vert_{\\infty }\n\\end{align*}\nApplying the above inequality recursively, we have that \n\\begin{align*}\n    R^\\top M_{k-1} \\ldots M_{1} D_1^0 \n    &\\leq \\frac{B_{k-1}}{2} \\left\\Vert R \\right\\Vert_{\\infty} \n    + R^\\top M^0_{k-1} M_{k-2} \\ldots M_{1} D_1^0\n    \\\\&\\leq \\frac{B_{k-1} + B_{k-2}}{2} \\left\\Vert R \\right\\Vert_{\\infty}\n    +  R^\\top M^0_{k-1} M^0_{k-2} M_{k-3} \\ldots M_{1} D_1^0\n    \\\\&~~~~~~~~~~~~~~~~~~~~~~~~~~~\\vdots\n    \\\\&\\leq \\frac{\\sum_t B_{t}}{2} \\left\\Vert R \\right\\Vert_{\\infty}\n    +  R^\\top M^0_{k-1} \\ldots M^0_{1} D_1^0\n    \\\\& = \\frac{B}{2} + W^0.\n\\end{align*}\n\\end{proof}\n\n\\subsubsection{Proof of Lemma~\\ref{lem: lb_maxmin}}\\label{app:lb_maxmin}\n\nFirst, we consider the case when $\\frac{B}{2w} \\leq 1$. We focus on the transition from the second-to-last layer $L_{k-1}$ to the last layer $L_k$. Remember that on layer $L_k$, $R(1) =\\Vert R \\Vert_{\\infty}$. We re-number (w.l.o.g.) the nodes on layer $k-1$ so that $M^0_{k-1}(1,i) \\geq \\frac{B}{2w}$ for all nodes $i \\in [l]$, and $M^0_{k-1}(1,i) < \\frac{B}{2w}$ for all nodes $i \\in \\{l+1,\\ldots,k\\}$, for some $l \\in \\{0,\\ldots,w\\}$. Let us set\n\\begin{align*}\nM_{k-1} = M_{k-1}^0 +  \\left(\n\\begin{matrix}\n0 & 0 & \\ldots & 0 & \\frac{B}{2w} - M^0_{k-1}(1,l+1) & \\ldots & \\frac{B}{2w} - M^0_{k-1}(1,w)\n\\\\0 & 0 & \\ldots & 0 & -\\alpha_{2,l+1} & \\ldots & -\\alpha_{2,w}& \n\\\\ \\vdots & \\vdots & \\vdots & 0 & \\vdots & \\cdots & \\vdots\n\\\\0 & 0 & \\ldots &  0 & -\\alpha_{w,l+1} & \\ldots & -\\alpha_{w,w}& \n\\end{matrix}\n\\right),\n\\end{align*} and\n\\begin{align*}\nM_t = M_t^0~\\forall t < k-1,\n\\end{align*}\nwhere the $\\alpha_{i,j}$'s are chosen to guarantee $M^0_{k-1}(i,j) \\geq \\alpha_{i,j} \\geq 0$ for all $i,j \\in [w]$ and $\\sum_{i=2}^w \\alpha_{i,j} = \\frac{B}{2w} - M^0_{k-1}(1,j) > 0$ for all $j > l$. Such a choice of $\\alpha_{i,j}$'s exists as \n\\[\n\\sum_{j=2}^w M^0_{k-1}(i,j) = 1 - M^0_{k-1}(1,j) \\geq \/(2w) - M^0_{k-1}(1,j).\n\\]\nNow, note that $M_{k-1} \\in \\mathcal{M}$, as all coefficients are between $0$ and $1$ and the elements within the same column still sum to $1$ by choice of $\\alpha_{i,j}$'s. Finally, \n\\[\nc(M_{k-1},M_{k-1}^0) \n= \\sum_{j = l+1}^w 2 \\left(\\frac{B}{2w} - M^0_{k-1}(1,j)\\right)\n\\leq \\sum_{j = l+1}^w 2 \\frac{B}{2w}\n\\leq B.\n\\]\nTherefore, $(M_{k-1},\\ldots,M_1)$ is a feasible solution for Programs~\\ref{SW_program} and~\\ref{maxmin_program} under budget $B$.\n\nNow, note that by construction, $M_{k-1}(1,j) \\geq \\frac{B}{2w}$ for all $j \\in [w]$. In turn, this implies that for any distribution $D_{k-1}$,\n\\begin{align*}\nR^\\top M_{k-1} D_{k-1} \n= \\sum_{i=1}^w R(i) \\left(M_{k-1} D_{k-1}\\right)(i)\n&= \\sum_{i=1}^w R(i) \\sum_{j=1}^w M_{k-1}(i,j) D_{k-1}(j)\n\\\\& \\geq R(1) \\sum_{j=1}^w M_{k-1}(1,j) D_{k-1}(j)\n\\\\&\\geq R(1) \\frac{B}{2w} \\sum_{j=1}^w D_{k-1}(j)\n\\\\&= \\frac{B}{2w} \\left\\Vert R \\right\\Vert_{\\infty}, \n\\end{align*}\nsince $\\sum_{j=1}^w D_{k-1}(j) = 1$ by virtue of $D_{k-1}$ being a probability distribution, and because $R(1) = \\left\\Vert R \\right\\Vert_{\\infty}$ by choice of node indexing. In particular, for all $j$, $D_{k-1} = M_{k-2} \\ldots M_1 e_j$ is a probability distribution, hence \n\\[\nR^\\top M_{k-1} \\ldots M_1 e_j \\geq \\frac{B}{2w} \\left\\Vert R \\right\\Vert_{\\infty}.\n\\]\nTherefore, there exists a solution to Program~\\eqref{maxmin_program} that has value $\\frac{B}{2w} \\left\\Vert R \\right\\Vert_{\\infty}$, implying any optimal solution to Program~\\eqref{maxmin_program} has value at least $\\frac{B}{2w} \\left\\Vert R \\right\\Vert_{\\infty}$. In turn, such an optimal solution $M^f_1, \\ldots,M^f_{k-1} \\in S^f$ must have welfare \n\\begin{align*}\nR^\\top M^f_{k-1} \\ldots M^f_1 D_1^0 \n= \\sum_{j=1}^w D_1^0(j) R^\\top M^f_{k-1} \\ldots M^f_1 e_j\n\\geq \\sum_{j=1} D_1^0(j) \\frac{B}{2w} \\left\\Vert R \\right\\Vert_{\\infty} \n=\\frac{B}{2w} \\left\\Vert R \\right\\Vert_{\\infty},\n\\end{align*}\nwhich concludes the proof when $B \\leq 2w$. \n\nWhen $B > 2w$, let $B' = 2w$. By the above, any optimal solution to maximin Program~\\eqref{maxmin_program} with budget $B' = 2w$ has value at least $\\frac{B'}{2w} \\left\\Vert R \\right\\Vert_{\\infty} = \\left\\Vert R \\right\\Vert_{\\infty}$. This immediately implies that any optimal solution to Program~\\eqref{maxmin_program} with budget $B$ also has value at least $\\left\\Vert R \\right\\Vert_{\\infty}$, since any feasible solution for budget $B'$ is feasible for budget $B$. In turn, any optimal solution to maximin Program~\\eqref{maxmin_program} under budget $B > 2w$ must have welfare at least $\\left\\Vert R \\right\\Vert_{\\infty}$. \n\n\n\\subsubsection{Proof of Lemma~\\ref{lem: lb_maxmin_w0}}\\label{app:lb_maxmin_w0}\n\nThe proof of the lemma uses the following Claim~\\ref{clm: maxmin_equalspending}, that shows that an optimal solution to Program~\\eqref{maxmin_program} spends all the budget $B$:\n\\begin{claim}\\label{clm: maxmin_equalspending}\nLet $(M_1,\\ldots,M_{k-1})$ be a solution to maximin Program~\\eqref{maxmin_program} with social welfare strictly less than $\\Vert R \\Vert_{\\infty}$. If all edges are malleable, it must be the case that $\\sum_{t=1}^{k-1} c(M_t,M_t^0) = B$.\n\\end{claim}\nThe proof idea is simple: if $\\sum_{t=1}^{k-1} c(M_t,M_t^0) < B$, the leftover budget can be used to improve the scial welfare, unless this social welfare already is the maximum achievable value of $\\Vert R \\Vert_{\\infty}$.\n\n\\begin{proof}\nBy contradiction, suppose that $B > \\sum_{t=1}^{k-1} c(M_t,M_t^0)$. Remember the numbering of nodes on layer $L_k$ is chosen such that $R(1) = \\left\\Vert R \\right\\Vert_{\\infty}$. Let us pick all $j$ such that $M_{k-1}(1,j) < 1$ (if such a $j$ exists); note that since $M_{k-1}$ is stochastic, there also exists $q \\neq 1$ such that $M_{k-1}(q,j) > 0$. For $\\varepsilon$ arbitrarily small, there hence exists a matrix $M'_{k-1}(\\varepsilon) \\in \\mathcal{M}$ such that for all such $j$, $M'_{k-1}(1,j) = M_{k-1}(1,j) + \\varepsilon$ and $\\sum_{q \\neq 1} M'_{k-1}(q,j) = \\sum_{q \\neq 1} M_{k-1}(q,j) - \\varepsilon$, and such that $M'_{k-1}(q,j) = M_{k-1}(q,j)$ for all $q \\in [w]$ and all $j$ with $M_{k-1}(1,j) = 1$.\n\nNow, note that $c(M'_{k-1},M^0_{k-1})\n\\leq c(M'_{k-1},M_{k-1}) + c(M_{k-1},M^0_{k-1})$ with \n$\n\\lim_{\\varepsilon \\to 0} c(M'_{k-1},M_{k-1}) = 0\n$. \nIn turn, this implies that for $\\varepsilon$ small enough,\n$\n\\sum_{t=1}^{k-2} c(M_t,M_t^0) + c(M_t',M_t^0) \\leq  B\n$, \nhence $\\left(M_1,\\ldots,M_{k-2},M_{k-2}'\\right)$ is feasible for Program~\\eqref{maxmin_program}. Further, by construction, for all $j$ with $M_{k-1}(1,j) < 1$, we have\n\\begin{align*}\nR^\\top M'_{k-1} e_j - R^\\top M_{k-1} e_j \n\\geq \\left(R(1) - \\max_{q \\neq 1} R(q)\\right) \\varepsilon\n> 0,\n\\end{align*}\nand for all $j$ with $M_{k-1}(1,j) = 1$, we have $R^\\top M'_{k-1} e_j = R^\\top M_{k-1} e_j$. Since for all starting nodes $i \\in L_1$ such that $R^\\top M_{k-1} \\ldots M_{1} e_i < \\left\\Vert R \\right\\Vert_{\\infty}$, there must exist $j$ such that $M_{k-1}(1,j) < 1$ and $\\left(M_{k-2} \\ldots M_1 \\right)(j,i) > 0$ (otherwise $\\left(M_{k-1} \\ldots M_1 \\right)(1,i) = 1$ and node $i$ obtains reward $\\left\\Vert R \\right\\Vert_{\\infty}$), it immediately follows that for all such $i$,\n\\[\nR^\\top M'_{k-1} M_{k-2} \\ldots M_1 e_i  > R^\\top M_{k-1} M_{k-2} \\ldots M_1 e_i.\n\\]\nThis contradicts $(M_1,\\ldots,M_{k-1})$ being an optimal solution for Program~\\eqref{maxmin_program}.\n\\end{proof}\n\nWe are now ready to prove Lemma~\\ref{lem: lb_maxmin_w0}. Let $M_1^f,\\ldots,M_{k-1}^f$ be an optimal solution for Program~\\eqref{maxmin_program}. Note that if the maximin value of this solution is $\\left\\Vert R \\right\\Vert_{\\infty}$, then the solution necessarily has welfare $\\left\\Vert R \\right\\Vert_{\\infty} \\geq W^0$, which concludes the proof. So, without loss of generality, we can assume there exists at least one starting node $q$ such that \n \\[\n R^\\top M_{k-1}^f \\ldots M_1^f e_q < \\left\\Vert R \\right\\Vert_{\\infty}.\n\\]\nFix a layer $t$, and let $R_{out}^\\top \\triangleq R^\\top M_k^f \\ldots M_{t+1}^f$ and $M_{in} = M_{t-1}^f \\ldots M_1^f$. Note that the utility obtained by the $q$-th node in the starting layer is immediately given by\n\\begin{align*}\nR_i = \\sum_{i=1}^w M_{in}(i,q) \\sum_{j=1}^w M^f_t(j,i) R_{out}(j).\n\\end{align*}\nIndeed, starting from node $q$ in the first layer, an individual transitions to node $i$ on layer $t$ with probability $M_{in}(i,q)$, then to node $j$ with reward $R_{out}(j)$ on layer $t+1$ with probability $M_t(j,i)$. \n\nSuppose by contradiction that for some $i'$, $\\sum_{j=1}^w M^f_t(j,i') R_{out}(j) < \\sum_{j=1}^w M^0_t(j,i') R_{out}(j)$ (necessarily, $M^f_t(j,i') \\neq M^0_t(j,i')$ for some $j$). We will construct a set of transition matrices that achieves the same maximin value, but requires budget strictly less than $B$. To do so, let $M_t'$ be such that $M_t'(j,i') = M_t^0(j,i')$ for all $j$ and $M_t'(j,i) = M_t^f(j,i)$ for all $i \\neq i'$, for all $j$. First, $c(M_t',M_t^0) < c(M_t^f,M_t^0)$, as\n\\begin{align*}\n\\sum_{i,j} \\left\\vert M_t'(j,i) - M_0(j,i)\\right\\vert\n&= \\sum_{i \\neq i', j} \\left\\vert M_t^f(j,i) - M_0(j,i)\\right\\vert\n\\\\&<  \\sum_{i \\neq i',j} \\left\\vert M_t^f(j,i) - M_0(j,i)\\right\\vert \n+\\sum_{j} \\left\\vert M_t^f(j,i') - M_0(j,i')\\right\\vert\n\\\\& = c(M_t,M_t^0)\n\\end{align*}\nwhere the strict inequality follows from the fact that $M^f_t(j,i') \\neq M^0_t(j,i')$ for some $j$. Second, for all $q$, we immediately have that as the $M_{in}(i,q)$ are non-negative, \n\\[\n\\sum_{i=1}^w M_{in}(i,q) \\sum_{j=1}^w M'_t(j,i) R_{out}(j) \\geq \\sum_{i=1}^w M_{in}(i,q) \\sum_{q=1}^w M^f_t(j,i) R_{out}(j),\n\\]\nsince for all $i \\neq i'$ we have $\\sum_{j=1}^w M'_t(j,i) R_{out}(j) = \\sum_{j=1}^w M^f_t(j,i) R_{out}(j)$, and by construction \n\\\\$\\sum_{j=1}^w M'_t(j,i') R_{out}(j) = \\sum_{j=1}^w M^0_t(j,i') R_{out}(j) > \\sum_{j=1}^w M^f_t(j,i') R_{out}(j)$ for $i'$. In particular, this implies that $(M_1^f,\\ldots,M_{t-1}^f,M_t',M_{t+1}^f,\\ldots,M_{k-1}^f)$ is an optimal solution to Program~\\eqref{maxmin_program} that uses budget strictly less than $B$. This contradicts Claim~\\ref{clm: maxmin_equalspending}, that shows that the leftover budget can then be used to increase the optimal value of Program~\\ref{maxmin_program}, implying that $M^f$ cannot be an optimal solution. Therefore, it must be the case that for all $i \\in [w]$, for all $t \\in [k-1]$,\n\\[\n\\sum_{j=1}^w M_t^f(j,i) R^\\top M_{k-1}^f \\ldots M_{t+1}^f e_j \\geq \\sum_{j=1}^w M_t^0(j,i) R^\\top M_{k-1}^f \\ldots M_{t+1}^f e_j,\n\\]\nor equivalently\n\\begin{align}\\label{eq: increase_utility}\nR^\\top M_{k-1}^f \\ldots M_{t+1}^f M_t^f e_i \\geq R^\\top M_{k-1}^f \\ldots M_{t+1}^f M_t^0 e_i. \n\\end{align}\nApplying this with $t = 1$, we have that for all starting $q$ on layer $L_1$,\n\\[\nR^\\top M_{k-1}^f \\ldots M_{1}^f e_q \\geq R^\\top M_{k-1}^f \\ldots M_{2}^f M_1^0 e_q.\n\\]\nNow, suppose by induction that for all $i \\in [w]$,\n\\[\nR^\\top M_{k-1}^f \\ldots M_1^f e_i \\geq R^\\top M_{k-1}^f \\ldots M_{t}^f  M_{t-1}^0 \\ldots M_1^0 e_i.\n\\]\nIt follows that\n\\begin{align*}\nR^\\top M_{k-1}^f \\ldots M_1^f e_q\n&\\geq R^\\top M_{k-1}^f \\ldots M_{t}^f  M_{t-1}^0 \\ldots M_1^0 e_q\n\\\\&= R^\\top M_{k-1}^f \\ldots M_{t}^f \\sum_i \\left(M_{t-1}^0 \\ldots M_1^0 e_q\\right)(i) e_i\n\\\\& = \\sum_i \\left(M_{t-1}^0 \\ldots M_1^0 e_q\\right)(i) R^\\top M_{k-1}^f \\ldots M_{t}^f e_i\n\\\\& \\geq \\sum_i \\left(M_{t-1}^0 \\ldots M_1^0 e_q\\right)(i) R^\\top M_{k-1}^f \\ldots M_{t}^0 e_i\n\\\\&=R^\\top M_{k-1}^f \\ldots M_{t+1}^f  M_t^0 \\ldots M_1^0 e_q\n\\end{align*}\nwhere the second-to-last equation follows from Equation~\\eqref{eq: increase_utility}. Therefore, by induction, we have that for all $q$,\n\\[\nR^\\top M_{k-1}^f \\ldots M_1^f e_q \\geq R^\\top M_{k-1}^0 \\ldots M_1^0 e_q,\n\\]\ndirectly implying that \n\\[\nR^\\top M_{k-1}^f \\ldots M_1^f D_1^0 \\geq R^\\top M_{k-1}^0 \\ldots M_1^0 e_i D_1^0 = W^0.\n\\]\n\n\n\n\n\n\\subsection{Solving the Ex-ante Maximization Problem Using Algorithm~\\ref{alg: max_SW}}\n\nBecause Program~\\ref{exante_maxmin_program} is a $\\max\\min$ problem over a polytope, we can view it as a zero-sum game, and the solution that we want corresponds to a maxmin equilibrium strategy of this game. As first shown by \\cite{FS96}, it is possible to compute an approximate equilibrium of a zero-sum game if we can implement a \\emph{no-regret} learning algorithm for one of the players, and an approximate best-response algorithm for the other player: if we simply simulate repeated play of the game between a no-regret player and a best-response player, then the empirical average of player actions in this simulation converges to the Nash equilibrium of the game. \n\nThis forms the basis of our algorithm. One player plays the ``multiplicative weights'' algorithm over the initial positions in layer 1 of the graph. This induces at every round a distribution over initial positions. The best response problem, which must be solved by the other player, corresponds to solving a welfare-maximization problem given the distribution over initial positions represented by the multiplicative weights distribution. Fortunately, this is exactly the problem that we have already given a dynamic programming solution for. The solution in the end corresponds to the uniform distribution over the solutions computed by the best-response player over the course of the dynamics. The algorithm is formally described below:\n\\begin{algorithm}\n\\KwIn{Time horizon $T$, reward vector $R$ on layer $L_k$, initial transition matrices $M_1^0,\\ldots,M_{k-1}^0$, budget $B$, discretization parameter $\\varepsilon$.}\n\\KwOut{$M^1,\\ldots,M^T \\in \\mathcal{F}\\left(B,M_1^0,\\ldots,M_{k-1}^0\\right)$.}\n\\textbf{Initialization:} The no-regret player picks $D^1 = \\left(\\frac{1}{w},\\ldots,\\frac{1}{w}\\right) \\in \\mathcal{D}$, the uniform distribution over $[w]$.\\\\\n\\For{$t = 1, \\ldots, T$}{\nThe no-regret player plays distribution $D^t \\in \\mathcal{D}$.\\\\\nThe best-response player chooses $M^t \\in \\mathcal{F}\\left(B,M_1^0,\\ldots,M_{k-1}^0\\right)$ such that \n    \\[\n    R^\\top M^t_{k-1} \\ldots M^t_1 D^t \\geq \\max_{M \\in \\mathcal{F}} R^\\top M_{k-1} \\ldots M_1 D^t - \\varepsilon \\left\\Vert R \\right\\Vert_{\\infty},\n    \\]\n    using Algorithm~\\ref{alg: max_SW}.\\\\\nThe no-regret player observes $u^t_i = \\frac{R^\\top M^t_{k-1} \\ldots M^t_1 e_i}{\\Vert R \\Vert_{\\infty}}$ for all $i \\in [w]$, and picks $D^{t+1}$ via multiplicative weight update, as follows:\n    \\[\n    D^{t+1}(i) = \\frac{D^t(i) \\beta^{u^t_i}}{\\sum_{j=1}^w D^t(j) \\beta^{u^t_j}}~\\forall i \\in [w],\n    \\]\n with $\\beta = \\frac{1}{1 + \\sqrt{2 \\frac{\\ln w}{T}}} \\in [0,1)$. \n}\n\\caption{2-Player Dynamics for the Ex-Ante Maximin Problem}\\label{def: dynamics}\n\\end{algorithm}\n\n\n\\begin{lemma}\\label{lem:exante_accuracy}\nLet $T > 0$, $\\overline{\\Delta M}$ be the probability distribution that picks $\\left(M_1,\\ldots,M_{k-1}\\right) \\in \\mathcal{F}(B,M_1^0,\\ldots,M_k^0)$ with probability \n\\[\n\\frac{1}{T} \\sum_{t=1}^T \\mathbbm{1} \\left\\{\\left( M_1,\\ldots,M_{k-1} \\right) = \\left( M^t_1,\\ldots,M^t_{k-1} \\right) \\right\\},\n\\]\nwhere $M^1,\\ldots,M^T$ are the outputs of Algorithm~\\ref{def: dynamics}. Then $\\overline{\\Delta M}$ $\\left(\\varepsilon\n    +\\sqrt{2 \\frac{\\ln w}{T}} + \\frac{\\ln w}{T} \\right) \\Vert R \\Vert_{\\infty}$-approximately optimizes Program~\\ref{exante_maxmin_program}.\n\\end{lemma}\n\nThe proof of Lemma~\\ref{lem:exante_accuracy} follows from interpreting Program~\\ref{exante_maxmin_program} as zero-sum game, noting that the best response problem for the maximization player corresponds to the welfare-maximization problem for which we have an efficient algorithm, and then applying the no-regret dynamic analysis from~\\citet{FS96}. The details are provided in  Appendix~\\ref{app:exante_accuracy}.\n\n\n\n\n\n\n\\subsection{Overview of Our Results}\nBriefly, our main contributions are the following:\n\\begin{enumerate}\n    \\item We define and formalize the \\emph{pipeline intervention problem} with the social welfare, ex-ante maximin, and ex-post maximin objectives. We also prove a separation between the ex-post and ex-ante maximin solutions. \n    \\item We give an additive fully polynomial-time approximation scheme (FPTAS) for both the social welfare and ex-post maximin objectives for networks of constant width (but arbitrarily long depth). \\label{list:2}\n    \\item We give an efficient reduction from the ex-ante maximin objective problem to the ex-post maximin objective problem via equilibrium computation in two-player zero-sum games. Combined with our results from \\ref{list:2}, this yields an additive FPTAS for the ex-ante maximin objective problem for constant width networks as well. \n    \\item We define and prove tight bounds on the ``price of fairness'', which compares the optimal social welfare that can be achieved with a given budget to the social welfare of ex-post maximin optimal solutions.\n    \\item Finally, we show that the pipeline intervention problem is NP hard even to approximate in the general case when the width $w$ is not bounded --- and hence that our efficient approximation algorithms cannot be extended to the general case (or even the case of constant depth, polynomial width networks).\n\\end{enumerate}\n\n\n\\subsection{Related Work}\nThere is an enormous literature in ``algorithmic fairness'' that has emerged over the last several years, that we cannot exhaustively summarize here --- but see \\citet{RC18} for a recent survey. Most of this literature is focused on the myopic effects of a single intervention, but what is more conceptually related to our paper is work focusing on the longer-term effects of algorithmic interventions.\n\n\\citet{DL18} and \\citet{pipelines} study the effects of imposing fairness constraints on machine learning algorithms that might be composed together in various ways to reach an eventual outcome. They show that generally fairness constraints imposed on constituent algorithms in a pipeline or other composition do not guarantee that the same fairness constraints will hold on the entire mechanism as a whole. (They also study conditions under which fairness guarantees \\emph{are} well behaved under composition).  Two recent papers~(\\citet{delayed,delayed2}) study parametric models by which classification interventions in an earlier stage can have effects on the data distribution at later stages, and show that for many commonly studied fairness constraints, their effects can either be positive or negative in the long term, depending on the functional form of the relationship between classification decisions and changes in the agent type distribution.\n\n\nThere is also a substantial body of work studying game theoretic models for how interventions affect ``fairness'' goals. This work dates back to \\citet{CL93,FV92} in the economics literature, who propose game theoretic models to rationalize how unequal outcomes might emerge despite two populations being symmetrically situated. More recently, in the computer science literature, several papers consider more complicated models that are similar in spirit to \\citet{CL93,FV92}. \\citet{HC18} propose a  two-stage  model of a labor market with a ``temporary'' (i.e. internship) and ``permanent'' stage, and study the equilibrium effects of imposing a  fairness constraint on the temporary stage. \\citet{fat20} consider a model of the labor market with higher dimensional signals, and study equilibrium effects of subsidy interventions which can lessen the cost of exerting effort. \\citet{downstream} study the effects of admissions policies on a two-stage model of education and employment, in which a downstream employer makes rational decisions. \\citet{endogenous} study a model of criminal justice in which crime rates are responsive to the classifiers used to determine criminal guilt, and study which fairness constraints are consistent with the goal of minimizing crime. \n\n\n\\section{Introduction}\n\n\\input{intro}\n\n\\section{Model}\n\n\\input{model}\n\n\n\\section{Algorithmic Preliminaries}\n\n\\input{prelims}\n\n\\section{Social Welfare Maximization}\\label{sec:SW_max}\n\n\\input{SW_maximization}\n\n\n\\section{(Ex-post) Maximin Value Maximization}\\label{sec:expost_maximin}\n\nAlthough social welfare maximization is a natural objective, it is well-known that it can be ``unfair'' in the sense that it explicitly prioritizes the welfare of larger populations (here represented as initial positions that have larger probability mass) over smaller populations. We can alternately evaluate a solution according to the welfare of the \\emph{least-well-off} population (here represented by the initial position with the smallest expected value) and ask to optimize \\emph{that} objective. We show how to optimize this objective in this section, when one demands a deterministic solution.\n\n\n\\subsection{A Dynamic Programming Algorithm for Computing an Ex-post Maximin Allocation}\n\n\\paragraph{Algorithm and proof:} \n\\input{maxmindp.tex}\n\n\\section{(Ex-ante) Maximin Value Maximization}\\label{sec:exante_maximin}\n\\input{ex-ante_fair}\n\n\\section{Price of Fairness}\n\\input{price_fairness}\n\n\\section{Hardness of Approximation}\n\\input{hardness}\n\n\\bibliographystyle{plainnat}\n\n\\subsection{Running Time and Ex-Post Maximin Value Guarantees}\n\nRemember that we let $OPT_{MM}$ denote the maximin value value of the given network. The running time and accuracy guarantees of Algorithm~\\ref{alg: max_MM} are provided below:\n\n\\begin{theorem}\\label{thm:MMW_approximation}\nAlgorithm~\\ref{alg: max_MM} with discretization parameter $\\varepsilon$ yields maximin value at least $OPT_{MM} - 3(k-1) \\varepsilon \\Vert  R \\Vert_{\\infty}$, and has running time $O\\left(k \\frac{B}{\\varepsilon} \\left(\\frac{1}{\\varepsilon}\\right)^{w^4} g(w) \\right)$, where $g(w)$ is any upper-bound on the running time for solving linear Program~\\ref{MMW_DP}, which is always polynomial in $w$. \n\\end{theorem}\n\nThis immediately induces the following corollary:\n\\begin{corollary}\\label{cor: SW_algo_guarantee_MM}\nAlgorithm~\\ref{alg: max_MM} with discretization parameter $\\varepsilon' = \\frac{\\varepsilon}{3(k-1)}$ yields maximin value at least $OPT_{MM} - \\varepsilon \\Vert R \\Vert_{\\infty}$, and has running time $O\\left(k^2 \\frac{B}{\\varepsilon} \\left(\\frac{k}{\\varepsilon}\\right)^{w^4} g(w)\\right)$, where $g(w)$ is a polynomial upper-bound on the running time of linear Program~\\ref{MMW_DP}.\n\\end{corollary}\n\nThe proof of Theorem~\\ref{thm:MMW_approximation} is almost identical to that of Theorem~\\ref{thm: SW_approximation}. We provide the full proof in Appendix~\\ref{app:maximin_program}.\n\n\n\n\\subsection{Optimization Problems of Interest}\nIn this paper, we will provide algorithms to solve the following three optimization problems. We note at the outset that these optimization problems are non-convex, due to the fact that our objective values are not convex for $k \\geq 2$. Hence we should not expect efficient algorithms in the fully general setting; we will give efficient algorithms for networks of constant width $w$ (i.e. algorithms whose running time is polynomial in the depth of the network $k$), and show that outside of this class, the problem is NP hard even to approximate.\n\n\\paragraph{Social welfare maximization}\nThe first optimization problem we aim to solve is that of maximizing the social welfare of our network, under our budget constraint:\n\\begin{align}\\label{SW_program}\n   \\begin{split}\n                OPT_{SW}=\n                \\max_{M_1, \\ldots, M_{k-1}}&~~R^\\top M_{k-1} \\ldots M_1 D_1^0\n                \\\\\\text{s.t.}&~~ \\left(M_1,\\ldots,M_{k-1}\\right) \\in \\mathcal{F}\\left(B,M_1^0,\\ldots,M_{k-1}^0\\right)\n        \\end{split}\n\\end{align}\n\n\\paragraph{Ex-post maximin problem}\nThe second optimization problem aims to maximize the minimum expected reward that a population can obtain, where the minimum is taken over all initial positions: \n\\begin{align}\\label{maxmin_program}\n\\begin{split}\nOPT_{MM} = \n\\max_{M_1,\\ldots,M_{k-1}}&~~ \\min_{j \\in [w]} R^\\top M_{k-1} \\ldots M_1 e_j\n\\\\\\text{s.t.}&~~ \\left(M_1,\\ldots,M_{k-1}\\right) \\in \\mathcal{F}\\left(B,M_1^0,\\ldots,M_{k-1}^0\\right)\n\\end{split}\n\\end{align}\n\n\\paragraph{Ex-ante maximin problem}\nThe third optimization problem has the same objective as Program~\\ref{maxmin_program}, but allows randomization over sets of transition matrices that satisfy the budget constraint. Note that the budget constraint must be satisfied \\emph{ex-post}, for \\emph{any realization} of the set of transition matrices. To define this optimization problem, we let $\\Delta \\mathcal{F}\\left(B,M_1^0,\\ldots,M_{k-1}^0\\right)$ the set of probability distributions with support $\\mathcal{F}\\left(B,M_1^0,\\ldots,M_{k-1}^0\\right)$. The optimization program is given by:\n\\begin{align}\\label{exante_maxmin_program}\n\\begin{split}\nOPT_{RMM} = \n\\max_{\\Delta M}&~~ \\min_{j \\in [w]} R^\\top \\mathbb{E}_{M \\sim \\Delta M} \\left[M_{k-1} \\ldots M_1\\right] e_j\n\\\\\\text{s.t.}&~~\\Delta M \\in \\Delta\\mathcal{F}\\left(B,M_1^0,\\ldots,M_{k-1}^0\\right),\n\\end{split}\n\\end{align}\nwhere the expectation is taken over the randomness of distribution $\\Delta M$. Note that where Program~\\ref{exante_maxmin_program} can be viewed as optimizing an \\emph{ex-ante} notion of fairness, in which we are evaluated on the minimum expected value of individuals starting at any initial position, \\emph{before the coins of $\\Delta M$ are flipped.} In contrast, Program~\\ref{maxmin_program} evaluates the minimum expected value of individuals starting at any initial position for an already established set of transition matrices. \n\n\\begin{remark}\nPrograms~\\eqref{SW_program},~\\eqref{maxmin_program} and~\\eqref{exante_maxmin_program} all have solutions, and as such the use of maxima instead of suprema is well defined. To see this, first note that the feasible sets are non-empty since $\\left(M_1^0,\\ldots,M_{k-1}^0\\right) \\in \\mathcal{F}\\left(B,M_1^0,\\ldots,M_{k-1}^0\\right)$ for all $B \\geq 0$. For Program~\\eqref{SW_program}, the existence of a maximum is an immediate consequence of the fact that the objective function is continuous in $\\left(M_1,\\ldots,M_{k-1}\\right)$ and $\\mathcal{F}$ and $\\mathcal{M}$ are compact sets. For Program~\\eqref{maxmin_program}, note that no solution can have $R^\\top M_{k-1} \\ldots M_1 e_j \\geq \\Vert R \\Vert_{\\infty}$ for any $j$, as $M_{k-1} \\ldots M_1 e_j$ is a probability distribution. Hence, we can rewrite the program as \n\\begin{align*}\n\\begin{split}\n\\max_{v,M_1,\\ldots,M_{k-1}}&~~v\n\\\\\\text{s.t.}&~~0 \\leq v \\leq \\Vert R \\Vert_{\\infty},\n\\\\&~~R^\\top M_{k-1} \\ldots M_1 e_j \\geq v~\\forall j \\in [w],\n\\\\&~~ \\left(M_1,\\ldots,M_{k-1}\\right) \\in \\mathcal{F}\\left(B,M_1^0,\\ldots,M_{k-1}^0\\right).\n\\end{split}\n\\end{align*}\nThis is an optimization problem with a continuous objective function over a compact set, so it admits a solution. A similar argument follows for Program~\\eqref{exante_maxmin_program}.\n\\end{remark}\n\n\n\n\n\\subsection{Lower Bounds on $P_f^+$}\\label{sec:lower_bounds_pof}\n\nOur lower bounds are based on the following construction:\n\\begin{example}\\label{ex: lb_construction}\nConsider a network with only two layers, $L_1$ and $L_2$, such that $L_1$ has $w$ nodes and $L_2$ has $2$ nodes. Suppose the starting distribution is given by $D_1^0 = \\left(1-(w-1) \\varepsilon,\\varepsilon,\\ldots,\\varepsilon\\right)^\\top$ for $\\varepsilon > 0$ small enough, the reward vector is given by $R = (1,0)^\\top$, and the initial transition matrix $M_1^0$ is given by\n\\begin{align*}\nM_1^0 = \\left(\n\\begin{matrix}\n0 & \\ldots & 0\n\\\\1 & \\ldots & 1\n\\end{matrix}\n\\right)\n.\n\\end{align*}\nI.e., in the initial transition matrix, every starting node transitions to the destination node that has reward $0$, and the welfare of the initial network is $0$. We assume all edges are malleable.\n\\end{example}\n\n\\begin{theorem}\\label{thm: fairness_lb}\nFor all $w \\in \\mathbb{N}$, for any $\\delta > 0$, there exists a network with $k=2$ with price of fairness\n\\begin{align*}\nP_f \\geq \n    \\begin{cases} \n        w -\\delta & \\mbox{if } 0 < B \\leq 2\\\\\n        \\frac{2w}{B} -\\delta &\\mbox{if } 2 < B \\leq 2w\\\\\n        1 &\\mbox{if } B \\geq 2w\n    \\end{cases} \n.\n\\end{align*}\n\\end{theorem}\n\nThe proof follows from solving the social welfare maximization problem and the maximin value problem on Example~\\ref{ex: lb_construction}. The full proof is provided in Appendix~\\ref{app:fairness_lb}.\n\n\n\n\\subsection{Upper Bounds on $P_f^-$}\n\nImportantly, in this section, we restrict ourselves to pipelines such that \\emph{all} edges are malleable. In this case, we show upper bounds that tightly match the lower bounds of Section~\\ref{sec:lower_bounds_pof}.\n\nOur upper bounds will make use of the following claim, which bounds the maximum social welfare that can be achieved under budget $B$. \n\n\\begin{lemma}\\label{lem: ub_OPT}\n\\[\nOPT_{SW} \\leq \\left\\Vert R \\right\\Vert_{\\infty}\n\\]\nand\n\\[\nOPT_{SW} \\leq  W^0 + \\frac{B}{2} \\left\\Vert R \\right\\Vert_{\\infty},\n\\]\nwhere $W^0 = R^\\top M_{k-1}^0 \\ldots M_1^0 D_1^0$ is the initial welfare.\n\\end{lemma}\n\nThe proof of this lemma is straightforward and is deferred to Appendix~\\ref{app:ub_OPT}. \n\nWe will also need lower bounds on the social welfare achieved by any optimal solution to the maximin program. The first lower bound is a function of $B$ and $w$, but is independent of $W^0$.\n\\begin{lemma}\\label{lem: lb_maxmin}\nWhen all edges are malleable, for any $\\left(M_1^f,\\ldots,M_{k-1}^f\\right) \\in S^f$,\n\\[\nW\\left(M_1^f,\\ldots,M_{k-1}^f\\right) \\geq \\min\\left(1,\\frac{B}{2w}\\right) \\left\\Vert R \\right\\Vert_{\\infty}.\n\\]\n\\end{lemma}\n\nThe proof of Lemma~\\ref{lem: lb_maxmin} is deferred to Appendix \\ref{app:lb_maxmin}. We provide a brief proof sketch below:\n\\begin{proof}[Proof sketch]\nThe budget $B$ can be fully invested in improving edges from the second-to-last layer $L_{k-1}$ to the last layer $L_k$. The idea is to increase the transition from any node $u \\in L_{k-1}$ to the best node $v \\in L_k$ with reward $\\Vert R \\Vert_{\\infty}$, by an amount of $B\/2w$ each. Doing so guarantees the result, noting that every starting node in the first layer transitions to a node in $L_{k-1}$ with probability $1$, then to reward $\\Vert R \\Vert_{\\infty}$ on the last layer $L_k$ with probability at least $B\/2w$. Importantly, note that this proof relies on the fact that the edges from the second-to-last to the last layer are malleable.\n\\end{proof}\n\n\nThe second lower bound we need shows that the social welfare achieved by a solution to Program~\\eqref{maxmin_program} is lower-bounded by the initial social welfare $W^0 = R^\\top M_{k-1}^0 \\ldots M_1^0 D_1^0$.\n\\begin{lemma}\\label{lem: lb_maxmin_w0}\nWhen all edges are malleable, for any $\\left(M_1^f, \\ldots, M_{k-1}^f\\right) \\in S^f$, \n\\[\nW\\left(M_1^f, \\ldots, M_{k-1}^f\\right) \\geq W^0.\n\\]\n\\end{lemma}\n\n We defer the full proof of Lemma~\\ref{lem: lb_maxmin_w0} to Appendix~\\ref{app:lb_maxmin_w0} and provide a proof sketch below:\n \n\\begin{proof}[Proof sketch]\n The proof follows from the fact that increasing the expected reward of any given node $u$ in any layer $L_t$ in the network can be done by taking some of the transition probability from $u$ to a low-reward node and re-allocating it to the transition between $u$ and a higher reward node. Doing so does not decrease the expected reward of any other vertex in the network. In turn, it is always sub-optimal to invest budget into decreasing the expected reward of any node in the network.\n\\end{proof}\n\nWe can now use Lemmas~\\ref{lem: ub_OPT},~\\ref{lem: lb_maxmin} and~\\ref{lem: lb_maxmin_w0} to derive nearly tight upper bounds on the price of fairness with respect to the worst maximin solution:\n\\begin{theorem}\nFor every instance of the problem in which edges are malleable, we have that\n\\begin{align*}\nP_f^- \\leq \n    \\begin{cases} \n        w + 1 & \\mbox{if } 0 < B \\leq 2\\\\\n        \\frac{2w}{B} &\\mbox{if } 2 < B \\leq 2w\\\\\n        1 &\\mbox{if } B \\geq 2w\n    \\end{cases} \n.\n\\end{align*}\n\\end{theorem}\n\n\\begin{proof}\nWe divide the proof in three cases:\n\\begin{enumerate}\n\\item $B \\geq 2w$. By Lemma~\\ref{lem: lb_maxmin}, it must be the case that any optimal solution to Program~\\eqref{maxmin_program} has welfare at least $\\min\\left(1,\\frac{B}{2w}\\right) \\left\\Vert R \\right\\Vert_{\\infty} = \\left\\Vert R \\right\\Vert_{\\infty}$. It is then immediately the case that $OPT_{SW} = \\left\\Vert R \\right\\Vert_{\\infty}$ by Lemma~\\ref{lem: ub_OPT} and $P_f = 1$.\n\\item $2 < B \\leq 2w$. By Lemma~\\ref{lem: ub_OPT}, we have $OPT_{SW} \\leq \\left\\Vert R \\right\\Vert_{\\infty}$. Further, by Lemma~\\ref{lem: lb_maxmin}, we have that any solution to Program~\\eqref{maxmin_program} has welfare at least $\\frac{B}{2w} \\left\\Vert R \\right\\Vert_{\\infty}$. This immediately yields the result.\n\\item $0 < B \\leq 2$. By Lemma~\\ref{lem: ub_OPT}, we have $OPT_{SW} \\leq W^0 + \\frac{B}{2} \\left\\Vert R \\right\\Vert_{\\infty}$. By Lemmas~\\ref{lem: lb_maxmin} and~\\ref{lem: lb_maxmin_w0}, we have that the social welfare of any maximin solution is at least $W^0$ and at least $\\frac{B}{2w} \\left\\Vert R \\right\\Vert_{\\infty}$. Therefore, the price of fairness is upper-bounded on the one hand by \n\\[\nP_f^- \\leq \\frac{W^0 + \\frac{B}{2} \\left\\Vert R \\right\\Vert_{\\infty}}{W^0} = 1 + \\frac{\\frac{B}{2} \\left\\Vert R \\right\\Vert_{\\infty}}{W^0}.\n\\]\nand on the other hand by \n\\[\nP_f^- \\leq \\frac{W^0 + \\frac{B}{2} \\left\\Vert R \\right\\Vert_{\\infty}}{\\frac{B}{2w} \\left\\Vert R \\right\\Vert_{\\infty}} = w + \\frac{W^0}{\\frac{B}{2w} \\left\\Vert R \\right\\Vert_{\\infty}}.\n\\]\nWhen $W^0 \\geq \\frac{B}{2w} \\left\\Vert R \\right\\Vert_{\\infty}$, the first bound gives\n\\[\nP_f^- \\leq 1 + \\frac{\\frac{B}{2} \\left\\Vert R \\right\\Vert_{\\infty}}{\\frac{B}{2w} \\left\\Vert R \\right\\Vert_{\\infty}} = w + 1,\n\\]\nand when $W^0 \\leq \\frac{B}{2w} \\left\\Vert R \\right\\Vert_{\\infty}$, the second bound yields\n\\[\nP_f^- \\leq w + \\frac{\\frac{B}{2w} \\left\\Vert R \\right\\Vert_{\\infty}}{\\frac{B}{2w} \\left\\Vert R \\right\\Vert_{\\infty}} = w + 1,\n\\]\nwhich concludes the proof.\n\\end{enumerate}\n\\end{proof}\n\\subsection{Proof of Lemma~\\ref{lemma:sumwelfarebound}}\\label{app:sumwelfarebound}\nWe prove the lemma by proving a similar result at each layer by induction, starting backward from the penultimate layer. \n\n\\begin{lemma}\n\\label{lemma:generalsumwelfarebound}\nFor any layer $L_i$ with $i \\in \\{1,2,3\\}$, we have \n\\begin{align*}w^i_u + w^i_v &\\le \\left( \\frac{1}{2^{4-i}} + \\left(\\frac{1}{2}+\\frac{B_{\\ge i}}{2(4-i)}\\right)^{4-i} \\right) \n\\\\ \\max\\{w^i_u,w^i_v\\} &\\le \\left(\\frac{1}{2}+\\frac{B_{\\ge i}}{2(4-i)}\\right)^{4-i}\n\\end{align*} \nThe first inequality is tight at layer $L_i$ only when $B_{\\ge i}$ is split equally across the transition between layers to the right of $L_i$\n\\end{lemma}\n\nWe note that the above lemma applied at layer $1$ directly gives that $w_u^1 + w_v^1 \\geq 2U$ only when $B_1 = B_2 = B_3 = \\frac{B}{3}$ when equality holds, and all vertices in $\\mathcal{I}_2$ have $0$ reward. The second part of Lemma~\\ref{lemma:sumwelfarebound} holds from applying Lemma~\\ref{lemma:generalsumwelfarebound} at layer $2$ with $B_2 = B_3 = \\frac{B}{3}$ or equivalently, $B_{\\geq 2} = \\frac{2B}{3}$.\n\n\\begin{proof}\nNote that this lemma is trivially true for any layer $L_i$ when $B_{\\ge i} = 0$. Henceforth, we only look at layer $L_i$ when $B_{\\ge i} > 0$.\n\nWe begin by proving the lemma statement for Layer $L_3$ and work backwards towards layer $L_1$. Note that $w^3_u + w^3_v$ is maximized only by spending $B_{\\ge 3} = B_3$ on edges from either $u_3$ or $v_3$ to the reward node $u_4$ with reward $1$. This gives us $w^3_u+w^3_v \\le \\frac{1}{2}+\\frac{1}{2}+\\frac{B_{\\ge 3}}{2}$. Without loss of generality (due to the symmetry in the instance), let $w^3_u \\ge w^3_v$. Then $w^3_u$ is maximized by spending all the budget $B_{\\ge 3}$ on edge $(u_3,u_4)$, i.e., path $P_1$ (in the other case, all the budget is spent on path $P_2$). Thus, $\\max\\{w^3_u,w^3_v\\} \\le \\frac{1}{2}+\\frac{B_{\\ge 3}}{2}$. Thus, both parts of the lemma are true for Layer $L_3$.\n\nNow, let us assume our induction hypothesis holds at layer $L_{i+1}$. Consider layer $L_i$. Let us assume w.l.o.g that $w^{i+1}_u \\ge w^{i+1}_v$. To maximize $w^i_u+w^i_v$, it is easy to see that all budget must be spent on edges from $u_i$ or $v_i$ to $u_{i+1}$ --- since $x$, $y$, and $z$ always have reward $0$. Thus, we get:\n\\begin{align}\\label{eq:bound_induction}\n\\begin{split}\n    w^i_u + w^i_v\n    &\\le \\left(\\frac{1}{2}+\\frac{B_i}{2}\\right) w^{i+1}_u + \\frac{1}{2} w^{i+1}_v \\\\\n    &\\le \\left(\\frac{1}{2}+\\frac{B_i}{2}\\right) w^{i+1}_u \n    + \\frac{1}{2} \n    \\left(\\frac{1}{2^{4-i-1}} + \\left(\\frac{1}{2} + \\frac{B_{\\geq i+1}}{2(4-i-1)}\\right)^{4-i-1} - w_u^{i+1}\\right)  \\\\\n    &= \\frac{B_i}{2}w_u^{i+1} + \\frac{1}{2^{4-i}}  \n    + \\frac{1}{2} \\left(\\frac{1}{2} + \\frac{B_{\\geq i+1}}{2(4-i-1)}\\right)^{4-i-1}\\\\\n    &\\leq \\frac{1}{2^{4-i}}   \n    +\\frac{B_i}{2} \\left(\\frac{1}{2} + \\frac{B_{\\geq i+1}}{2(4-i-1)}\\right)^{4-i-1}\n    + \\frac{1}{2} \\left(\\frac{1}{2} + \\frac{B_{\\geq i+1}}{2(4-i-1)}\\right)^{4-i-1}\\\\\n    &= \\frac{1}{2^{4-i}}   \n    +\\left(\\frac{1}{2} + \\frac{B_i}{2}\\right) \\left(\\frac{1}{2} + \\frac{B_{\\geq i + 1}}{2(4-i-1)}\\right)^{4-i-1},\n    \\end{split}\n\\end{align}\nwhere the second and second-to-last inequalities follow from our induction hypothesis. When $i = 2$, the above bound becomes \n\\[\n\\frac{1}{4} + \\left(\\frac{1}{2} + \\frac{B_2}{2}\\right) \\left(\\frac{1}{2} + \\frac{B_3}{2}\\right),\n\\]\nwhich is uniquely maximized (given total budget $B_{\\geq 2}$ for layers more than $2$) if and only if $B_2 = B_3 = B_{\\geq 3}$. When $i = 1$, this bound becomes \n\\[\n\\frac{1}{8} + \\left(\\frac{1}{2} + \\frac{B_1}{2}\\right) \\left(\\frac{1}{2} + \\frac{B_2 + B_3}{4}\\right)^2,\n\\]\nwhich is similarly uniquely maximized (when $B = B_1 + B_2 + B_3$) if and only if $B_1 = \\frac{B_2 + B_3}{2}$, i.e. only if $B_1 = B\/3$, $B_2 + B_3 = 2B\/3$. In both cases, the unique maximizer satisfies $B_i = \\frac{B_{\\geq i}}{4-i}$ and $\\frac{B_{\\geq i+1}}{4-i-1} = B_i$. Therefore,\n\\begin{align*}\n    w^i_u + w^i_v\n    &\\leq \\frac{1}{2^{4-i}}   \n    +\\left(\\frac{1}{2} + \\frac{B_{\\geq i}}{2(4-i)}\\right) \\left(\\frac{1}{2} + \\frac{B_{\\geq i}}{2(4-i)}\\right)^{4-i-1}\n    \\\\&= \\frac{1}{2^{4-i}}   \n    +\\left(\\frac{1}{2} + \\frac{B_{\\geq i}}{2(4-i)}\\right)^{4-i},\n\\end{align*}\nand this equality can only be tight when i) $B_i = \\frac{B_{\\geq i}}{4-i}$ (by the unique maximizer argument above) and ii) the second inequality in Equation~\\eqref{eq:bound_induction} is tight, which means \n\\[\nB_{\\geq i+1} = B_{\\geq i} - B_i = \\frac{4-i-1}{4-i} B_{\\geq i}\n\\]\nis split equally across the $4-i-1$ transitions between layers to the right of $L_{i+1}$, by induction hypothesis. This immediately implies that $B_i = \\ldots = B_3 = \\frac{B_{\\geq i}}{4-i}$ when the inequality is tight.\n\nWe conclude our proof by showing an upper bound on $w_u^i,w_v^i$. By induction hypothesis, \n\\[\nw_u^{i+1},w_v^{i+1} \\leq \\left(\\frac{1}{2}+\\frac{B_{\\ge i+1}}{2(4-i-1)}\\right)^{4-i-1}.\n\\]\nIt immediately implies that \n\\[\nw_u^{i+},w_v^{i} \\leq \\left(\\frac{1}{2} + \\frac{B_{i}}{2}\\right) \\left(\\frac{1}{2}+\\frac{B_{\\ge i+1}}{2(4-i-1)}\\right)^{4-i-1},\n\\]\nnoting that the maximum transition probability from either $w_u^i$ or $w_v^i$ to the best of $w_u^{i+1},~w_v^{i+1}$ is at most $\\frac{1}{2} + \\frac{B_i}{2}$ (half the budget must be spent decreasing other edges, and half the budget $B_i$ is spent increasing transitions to the best node in $\\mathcal{I}_1$ on layer $i+1$). By the exact same argument as for the first part of the lemma, this is upper-bounded by $\\left(\\frac{1}{2} + \\frac{B_{\\geq i}}{2(4-i)}\\right)^{4-i}$. Hence the induction hypothesis holds at layer $i$.\n\\end{proof}\n\n\n\\subsection{A Dynamic Programming Algorithm for Social Welfare Maximization}\n\\label{subsection:dpsection}\nIn this section, we describe a dynamic programming algorithm for approximately solving the problem above on long skinny networks. The algorithm will run in polynomial time when the width $w$ of the network is small; its running time is polynomial in the depth $k$ of the network, but exponential in the width $w$. The formal description is given in Algorithm~\\ref{alg: max_SW}. Our algorithm works backwards, starting from the final transition matrix from layer $L_{k-1}$ to $L_{k}$. It builds up the solutions to sub-problems parameterized by \nthree parameters --- a layer $t$, a starting distribution over the vertices in layer $t$, and a budget $B_{\\geq t}$ that can be used at layers $\\geq t$. For each sub-problem, it computes an approximately welfare-optimal solution. Once all of these sub-problems have been solved, the optimal solution to the original problem can be read off from the ``sub-problem'' in which $t = 1$, the starting distribution is the distribution on initial positions, and $B_{\\geq 1} = B$. Here is the informal description of the algorithm: \n\\begin{enumerate}\n\n    \\item For $t$ going backwards from $k-1$ to $1$, the algorithm does the following exploration over budget splits and  probability distributions $D_t, D_{t+1} \\in \\mathcal{D}(\\varepsilon)$ (an $\\varepsilon$-net for the $w$-dimensional simplex in $\\ell_1$ norm) on $L_t$:\n    \\begin{enumerate}\n    \\item The algorithm explores all discretized splits of a budget $B_{\\geq t}$ to be used for layers $t$ to $k-1$ into a budget $B_t$ to expend on layer $t$ and a budget $B_{\\geq t + 1}$ to expend on the remaining layers $t+1$ to $k-1$, as well as all choices of target output probability distribution $D_{t+1} \\in \\mathcal{D}(\\varepsilon)$ on layer $L_{t+1}$ and the starting probability distribution $D_{t} \\in \\mathcal{D}(\\varepsilon)$. Informally, we can think of these ``target'' and ``initial'' probability distributions as guesses for what the distribution on vertices in layer $t+1$ and layer $t$ look like in the optimal solution. Recall that for each $D_{t+1}$ and $B_{\\geq t+1}$, our algorithm has already computed a near-optimal solution for a smaller sub-problem, which we will utilize in the next step. \n    \\item\\label{step2} The algorithm then finds a transition matrix  from $L_t$ to $L_k$ that maximizes welfare when the starting distribution on layer $t$ is $D_t$ and the remaining transition matrices are fixed as in the solution to the corresponding sub-problem. Although the overall welfare-maximization problem is non-convex, this sub-problem can be solved as a linear program (Program~\\ref{SW_DP}) because all transition matrices except for one have been fixed as the solution to our sub-problem. \n    \\item Finally, the algorithm picks and stores the recovered transition matrices from layer $L_t$ to $L_k$ that yield the highest reward, among all the transition matrices recovered from step~\\ref{step2}.  \n    \\end{enumerate}\n\\end{enumerate}\n\nWe remark that while (for notational simplicity) our algorithm is written as if all layers have size exactly $w$, it can easily be extended to the case in which all layers have size \\emph{at most} $w$.\n\n\\begin{algorithm}[ht!]\n\\SetAlgoLined\n\\KwIn{Input distribution $D_1$, reward vector $R$, initial transition matrices $M_1^0,\\ldots,M_{k-1}^0$, budget $B$, discretization parameter $\\varepsilon$.}\n\\KwOut{Transition $M(B^\\varepsilon,D_1^0)$ from $L_1$ to $L_k$.}\n\n\\textbf{Initialization:} Let $B_{\\geq k} = 0$, $M(B_{\\geq k}, D_{k}) = I$, $B^\\varepsilon = \\max \\{x \\in \\mathcal{B}(\\varepsilon):~x \\leq B\\}$.\\\\\n    \\For{layer $t = k-1, \\ldots, 1$}{\n        \\For{all distributions $D_t \\in \\mathcal{D}(\\varepsilon)$ if $t \\neq 1$ ($D_t = D_1^0$ if $t = 1$) and budgets $B_{\\geq t} \\in \\mathcal{B}(\\varepsilon)$ with $B_{\\geq t} \\leq B$}{\n            \\For{all distributions $D_{t+1} \\in \\mathcal{D}(\\varepsilon)$ and budgets $B_{\\geq t+1} \\leq B_{\\geq t}$ such that $B_{\\geq t+1} \\in \\mathcal{B}(\\varepsilon)$}{\n            Solve linear program\n            \n            \\begin{align}\\label{SW_DP}\n                \\begin{split}\n                M_t(B_{\\geq t},B_{\\geq t+1},D_t,D_{t+1}) \n                = \\argmax_{M_t}&~~R^\\top M(B_{\\geq t+1}, D_{t+1}) M_t D_t\n                \\\\\\text{s.t.}&~~ c\\left( M_t, M_t^0 \\right) \\leq B_{\\geq t} - B_{\\geq t+1} \n                \\\\&~~M_t(i,j) = M_t^0(i,j)~\\forall (i,j) \\in \\overline{E_t^{mal}}\n                \\\\&~~M_t \\in \\c\n                \\end{split}\n                \\end{align}\n            }\n            Pick $B_{\\geq t+1}, D_{t+1}$ leading to the highest objective value in Program~\\ref{SW_DP}, and set $M(B_{\\geq t},D_{t}) = M(B_{\\geq t+1}, D_{t+1}) M_t(B_{\\geq t},B_{\\geq t+1},D_t,D_{t+1})$.\\\\ \n        }\n    }\n\\textbf{Return} $M(B^\\varepsilon,D_1)$.\n\\caption{Dynamic Program for (Approximate) Social Welfare Maximization.}\\label{alg: max_SW}\n\\end{algorithm}\n\n\n\n\nWe briefly note why Program~\\eqref{SW_DP} is a linear program. The objective is linear because only the matrix $M_t$ represents variables. Thus we simply need to verify that the constraint on the cost is linear.\n\n\\begin{definition}\nWe say that a transition matrix $M_t \\in \\mathcal{M}$ is feasible with respect to a budget split $B_{\\geq t}, B_{\\geq t+1}$ if and only if \n\\[\nc\\left(M_t,M_t^0 \\right) \\leq B_{\\geq t+1} - B_{\\geq t}. \n\\]\nand $M_t(i,j) = M_t^0(i,j)$ for every non-malleable edge (i,j).\n\\end{definition}\nNote that saying that $M_t$ feasible with respect to $B_{\\geq t}, B_{\\geq t+1}$ is equivalent to saying that $M_t$ is a feasible solution to Program~\\eqref{SW_DP} with parameters $B_{\\geq t},B_{\\geq t+1},D_t,D_{t+1}$ for any $D_t, D_{t+1} \\in \\mathcal{D}(\\varepsilon)$.\nThe constraint $c\\left(M_t,M_t^0 \\right) \\leq B_{\\geq t+1} - B_{\\geq t}$ can be equivalently replaced by $2w^2 +1$ linear constraints. To do so, we introduce $w^2$ variables - $a_1,a_2,\\cdots a_{w^2}$. The constraint can then be rewritten in the form $\\sum_{i=1}^{w^2} |f_i| \\le B_{\\geq t+1} - B_{\\geq t}$, where each $f_i$ is a linear combination of the variables. We can thus express the budget constraint of Program~\\ref{SW_DP} by the following set of linear constraints:\n\\begin{enumerate}\n    \\item $f_i \\le a_i~~\\forall i \\in [w^2]$\n    \\item$-f_i \\le a_i~~\\forall i \\in [w^2]$\n    \\item $\\sum_{i=1}^{w^2} a_i \\le B_{\\geq t+1} - B_{\\geq t}$.\n\\end{enumerate} \nThus, Program~\\ref{SW_DP} can be written as a linear program with the number of constraints and variables being polynomial in $w$. \n\n\n\n\n\\subsection{Running Time and Social Welfare Guarantees}\n\nWe provide the running time and social welfare guarantees of Algorithm~\\ref{thm: SW_approximation} below. \n\n\\begin{theorem}\\label{thm: SW_approximation}\nAlgorithm~\\ref{alg: max_SW} instantiated with discretization parameter $\\varepsilon$ yields a solution achieving social welfare at least $OPT - 3(k-1) \\varepsilon \\Vert  R \\Vert_{\\infty}$, and has running time $O\\left(k \\frac{B}{\\varepsilon} \\left(\\frac{1}{\\varepsilon}\\right)^{w^2} f(w) \\right)$, where $f(w)$ is any upper-bound on the running time for solving linear Program~\\ref{SW_DP}, which is always polynomial in $w$.\n\\end{theorem}\n\n\nThis immediately yields the following corollary:\n\\begin{corollary}\\label{cor: SW_algo_guarantee}\nAlgorithm~\\ref{alg: max_SW} with discretization parameter $\\varepsilon' = \\frac{\\varepsilon}{3(k-1)}$ yields social welfare at least $OPT - \\varepsilon \\Vert R \\Vert_{\\infty}$, and has running time $O\\left(k^2 \\frac{B}{\\varepsilon} \\left(\\frac{k}{\\varepsilon}\\right)^{w^2} f(w) \\right)$, where $f(w)$ is any upper-bound on the running time for solving linear Program~\\ref{SW_DP}, which is always polynomial in $w$.\n\\end{corollary}\n\nWe observe that this running time is polynomial in $k$ (the depth of the network) and $1\/\\varepsilon$ (the inverse additive error tolerance), but exponential in $w$ (the width of the network). Hence our algorithm runs in polynomial time for the class of constant width networks. \n\n\\begin{remark}\nWe note that our additive near-optimality guarantee can be translated into a multiplicative guarantee. In the case where \\emph{all edges are malleable}, this follows from noting that given budget $B$, $OPT \\geq \\frac{B}{2w} \\Vert R \\Vert_{\\infty}$: this can be reached by investing the totality of the budget into transitioning every node in the second-to-last layer to the highest reward node in the last layer, with probability $\\frac{B}{2w}$ for each such node. Taking $\\varepsilon = \\delta \\cdot \\frac{B}{6(k-1) w}$ for some constant $\\delta < 1$ gives a multiplicative approximation to the optimal social welfare with approximation factor $1-\\delta$. \n\nFor the case in which non-malleable edges are allowed, a lower bound on $OPT$ is given by $OPT \\geq W_0$. Taking $\\varepsilon = \\delta \\cdot \\frac{W_0}{3(k-1) \\Vert R \\Vert_{\\infty}}$ yields a multiplicative $1-\\delta$ approximation still.\n\\end{remark}\n\n\n\\paragraph{Proof of Theorem~\\ref{thm: SW_approximation}} The proof of Theorem~\\ref{thm: SW_approximation} relies on the following lemma, and its corollary:\n\\begin{lemma}\nLet $M \\in \\mathbb{R}^{w \\times w}$ be a left stochastic matrix, and let $D, D' \\in \\mathcal{D}$ be probability distributions.\n\\[\n\\Vert M D - M D' \\Vert_1 \\leq \\Vert D - D' \\Vert_1.\n\\]\n\\end{lemma}\n\n\\begin{proof}\nNote that \n\\begin{align*}\n\\Vert M (D-D') \\Vert_1\n= \\sum_{i=1}^w \\left\\vert \\left(M(D-D')\\right)(i) \\right\\vert\n&= \\sum_{i=1}^w \\left\\vert \\sum_{j=1}^w M(i,j) (D(j) - D'(j)) \\right\\vert\n\\\\&\\leq \\sum_{i=1}^w \\sum_{j=1}^w \\left\\vert M(i,j) (D(j) - D'(j)) \\right\\vert\n\\\\&= \\sum_{j=1}^w \\left\\vert D(j) - D'(j) \\right\\vert \\sum_{i=1}^w \\left\\vert M(i,j) \\right\\vert\n\\\\&= \\sum_{j=1}^w \\left\\vert D(j) - D'(j) \\right\\vert\n\\\\&= \\Vert D - D' \\Vert_1,\n\\end{align*}\nwhere the inequality follows from the triangle inequality, and the second-to-last equality from the fact that\n\\[\n\\sum_{i=1}^w \\left\\vert M(i,j) \\right\\vert = \\sum_{i=1}^w  M(i,j) = 1~~\\forall j \\in [w]\n\\]\nas $M$ is a left stochastic matrix.\n\\end{proof}\n\n\\begin{corollary}\\label{cor:approx_loss}\nLet $R \\in \\mathbb{R}^w$ be a real vector and $D,D' \\in \\mathcal{D}$ be probability distributions such that $\\Vert D - D' \\Vert_1 \\leq \\varepsilon$, and $M \\in \\mathbb{R}^{w \\times w}$ a left stochastic matrix. Then\n\\[\nR^\\top M D \\geq R^\\top M D' - \\Vert R \\Vert_{\\infty}  \\cdot \\varepsilon.\n\\]\n\\end{corollary}\n\n\\begin{proof}[Proof of Corollary~\\ref{cor:approx_loss}]\n$\n\\Vert R^\\top M (D'-D)\\Vert_1 \n\\leq \\Vert R \\Vert_{\\infty} \\Vert M (D'-D)\\Vert_1 \n\\leq \\Vert R \\Vert_{\\infty} \\Vert D'-D\\Vert_1 \n\\leq \\Vert R \\Vert_{\\infty} \\cdot  \\varepsilon\n$\n, where the first step follows from Holder's inequality.\n\\end{proof}\n\n\nWe are now ready to prove Theorem~\\ref{thm: SW_approximation}:\n\n\n\\begin{proof}[Proof of Theorem~\\ref{thm: SW_approximation}]\nLet us denote by $B_1^\\varepsilon,\\ldots, B_{k-1}^\\varepsilon$  a split of the budget for the discretized problem with $B^\\varepsilon_{\\geq t} = B_t^\\varepsilon + \\ldots + B_{k-1}^\\varepsilon$. Let $M_1^\\varepsilon, \\ldots, M_{k-1}^\\varepsilon$ a set of transition matrices achieving welfare $R^\\top M_{k-1}^\\varepsilon \\ldots M_1^\\varepsilon D_1^0 \\geq OPT^\\varepsilon \\triangleq OPT - (k-1) \\varepsilon \\left\\Vert R \\right\\Vert_{\\infty}$ that is feasible with respect to budget split $B_1^\\varepsilon,\\ldots, B_{k-1}^\\varepsilon$. Note that such a budget split and matrices exist by Claim~\\ref{clm: budget_loss}. Let $D_t^\\varepsilon$ the probability distribution on layer $t$ defined by these transition matrices, i.e.\n\\[\nD_t^\\varepsilon = M^\\varepsilon_{t-1} \\ldots M^\\varepsilon_1 D_1^0.\n\\]\nTo prove the result, we will show by induction that for all $B_{\\geq t} \\geq B^\\varepsilon_{\\geq t}$, and for $D_t \\in \\mathcal{D}(\\varepsilon)$ such that $\\Vert D_t - D_t^\\varepsilon \\Vert_1 \\leq \\varepsilon$,\n\\[\nR^\\top M(B_{\\geq t},D_t) D_t \\geq OPT^\\varepsilon - 2(k - t) \\varepsilon \\Vert  R \\Vert_{\\infty}.\n\\]\nThis will directly imply that as $B^\\varepsilon$ is one of the possible values of $B_{\\geq 1}$,\n\\[\nR^\\top M(B^\\varepsilon,D_1) D_1 \\geq OPT^\\varepsilon - 2(k-1) \\varepsilon \\left\\Vert R \\right\\Vert_{\\infty}.\n\\]\nCombined with Claim~\\ref{clm: budget_loss} that states $OPT_\\varepsilon \\geq OPT - (k-1) \\varepsilon \\left\\Vert R \\right\\Vert_{\\infty}$, we will obtain the result. \n\nLet us now provide our inductive proof. First, consider the transition from layer $L_{k-1}$ to layer $L_k$. Note that \n\\[\nOPT^\\varepsilon \\le  R^\\top M_{k-1}^\\varepsilon\\ldots M_1^\\varepsilon D_1^0 = R^\\top M_{k-1}^\\varepsilon D_{k-1}^\\varepsilon.\n\\]\nLet $D_{k-1} \\in \\mathcal{D}(\\varepsilon)$ be such that $\\Vert D_{k-1} - D^\\varepsilon_{k-1} \\Vert \\leq \\varepsilon$. Note then that by Corollary~\\ref{cor:approx_loss}, \n\\[\nR^\\top M_{k-1}^\\varepsilon D_{k-1} \\geq R^\\top M_{k-1}^\\varepsilon D^\\varepsilon_{k-1} - \\varepsilon \\Vert  R \\Vert_{\\infty}.\n\\]\nFurther, $M_{k-1}^\\varepsilon$ is feasible for Program~\\eqref{SW_DP} with respect to $B_{\\geq k-1}, B_{\\geq k} = 0$, given $B_{\\geq k-1} \\geq B_{\\geq k-1}^\\varepsilon$. As such, for $B_{\\geq k-1} \\geq B_{\\geq k-1}^\\varepsilon$, we have that\n\\[\nR^\\top M(B_{\\geq k-1},D_{k-1}) D_{k-1}  \\geq R^\\top M_{k-1}^\\varepsilon D_{k-1},\n\\]\nand in turn\n\\[\nR^\\top M(B_{\\geq k-1},D_{k-1}) D_{k-1} \n\\geq OPT^\\varepsilon - \\varepsilon \\Vert  R \\Vert_{\\infty}.\n\\]\n\nNow, suppose the induction hypothesis holds at layer $t+1$. I.e., for all $B_{\\geq t+1} \\geq B^\\varepsilon_{\\geq t+1}$, for $D_{t+1} \\in \\mathcal{D}(\\varepsilon)$ such that $\\Vert D_{t+1} - D_{t+1}^\\varepsilon \\Vert_1 \\leq \\varepsilon$,\n\\[\nR^\\top M(B_{\\geq t+1},D_{t+1}) D_{t+1} \\geq OPT^\\varepsilon - 2(k - t - 1) \\varepsilon \\Vert  R \\Vert_{\\infty}.\n\\]\nFor any $B_{\\geq t} \\geq B_{\\geq t}^\\varepsilon$, note that one can set $B_{\\geq t+1} = B_{\\geq t+1}^\\varepsilon$ and $B_t \\geq B_t^\\varepsilon$; hence, $M_t^\\varepsilon$ is feasible for Program~\\eqref{SW_DP} with respect to $B_t \\geq B_t^\\varepsilon,B^\\varepsilon_{\\geq t+1}$. Since $\\Vert D_t -D_t^\\varepsilon \\Vert_1 \\leq \\varepsilon$ and $\\Vert D_{t+1} - M_t^\\varepsilon D_t^\\varepsilon \\Vert_1 \\leq \\varepsilon$, we have that by Corollary~\\ref{cor:approx_loss},\n\\[\nR^\\top M(B^\\varepsilon_{\\geq t+1}, D_{t+1}) M^\\varepsilon_t D_t \n\\geq R^\\top M(B^\\varepsilon_{\\geq t+1}, D_{t+1}) M^\\varepsilon_t D_t^\\varepsilon - \\varepsilon \\Vert  R \\Vert_{\\infty}\n\\geq R^\\top M(B^\\varepsilon_{\\geq t+1}, D_{t+1}) D_{t+1} - 2 \\varepsilon \\Vert  R \\Vert_{\\infty}.\n\\]\nUsing the induction hypothesis, we obtain that\n\\[\nR^\\top M(B^\\varepsilon_{\\geq t+1}, D_{t+1}) M^\\varepsilon_t D_t \\geq OPT^\\varepsilon - 2(k - t) \\varepsilon \\Vert  R \\Vert_{\\infty}.\n\\]\nIn particular,\n\\[\nR^\\top M(B_{\\geq t}, D_{t}) D_t \\geq OPT^\\varepsilon - 2(k - t) \\varepsilon \\Vert  R \\Vert_{\\infty},\n\\]\nwhich concludes the proof of the social welfare guarantee.\nFor the running time, we note that at each time step $t$, we solve one instance of Program~\\ref{SW_DP} for each of the (at most) $\\frac{B}{\\varepsilon}$ possible budget splits of $B_{\\geq t}$ and for each of the $\\left(\\frac{1}{\\varepsilon}\\right)^w$ (by Claim~\\ref{clm:eps_net}) probability distributions in $\\mathcal{D}(\\varepsilon)$ in layer $L_t$ and layer $L_{t+1}$; i.e., for each $t$, the algorithm solves $O \\left(\\frac{B}{\\varepsilon} \\left(\\frac{1}{\\varepsilon}\\right)^{w^2}\\right)$ optimization programs. Then, the algorithm finds the solution of all of these programs with the best objective value, which can be done in time linear in the number of such solutions, i.e. $O \\left(\\frac{B}{\\varepsilon}\\left(\\frac{1}{\\varepsilon}\\right)^{w^2}\\right)$. This is repeated for $k-1$ values of $t$.\n\\end{proof}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}