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+{"text":"\\section{Introduction}\n\nThe phenomenon of persistent current in mesoscopic normal metal rings has \ngenerated a lot of excitement as well as controversy over the past years. \nIn a pioneering work, B\\\"{u}ttiker, Imry and Landauer~\\cite{butt} predicted \nthat, even in the presence of disorder, an isolated one-dimensional metallic \nring threaded by a magnetic flux $\\phi$ can support an equilibrium persistent \ncurrent with periodicity $\\phi_0=ch\/e$, the elementary flux quantum. Later, \nthe existence of persistent current was further confirmed by several \nexperiments~\\cite{chand,maily,levy,deb,reul,jari,rab}. However, these \nexperiments yield many results those are not well-understood theoretically \neven today~\\cite{cheu1,cheu2,mont,bouc,alts,von,schm,ambe,abra,bouz,giam,\nwain,avishai,weiden,orella1,kulik,mos}. The results of the single loop \nexperiments are significantly different from those for the ensemble of \nisolated loops. Persistent currents with expected $\\phi_0$ periodicity \nhave been observed in isolated single Au rings~\\cite{chand} and in a \nGaAs-AlGaAs ring~\\cite{maily}. Levy {\\em et al.}~\\cite{levy} found \noscillations with period $\\phi_0\/2$ rather than $\\phi_0$ in an ensemble \nof $10^7$ independent Cu rings. Similar $\\phi_0\/2$ oscillations were \nalso reported for an ensemble of disconnected $10^5$ Ag rings~\\cite{deb} \nas well as for an array of $10^5$ isolated GaAs-AlGaAs rings~\\cite{reul}. \nIn an experiment, Jariwala {\\em et al.}~\\cite{jari} obtained both \n$\\phi_0$ and $\\phi_0\/2$ periodic persistent currents for an array of thirty \ndiffusive mesoscopic Au rings. Except for the case of nearly ballistic \nGaAs-AlGaAs ring~\\cite{maily}, all the measured currents are in general \none or two orders of magnitude larger than those expected from theory. \nThe diamagnetic response of the measured $\\phi_0\/2$ oscillations of \nensemble-averaged persistent currents near zero magnetic field also \ncontrasts with most predictions~\\cite{schm,ambe}. \n\nFree electron theory predicts that, at zero temperature, an ordered \none-dimensional metallic ring threaded by a magnetic flux $\\phi$ supports \npersistent current with maximum amplitude $I_0=ev_F\/L$, where $v_F$ is \nthe Fermi velocity of an electron and $L$ is the circumference of the \nring. Metals are intrinsically disordered which tends to decrease persistent \ncurrent, and calculations show that the magnitude of the currents reduces \nto $I_0l\/L$, where $l$ is the elastic mean free path of the electrons. \nThis expression remains valid even if one takes into account the finite \nwidth of the ring by adding contributions from the transverse channels, \nsince disorder leads to a compensation between the channels~\\cite{cheu2,mont}. \nHowever, measurements on single isolated mesoscopic rings~\\cite{chand,maily} \ndetected $\\phi_0$-periodic persistent currents with amplitudes of the order \nof $I_0\\sim ev_F\/L$, (close to the value for an ordered ring). Though theory \nseems to agree with experiment~\\cite{maily} only when disorder is weak, the \namplitudes of the currents in single-isolated-diffusive gold \nrings~\\cite{chand} are two orders of magnitude larger than the theoretical \nestimates. This discrepancy initiated intense theoretical activity, and it \nis generally believed that the electron-electron correlation plays an \nimportant role in the disordered rings~\\cite{abra,bouz,giam}, though the \nphysical origin behind this enhancement of persistent current is still \nunclear. \n\nIn this article we investigate a detailed study of persistent current and \nlow-field magnetic susceptibility in single channel rings in the \ntight-binding framework considering long-range hopping of electrons in the\n{\\em shortest} path. Our calculations show that in a disordered ring \nwith higher order hopping integrals, current amplitude is comparable to \nthat of an ordered ring. This is due to the fact that higher order hopping\nintegrals try to delocalize the energy eigenstates and thus compensate\nthe effect of disorder. In the rest part of this article, we describe the\ndependences of the sign of low-field currents as a function of the total\nnumber of electrons $N_e$, and also discuss the effect of temperature \non these low-field currents.\n\nThe plan of the paper is as follow. Section $2$ relates the behavior of\npersistent current in the presence of long-range hopping integrals and\nclearly describes how current amplitude in disordered rings becomes \ncomparable to that of an ordered ring. In Section $3$, we investigate \nthe behavior of low-field magnetic susceptibility at absolute zero \ntemperature both for ordered and disordered rings as a function of $N_e$.\nSection $4$ focuses the effect of temperature on the low-field currents \nand determines the critical value of magnetic flux $\\phi_c(T)$ where \ncurrent changes its sign from the paramagnetic to diamagnetic phase for \nthe rings with even number of electrons. Finally, the conclusions of our \nstudy can be found in Section $5$.\n\n\\section{Persistent current}\n\nWe describe a $N$-site ring (see Fig.~\\ref{ring}) enclosing a magnetic flux \n$\\phi$ (in units of the elementary flux quantum $\\phi_0=ch\/e$) by the \nfollowing tight-binding Hamiltonian in the Wannier basis,\n\\begin{equation}\nH=\\sum_i\\epsilon_i c_{i}^{\\dagger}c_{i} +\\sum_{i\\ne j} v_{ij} \n\\left[e^{-i \\theta} c_{i}^{\\dagger} c_{j}+ h.c. \\right]\n\\label{hamil}\n\\end{equation} \nwhere $\\epsilon_i$'s are the site energies, $v_{ij}$'s are the hopping \nintegrals between the sites $i$ and $j$, and $\\theta={2\\pi\\phi}(|i-j|)\/N$. \nThe long-range hopping (LRH) integrals are taken as, \n\\begin{equation}\nv_{ij}=\\frac{v}{\\left|\\frac{N}{\\pi} \\sin\\left[\\frac{\\pi}{N}(i-j)\\right]\n\\right|^{\\alpha}} \n\\end{equation}\nwhere, $v$ being a constant representing the nearest-neighbor hopping (NNH) \nintegral. In the present work, electron-electron interaction is not included, \nand therefore, we do not consider the spin of the electrons since it will \nnot change any qualitative behavior of persistent currents. Throughout \nour study we set $v=-1$, and use the units where $c=e=h=1$.\n\nAn electron in an eigenstate with energy $E_n$ carries a persistent\n\\begin{figure}[ht]\n{\\centering \\resizebox*{5cm}{4cm}{\\includegraphics{ring.eps}}\\par}\n\\caption{One-dimensional mesoscopic normal metal ring threaded by a \nmagnetic flux $\\phi$. A persistent current $I$ is established in the ring.}\n\\label{ring}\n\\end{figure}\ncurrent $I_n(\\phi)=-\\partial E_n\/\\partial \\phi$, and at zero\ntemperature total persistent current is given by $I(\\phi)=\\sum_n I_n(\\phi)$,\nwhere the summation is over all the states below the Fermi level. \n\nFor an ordered ring, setting $\\epsilon_i=0$ for all $i$, the energy of the\n$n$-th eigenstate can be expressed as,\n\\begin{equation}\nE_n(\\phi)=\\sum_m \\frac{2v}{m^{\\alpha}}\\cos\\left[\\frac{2\\pi m}{N}\n(n+\\phi)\\right]\n\\label{engy}\n\\end{equation}\nwhere $m$ is an integer and it runs from $1$ to $N\/2$ for the rings with \neven $N$, while it goes from $1$ to $(N-1)\/2$ for those rings described \nby odd $N$. Now the persistent current carried by this $n$-th\neigenstate becomes,\n\\begin{equation}\nI_n(\\phi)=\\left(\\frac{4\\pi v}{N}\\right)\\sum_m m^{1-\\alpha}\\sin\\left[\n\\frac{2\\pi m}{N}(n+\\phi)\\right]\n\\label{curr}\n\\end{equation}\nFor very large value of $\\alpha$, Eqs.~(\\ref{engy}) and (\\ref{curr})\nessentially reduce to the expressions for the energy spectrum and persistent\ncurrent of an ordered ring described by the nearest-neighbor tight-binding\nHamiltonian. As we decrease $\\alpha$, contributions from the higher order\nhopping integrals become appreciable which modify the energy spectrum and\npersistent current, and we will see that the modifications are quite \nsignificant in the presence of disorder. Figure~\\ref{longorder} shows the\nvariation of persistent current as a function of magnetic flux $\\phi$ for \nsome perfect rings ($N=120$). The solid and dotted curves represent the \nresults for the rings described by LRH ($\\alpha=1.3$) and NNH integrals \nrespectively, where the curves plotted in (a) give the variation of the \ncurrents for the rings with odd $N_e$ \n\\begin{figure}[ht]\n{\\centering \\resizebox*{7.5cm}{10cm}{\\includegraphics{longorder.eps}}\\par}\n\\caption{Current-flux characteristics for some perfect rings with size \n$N=120$. The solid and dotted curves are respectively for the rings with LRH \n($\\alpha=1.3$) and NNH integrals, where (a) $N_e=35$ (odd) and (b) $N_e=40$\n(even).}\n\\label{longorder}\n\\end{figure}\n($N_e=35$) and the same are plotted for the rings with even $N_e$ ($N_e$=40) \nin (b). The results predict that the current amplitude increases in the \npresence of LRH integrals, compared to the NNH integral. In this context \nit has been examined that, in the presence of LRH integrals, the amplitude \nof the current initially increases (not shown here in the figure) as we \nincrease the ring \nsize, but eventually it falls when the ring becomes larger. This is due\nto the fact that as we increase the number of sites, the Hamiltonian\nEq.~(\\ref{hamil}) includes additional higher order hopping integrals which \ncause an increase in the net velocity of the electrons, but after certain\nring size the increment in velocity drops to zero since the additional\nhopping integrals are then between far enough sites giving negligible\ncontributions. \n\nNow we address the problem of persistent current in the presence of disorder.\nIn order to introduce the disorder in the ring, we choose the site energies\n$\\epsilon_i$'s randomly from a ``Box\" distribution function of width $W$,\nwhich reveal that the ring is subjected to the diagonal disorder. As \nrepresentative examples of persistent current in disordered rings, we plot \n\\begin{figure}[ht]\n{\\centering \\resizebox*{7.5cm}{10cm}{\\includegraphics{longdisorder.eps}}\\par}\n\\caption{Current-flux characteristics for some typical disordered rings \n($W=1$) with size $N=120$. The solid and dotted curves correspond to the \nrings with LRH ($\\alpha=1.3$) and NNH integrals respectively, where \n(a) $N_e=35$ (odd) and (b) $N_e=40$ (even).}\n\\label{longdisorder}\n\\end{figure}\nthe results in Fig.~\\ref{longdisorder} for some $120$-site rings taking \n$W=1$. All the curves shown in Fig.~\\ref{longdisorder} are performed for \nthe distinct disordered configurations of the rings and no averaging is done \nhere since in the averaging process several mesoscopic phenomena disappear. \nThe solid and dotted curves correspond to the rings with LRH and NNH \nintegrals respectively, and our results predict that current amplitude gets \nan order of magnitude enhancement in the rings described by the LRH \nintegrals compared to those rings described by the NNH integral. \nFigure~\\ref{longdisorder}(a) shows the variation of the persistent current \nfor the rings with odd $N_e$ ($N_e=35$) and Fig.~\\ref{longdisorder}(b) gives \nthe variation of the currents for the rings with even $N_e$ ($N_e=40$).\nIt is apparent from Figs.~\\ref{longorder} and \\ref{longdisorder} that the \ncurrent amplitudes in disordered rings with LRH integrals are of the same \norder of magnitude as observed in ordered rings. We have also seen that \nthe decrease in amplitude of the current is quite small even if we increase \nthe strength of disorder. In the NNH models current amplitudes are \nsuppressed due to the localization of the energy eigenstates~\\cite{lee1}. \nOn the other hand, the present tight-binding model with LRH integrals \nsupports extended electronic eigenstates even in the presence of disorder \nand for this reason persistent currents are not reduced by the impurities.\n\n\\section{Magnetic susceptibility at $T=0$ K}\n\nThe sign of persistent currents can be determined exactly by calculating\nmagnetic susceptibility, and here we investigate the properties of low-field\n\\begin{figure}[ht]\n{\\centering \\resizebox*{7.5cm}{5.5cm}{\\includegraphics{longsusceporder.eps}}\n\\par}\n\\caption{$\\chi(\\phi)$ versus $N_e$ curves for some perfect rings with \n$N=150$. The solid and dotted curves represent the rings with LRH \n($\\alpha=1.4$) and NNH integrals respectively.}\n\\label{longsusceporder}\n\\end{figure}\ncurrents for the rings with fixed number of electrons $N_e$. The general \nexpression of magnetic susceptibility is expressed in the form,\n\\begin{equation}\n\\chi(\\phi)=\\frac{N^3}{16\\pi^2}\\left(\\frac{\\partial I(\\phi)}{\\partial \\phi}\n\\right)\n\\end{equation}\nThus by calculating the sign of $\\chi(\\phi)$ we can predict whether the \ncurrent is diamagnetic or paramagnetic.\n\nLet us first discuss the properties of low-field currents in perfect rings\nat absolute zero temperature ($T=0$ K). In Fig.~\\ref{longsusceporder}, we \nplot $\\chi(\\phi)$ as a function of $N_e$ in the limit $\\phi\\rightarrow 0$ \nfor some perfect rings with $N=150$. The solid and dotted curves represent \nthe variation of $\\chi$ for the rings described by LRH and NNH integrals\nrespectively. The results show that, for perfect rings, low-field currents \nexhibit only the diamagnetic sign irrespective of the total number of \nelectrons $N_e$ in the ring. This variation can be clearly understood \nif we consider the slope of the curves as plotted in \nFigs.~\\ref{longorder}(a) and (b) \n\\begin{figure}[ht]\n{\\centering \\resizebox*{7.5cm}{10cm}\n{\\includegraphics{longsuscepdisorder.eps}}\\par}\n\\caption{$\\chi(\\phi)$ versus $N_e$ curves of some disordered rings ($W=1$) \nwith size $N=150$. The solid and dotted lines are respectively for the \nrings with even and odd $N_e$, where (a) NNH and (b) LRH ($\\alpha=1.4$) \nintegrals.} \n\\label{longsuscepdisorder}\n\\end{figure}\nnear $\\phi=0$. Thus, both for the perfect rings with odd and even $N_e$, \ncurrent has only negative slope which predicts the diamagnetic persistent \ncurrent.\n\nThe effect of impurities on the sign of low-field currents is quite \ninteresting. As representative examples, in Fig.~\\ref{longsuscepdisorder}\nwe display the variation of $\\chi$ as a function of $N_e$, where\n(a) and (b) represent the rings with NNH and LNH integrals respectively. The \nsolid and dotted lines correspond to the results for the rings containing \neven and odd number of electrons respectively. These results emphasize that, \nin the disordered rings the low-field currents exhibit the diamagnetic sign \nfor odd $N_e$, while we get the paramagnetic response for the rings with \neven $N_e$. The diamagnetic and the paramagnetic natures of the low-field \ncurrents in the presence of impurity in the rings can be understood easily \nif we take the slope of the curves as given in Figs.~\\ref{longdisorder}(a) and\n(b). Such an effect of disorder on the low-field currents is true for any \ndisordered configuration. Accordingly, in the presence of impurity one can \neasily predict the sign of the low-field currents both for the rings with \nodd and even $N_e$, irrespective of the specific realization of disordered \nconfiguration of the rings. \n\n\\section{Magnetic susceptibility at finite temperature}\n\nThis section focuses the effect of temperature on the low-field currents. \n\\begin{figure}[ht]\n{\\centering \\resizebox*{7.5cm}{10cm}{\\includegraphics{longtemp.eps}}\\par}\n\\caption{Variation of $\\phi_c(T)$ with $N_e$ (even $N_e$ only) for some\ndisordered rings ($W=1$) taking $N=60$, where (a) rings with NNH integrals \nand (b) rings with LRH ($\\alpha=1.6$) integrals. The upper and lower curves \nboth in (a) and (b) are respectively for the rings with $T\/T^{\\star}=1.0$ \nand $T\/T^{\\star}=0.5$.}\n\\label{longtemp}\n\\end{figure}\nAs temperature increases the probability that electrons occupy higher \nenergy levels, those may carry larger currents, increases. But if we increase \nthe temperature in such a way that crosses the energy gap between two \nsuccessive energy levels, which carry currents in opposite directions,\nthen mutual cancellations of the positive and negative currents decrease the\nnet current amplitude. Therefore, it is necessary to specify a characteristic \ntemperature $T^{\\star}$, which is determined by the energy level spacing \n$\\Delta$. At finite temperature, thermal excitations, such as phonons, \nare present which interact with the electrons inelastically and thus \nrandomizes the phase of the electronic wave functions. These interactions \ntry to destroy the phase coherence of the electrons which can remove the \nquantum effects. Hence, it is necessary to do the calculations at \nsufficiently low temperatures such that the phase coherence length of an \nelectron exceeds the circumference $L$ of the ring.\n\nIn our calculations of low-field magnetic susceptibility at absolute zero \ntemperature ($T=0$ K), we see that the current has only diamagnetic sign \nfor perfect rings irrespective of the total number of electrons, while in \nthe presence of impurity, it exhibits respectively the diamagnetic and \nparamagnetic sign for the rings with odd and even $N_e$. It is well known \nthat at any finite temperature ($T \\ne 0$ K), low-field current has a \nparamagnetic sign for the rings with even $N_e$ and in this section we \ndetermine that critical value of magnetic flux, $\\phi_c(T)$, where \nthe low-field current changes its sign from the paramagnetic to diamagnetic \nnature. In Fig.~\\ref{longtemp} we display the variation of $\\phi_c(T)$ as \na function of $N_e$ (here $N_e$ is even only) for $60$-site disordered rings\n($W=1$) at two different temperatures. The upper and lower curves in \nFigs.~\\ref{longtemp}(a) and (b) are respectively for the rings with \n$T\/T^{\\star}=1.0$ and $0.5$. Figure~\\ref{longtemp}(a) shows the results for \nthe rings with NNH integral, while the same are plotted for the rings with \nLRH ($\\alpha=1.6$) integrals in Fig.~\\ref{longtemp}(b). From these results \nwe can emphasize that, as the temperature increases the critical value of\nmagnetic flux $\\phi_c(T)$, where the low-field current changes its sign from\nthe paramagnetic phase to the diamagnetic one, increases. \n\n\\section{Concluding remarks}\n\nIn conclusion, we have investigated the behavior of persistent current in \nsingle-isolated mesoscopic rings subjected to both NNH and LRH integrals\nwithin the tight-binding framework. Our exact numerical calculations have\nshown that the current amplitude in disordered rings are comparable to that \nof ordered rings if we consider the model with LRH integrals instead of \nusual NNH integral models. This is due to the fact that higher order hopping \nintegrals try to delocalize the energy eigenstates and thus prevents the \nreduction of current due to disorder in the rings. Later, we have studied \nthe low-field magnetic response at $T=0$ K both for the perfect and \ndisordered rings and our results have predicted that the sign of the currents \ncan be mentioned precisely even in the presence of impurity in the rings. At \nthe end, we have calculated the magnetic response at finite temperatures \n($T\\ne 0$ K) and estimated the critical value of magnetic flux $\\phi_c(T)$ \nwhere the low-field current changes its sign from the paramagnetic to \nthe diamagnetic nature.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Renormalization of noncommutative euclidean scalar field theory\\label{RNCESFT}}\n\n\\noindent\nIn this article we will review recent progress in understanding how to\nrenormalize field theories on noncommutative euclidean space, and some\nnew advances into how these models may be analytically continued to\nMinkowski signature. Here we will be exclusively interested in scalar\nfield theories on Moyal spacetimes of even dimension. If $\\phi$ is a\nreal scalar field on ${\\mathbb{R}}^{2d}$ with Fourier transform\n$\\widetilde\\phi$, then the interactions in noncommutative field theory\non this space can be encoded by modifying the pointwise products\n$\\phi\\cdot\\phi$ to star-products $\\phi\\star\\phi$, which in momentum\nspace amounts to altering the Fourier convolution products as\n\\begin{equation}\n\\widetilde\\phi(k)\\,\\widetilde\\phi(q)~\\longrightarrow~\n\\widetilde\\phi(k)\\,\\widetilde\\phi(q)~\n\\epsilon^{{\\,{\\rm i}\\,} k\\times q} \\ , \\qquad k\\times q\\=\\mbox{$\\frac12$}\\,\nk_\\mu\\,\\theta^{\\mu\\nu}\\,q_\\nu\n\\end{equation}\nwhere $\\theta^{\\mu\\nu}$ is a constant antisymmetric matrix which we\nassume is of maximal rank for simplicity. Foundational aspects of the\ntheory are covered in~\\cite{Szabo1}.\n\\unitlength=1.00mm\n\\linethickness{0.4pt}\n\nIn $\\lambda\\,\\phi_{2d}^{\\star n}$-theory, the interaction vertex in\nmomentum space is thus modified to\n\\begin{equation}\n\\begin{picture}(100.00,25.00)\n\\thinlines\n\\put(32.00,12.00){\\line(-1,1){10.00}}\n\\put(32.00,12.00){\\line(-1,-1){10.00}}\n\\put(32.00,12.00){\\circle*{1.50}}\n\\put(18.00,22.00){\\makebox(0,0)[l]{{$k_1$}}}\n\\put(18.00,2.00){\\makebox(0,0)[l]{{$k_2$}}}\n\\put(26.00,12.00){\\makebox(0,0)[l]{{$\\lambda$}}}\n\\put(32.00,12.00){\\line(1,1){10.00}}\n\\put(32.00,12.00){\\line(1,-1){10.00}}\n\\put(40.00,12.00){\\makebox(0,0)[l]{$\\vdots$}}\n\\put(44.00,2.00){\\makebox(0,0)[l]{{$k_3$}}}\n\\put(44.00,22.00){\\makebox(0,0)[l]{{$k_n$}}}\n\\put(48.00,12.00){\\makebox(0,0)[l]{{$~=~\\lambda\n\\exp\\Big({\\,{\\rm i}\\,}\\sum\\limits_{I<J}\\, k_I\\times k_J\\Big) \\ . $}}}\n\\end{picture}\n\\label{intvertex}\\end{equation}\nThe perturbative quantum field theory suffers from the infamous UV\/IR\nmixing problem~\\cite{Minwalla99}. While the convergence properties of\nplanar graphs are the same as in the corresponding commutative quantum\nfield theory at $\\theta=0$, the phase factors in (\\ref{intvertex})\ndrastically alter the properties of non-planar graphs. Although this\nphase factor improves the ultraviolet behaviour of amplitudes,\ndivergences reappear as poles at vanishing external momenta. This can\nbe summarized by saying that an ultraviolet cutoff {$\\Lambda$}\nnecessarily induces an effective infrared cutoff\n  {$\\Lambda_0\\=\\frac1{\\theta\\,\\Lambda}$}. This would seem to ruin\n  standard renormalization schemes, such as the wilsonian prescription\n  which requires a clear separation of energy scales. More precisely,\n  although at one-loop order the on-shell amplitudes for massive\n  particles are all finite, when these graphs are inserted as\n  sub-graphs at higher-loop orders virtual particles of vanishing\n  momentum produce uncontrollable divergences in the amplitudes. Thus the\n  field theory cannot be renormalized. UV\/IR mixing also occurs on\n  more complicated noncommutative spaces such as $\\kappa$-deformed\n  spaces~\\cite{Grosse06}, and thus appears to be a generic feature of\n  noncommutative field theories.\n\nThe cure to this problem~\\cite{Langmann02,Grosse04} is to consider\ninstead a covariant version of the field theory which renders its\nultraviolet and infrared regimes indistinguishable. In momentum space,\nthis can be regarded as a modification of the Fourier momenta as\n\\begin{equation} \nk_\\mu~\\longmapsto~K_\\mu\\=k_\\mu+B_{\\mu\\nu}\\,x^\\nu \\ ,\n\\label{Landaumom}\\end{equation}\nwhere $B_{\\mu\\nu}$ is another antisymmetric constant matrix of maximal\nrank which is generically independent of $\\theta^{\\mu\\nu}$. We may\nthink of this matrix as a ``magnetic'' background, so that the momenta\n(\\ref{Landaumom}) can be regarded as ``Landau'' momenta. If\n$(k_\\mu,x^\\nu)$ are canonically conjugate variables, then these new momenta generate\na ``noncommutative momentum space'',\n\\begin{equation}\n[K_\\mu,K_\\nu]\\=2{\\,{\\rm i}\\,} B_{\\mu\\nu} \\ ,\n\\end{equation}\nthe familiar feature of physical momenta for charged particles\npropagating in a constant magnetic field. Thus the position and momentum spaces are formally identical, and there is no longer any distinction between what is meant by ultraviolet or infrared.\n\nThe {\\it Grosse--Wulkenhaar model} is then the real euclidean scalar\n   {$\\lambda\\,\\phi_{2d}^{\\star4}$}-theory in a background harmonic\n   oscillator potential. Analogously to (\\ref{Landaumom}), this amounts to replacing the Laplace operator giving the kinetic term in the scalar field action according to \n\\begin{equation}\n\\partial_\\mu^2~\\longmapsto~\\partial_\\mu^2+\\frac{\\omega^2}2\\,\\widetilde\n x_\\mu^{\\,2} \\ , \\qquad \\widetilde x_\\mu\\=2\\theta_{\\mu\\nu}^{-1}\\,x^\\nu \\ .\n\\end{equation}\nThe quantum field theory is then symmetric under Fourier transformation of fields, which amounts to exchanging momenta with positions as {$k_\\mu~\\leftrightarrow~\\widetilde x_\\mu$}.\n\nThe renormalization properties of this modified noncommutative field theory may then be summarized as follows~\\cite{Grosse04}--\\cite{Gurau09}. The covariant model is \\emph{renormalizable to all orders in\n{$\\lambda$}}. A crucial ingredient of the original renormalizability proof~\\cite{Grosse04} is the fact that the quantum field theory can regulated and described by a {matrix model}, with natural cutoff the matrix size\n  {$N$}. At {$\\omega\\=1$} the field theory is completely invariant under the UV\/IR duality transformation, without any rescalings of the parameters, whence this point in parameter space is called the self-dual point. At this point, the beta-functions in both couplings $\\lambda$ and $\\omega$ vanish to all orders of perturbation theory, and thus the renormalized coupling flows to a finite\n  bare coupling. This is analogous to what happens in a conformally\n  invariant quantum field theory. It implies, in particular, that the\n  duality covariant noncommutative field theory contains no Landau\n  ghost (or renormalons), contrary to the usual commutative\n  $\\phi_4^4$-theory, and unlike non-abelian gauge theories this\n  elimination is achieved without asymptotic freedom (but instead with\n  ``asymptotic safety''). For these reasons, a {non-perturbative} completion of the quantum field theory is believed possible. See~\\cite{Rivasseau07} for further details.\n\n\\bigskip\n\n\\section{UV\/IR duality on noncommutative euclidean space\\label{UVIRDIES}}\n\n\\noindent\nIn this article we will demonstrate how to construct an analogous\nduality covariant scalar field theory on noncommutative Minkowski\nspace. For later comparison, and because some of the duality proofs\ntranscribe immediately to lorentzian signature, we will first review\nthe proof of euclidean duality in some detail and the ensuing matrix model\nrepresentation, following~\\cite{Langmann02,Langmann04}. The\nGrosse--Wulkenhaar model has also been formulated on solvable\nsymmetric spaces in~\\cite{Bieliavsky08}, where the UV\/IR duality is\ninterpreted in terms of metaplectic representations of the Heisenberg group.\n\n\\subsection{Classical duality}\n\nThe original duality covariant model considers charged scalar fields {$\\phi(x)$} on euclidean space {$\\mathbb{R}^{2d}$} with action\n\\begin{equation}\nS[\\phi]\\=\\int\\,{\\rm d}^{2d}x~\\Big(\\phi^\\dagger\\,\\left(D_\\mu^2+\\mu^2\\right)\\,\\phi\n+g^2\\,\\phi^\\dagger\\star\\phi\\star\\phi^\\dagger\\star\\phi\\Big) \\ ,\n\\label{Sphi}\\end{equation}\nwhere\n\\begin{equation}\nD_{\\mu}\\=\\mbox{$\\frac{1}{\\sqrt{2}}$}\\,\\left(-{\\,{\\rm i}\\,}\n\\partial_{\\mu}+B_{\\mu\\nu}\\,x^\\nu\\right)\n\\end{equation}\nis the gauge-covariant derivative in the magnetic background. All of our considerations also apply to the other inequivalent quartic interaction $\\phi^\\dag\\star\\phi^\\dag\\star\\phi\\star\\phi$, but for simplicity we focus only on the one given in (\\ref{Sphi}). This action is invariant under the {duality transformation} of order~two given by\n\\begin{eqnarray}\n\\phi({x})&\\longrightarrow& \\widehat{\\phi}({x})\\=\\sqrt{|\\det(B)|}~\n\\widetilde\\phi(B\\cdot{x}) \\ , \\nonumber \\\\[4pt]\n\\theta&\\longrightarrow&\n\\widehat{\\theta}\\=-4B^{-1}\\,\\theta^{-1}\\,B^{-1} \\ , \\nonumber \\\\[4pt]\ng&\\longrightarrow&\\widehat g\\=2^d\\,\\big|\\det(B\\,\\theta)\\big|^{-1\/2}\\,g \\ .\n\\label{dualitytransf}\\end{eqnarray}\nThe proof is an elementary, but somewhat lengthy, calculation~\\cite{Langmann02}. The statement for the quadratic terms is a straightforward application of the Parseval identity, while for the quartic terms it follows from the integral kernel representation of the star-product (valid on the space of Schwartz fields $\\phi$). With the normalizations above the {self-dual} point, where the action is completely invariant on parameter space, is given by {$\\theta\\=2B^{-1}$}.\n\n\\subsection{Quantum duality}\n\nWhile the UV\/IR duality is relatively straightforward at the classical level, it is somewhat more involved at the quantum level. The quantum field theory is defined by the generating functional of connected Green's functions\n\\begin{equation}\n\\mathcal{G}(J)\\=-\\log\\frac{Z[J]}{Z[0]} \\ ,\n\\end{equation}\nwhere $Z[J]$ is the partition function for the field theory coupled to external sources $J$ and $J^\\dag$ given by the functional integral\n\\begin{equation}\nZ[J]\\=\\int\\,\\mathcal{D}\\phi~\\mathcal{D}\\phi^\\dagger~\\exp\\Big(-S[\\phi]-\n\\int\\,{\\rm d}^{2d}x~\\big(\\phi^\\dag\\,J+\\phi\\,J^\\dag\\big)\\Big) \\ .\n\\label{ZJ}\\end{equation}\nCorrelation functions are obtained by taking suitable functional derivatives with respect to $J$ and $J^\\dag$. By Parseval's identity, the extra source couplings in (\\ref{ZJ}) are invariant under (\\ref{dualitytransf}), and so is the functional integration measure. Thus the generating functional is {\\it formally} invariant under the duality transformation of Schwartz functions {$\\phi~\\longmapsto~\\widehat\\phi$} on {$\\mathbb{R}^{2d}$},\n\\begin{equation}\n\\mathcal{G}(J;B,g,{\\theta})\\= \\mathcal{G}(\\,\\widehat J;B,\\widehat\ng,\\widehat{\\theta}\\,) \\ .\n\\end{equation}\nHowever, to substantiate this claim we need to make sense of the functional integration in (\\ref{ZJ}), which nessitates regulated perturbation theory. This requires a duality invariant regularization {$\\mathcal{G}~\\longrightarrow~\\mathcal{G}_\\Lambda$} which we now describe.\n\nFor simplicity, let us consider the case $d=1$. The crux of the proof is to expand the scalar fields not in the standard plane wave eigenstates of the momentum operator $-{\\,{\\rm i}\\,}\\partial_\\mu$ (which requires distinguishing between ultraviolet and infrared), but instead in the ``matrix basis'' {$f_{n,m}\\in L^2(\\mathbb{R}^2)$,\n    $n,m\\=0,1,\\dots$} of {Landau wavefunctions}\n\\begin{equation}\n\\phi(x)\\=\\sum_{n,m=0}^\\infty\\,f_{n,m}(x)~\\phi_{n,m} \\ ,\n\\end{equation}\nwhich are eigenstates of the Landau hamiltonian with\n\\begin{equation}\nD_\\mu^2f_{n,m}\\=2B\\,\\big(n+\\mbox{$\\frac12$}\\big)\\,f_{n,m}~=:E_n\\,f_{n,m} \\ .\n\\end{equation}\nNote that each Landau level is infinitely degenerate. The second quantum number features because the Landau wavefunctions are simultaneously eigenstates of the Landau hamiltonian with the reflected magnetic field,\n\\begin{equation} \nD_\\mu^2\\big|_{B\\to-B}f_{n,m}\\=E_m\\,f_{n,m} \\ ,\n\\end{equation}\nsuch that the sum $D_\\mu^2+D_\\mu^2\\big|_{B\\to-B}\\=-\\partial_\\mu^2+(B\\cdot x)_\\mu^2$ gives the background harmonic oscillator potential which figures in the Grosse--Wulkenhaar model.\n\nFor a suitable cut-off function {$F:[0,\\infty)\\to[0,\\infty)$}, with\n$F(0)\\=1$ and $F(s)\\=0$ for all $s>s_0$ and some finite $s_0$, we now replace the free\n  propagator in the Landau basis,\n\\begin{equation}\nC(n,m)\\=\\big(E_n+\\mu^2\\big)^{-1} \\ ,\n\\end{equation}\nwith the regulated propagator\n\\begin{equation}\nC_\\Lambda(n,m)\\=\n\\big(E_n+\\mu^2\\big)^{-1}\\,F\\big(\\Lambda^{-2}\\,(E_n+E_m)\\big)\n\\label{CLambda}\\end{equation}\nwhere $\\Lambda\\in{\\mathbb{R}}$ provides an ultraviolet cutoff in $(n,m)$-space and $C_\\Lambda\\to C$ as $\\Lambda\\to\\infty$. The argument of the cut-off function is the eigenvalue of the differential operator $D_\\mu^2+D_\\mu^2\\big|_{B\\to-B}$. In particular, the cut-off on the operator $-\\partial_\\mu^2$ truncates all high-momentum modes while the cut-off on the operator $(B\\cdot x)_\\mu^2$ truncates the long-distance modes. \n\nSince $F(s)\\=0$ for $s>s_0$ and some finite $s_0\\in(0,\\infty)$, for $\\Lambda$ finite the propagators (\\ref{CLambda}) are non-zero only for $E_n+E_m$ smaller than a uniform upper bound, which from the forms of the Landau eigenvalues can happen only for finitely many values of $n,m$.\nWith this regularization, every Feynman diagram is of the schematic form \n\\begin{equation}\n\\sum_{n_1,m_1,\\ldots,n_K,m_K\\geq0}~\\prod_{k=1}^K\\,\nC_\\Lambda(n_k,m_k)~\\times~({\\rm vertices})\n\\label{finitesums}\\end{equation}\nwhere the vertex factors will in general be complicated combinatorial quantities, but their explicit form is immaterial for our argument. Since the propagators in (\\ref{finitesums}) are non-zero for only finitely-many $n_k,m_k$, every Feynman diagram is given by a finite sum, i.e. all Feynman amplitudes converge. This completes the proof of quantum duality.\n\n\\subsection{Matrix model representation}\n\nA beautiful feature of the covariant quantum field theory is that it can be mapped exactly onto a matrix model, a manipulation which has no counterpart in ordinary field theory. For simplicity, we consider the two-dimensional model at the self-dual point, $d\\=1$ and  $\\theta\\=2B^{-1}>0$ (see~\\cite{Langmann04} for the general case). The crucial feature is the projector property of the Landau wavefunctions with respect to the star-product,\n\\begin{equation}\nf_{n,m}\\star f_{n',m'}\\=\\sqrt{\\frac B{4\\pi}}~ \\delta_{m,n'}~f_{n,m'} \\ , \n\\end{equation}\ntogether with $f_{n,m}{}^*\\=f_{m,n}$ and the normalization\n\\begin{equation}\n\\int\\,{\\rm d}^2x~f_{n,m}(x) \\=\\sqrt{\\frac{4\\pi}B}~\\delta_{n,m} \\ .\n\\end{equation}\n\nThe action functional (\\ref{Sphi}) can thereby be expressed in the form\n\\begin{equation}\nS[\\phi]\\={\\rm Tr}\\Big(\\phi^\\dag\\,\\mathcal{B}\\,\\phi+\\mu^2\\,\\phi^\\dag\\,\n\\phi+\\tilde g^2\\,\\big(\\phi^\\dag\\,\n\\phi\\big)^2\\Big) \\ ,\n\\end{equation}\nwhere in this formula $\\phi\\=(\\phi_{n,m})$ is an infinite matrix,\n$\\tilde g^2\\=B\\,g^2\/4\\pi$, and $\\mathcal{B}_{n,m}\\=2B\\,\\big(n+\\frac12\\big)~\\delta_{n,m}$. The quantum field theory thus has a {$U(\\infty)$} symmetry {$\\phi~\\longrightarrow~U^\\dag\\,\\phi\\,U$}, and is the {$N~\\longrightarrow~\\infty$} limit of the {$N\\times N$} complex \n  matrix model in an external field whose partition function is given by\n\\begin{equation}\nZ_N\\=\\int~\\prod_{n,m=1}^N\\,{\\rm d}\\phi_{n,m}~{\\rm d}\\phi^\\dag_{n,m}~\n\\epsilon^{-S[\\phi]} \\ .\n\\end{equation}\nThis is an integral over a finite-dimensional space, and thus gives a constructive non-perturbative definition of the quantum field theory. Various exact integrability properties of the model in this representation, which is related to the Kontsevich--Penner matrix model (a hermitean matrix model in an external field with logarithmic potential), are described in~\\cite{Langmann04}.\n\n\\bigskip\n\n\\section{Analytic continuation to Minkowski signature}\n\n\\noindent\nThe continuation of the duality covariant field theory to noncommutative Minkowski space is naively obtained by Wick rotation {$x^0~\\longrightarrow~\\pm{\\,{\\rm i}\\,} t$ plus an additional change $B_{0i}~\\longrightarrow~\\pm{\\,{\\rm i}\\,} E_i$}, giving the dynamics in an electromagnetic background. While this is wrong for a number of reasons, we shall see that many of our results can be obtained in hindsight via a careful continuation of this sort. The reason that this naive approach is not expected to work is that the perturbative dynamics of (non-covariant) noncommutative field theory {\\it cannot} be\n  obtained simply by Wick rotation~\\cite{Bahns02}--\\cite{Rim02}. In contrast to the commutative case, time-ordering factors and the two-point function do not combine into Feynman\n  propagators in non-planar graphs with a noncommuting time direction. Because of this complication, the Dyson and Feynman expansions are distinct, and the renormalization properties in the Dyson series are very\n  different. By developing the complicated rules of time-ordered perturbation theory on noncommutative spacetime, one can restore unitarity and causality of the quantum field theory. In fact, it has been suggested that UV\/IR mixing may be far less severe or even\n  absent in this case~\\cite{Bahns09}. In the following we shall summarize results from the analysis of~\\cite{Fischer08} which are carried out by defining the quantum field theory using a functional integral framework, rather than time-ordered perturbation theory.\n\n\\subsection{Results}\\label{results}\n\nWe will begin by stating the main results from~\\cite{Fischer08}, before going into the technical details of their derivation, which requires various notions from functional analysis and the theory of generalized functions. Again we restrict to the case of $1+1$ dimensions, corresponding to a pure electric background, the general case being a straightforward combination with the earlier euclidean analysis~\\cite{Fischer08}.\n\n\\subsubsection*{Matrix basis}\n\nThere is a dense domain of scalar fields {$\\phi\\in\\Phi\\subset L^2(\\mathbb{R}^2)$}\n  and ``electric Landau wavefunctions'' {$f_{n,m}^\\pm\\in\\Phi', \\\n    n,m\\=0,1,\\dots$} such that \n\\begin{equation}\n\\phi(x)\\=\\sum_{n,m=0}^\\infty\\,f^+_{n,m}(x)~\\phi^-_{n,m} \\= \\sum_{n,m=0}^\\infty\\,\nf^-_{n,m}(x)~\\phi^+_{n,m} \\ ,\n\\label{phix}\\end{equation}\nand\n\\begin{equation}\nD_\\mu^2f^\\pm_{n,m}\\=\\pm{\\,{\\rm i}\\,} E_n\\,f_{n,m}^\\pm\n\\ , \\qquad D_\\mu^2\\big|_{B\\to-B}f^\\pm_{n,m}\\=\\pm{\\,{\\rm i}\\,} E_m\\,f^\\pm_{n,m} \\ .\n\\label{Dfpmnm}\\end{equation}\nThe $\\pm$ labels here correspond to the two choices of sign in the Wick rotation. In fact, although we shall not derive them in this way, the wavefunctions $f_{n,m}^\\pm$ can be obtained merely by Wick rotating the standard Landau wavefunctions. Nevertheless, this result should look somewhat odd to the reader, since (\\ref{Dfpmnm}) seems to assert that $f_{n,m}^\\pm$ are eigenfunctions of self-adjoint operators with \\emph{imaginary} eigenvalues. However, the crucial point is that these functions live in the topological dual $\\Phi'$ of the domain $\\Phi$, which is much larger than the domain of these differential operators. Below we will see how this can be used to define notions of generalized eigenfunctions with generalized eigenvalues, which can be complex. The electric Landau wavefunctions obey the $L^2$-orthonormality and star-product projector relations\n\\begin{equation}\nf_{n,m}^\\pm\\,^*\\=f_{m,n}^\\mp \\ , \\quad \\big(f_{n,m}^\\pm\\,,\\,\nf_{n',m'}^\\mp\\big)_{L^2}\\=\\delta_{m,n'}\\,\\delta_{n,m'} \\ , \\quad\nf^\\pm_{n,m}\\star f^\\pm_{n',m'}\\=\\sqrt{\\frac B{4\\pi}} ~\\delta_{m,n'}\\,f^\\pm_{n,m'} \\ ,\n\\end{equation}\ntogether with the normalization condition\n\\begin{equation}\n\\int\\,{\\rm d}^2x~f_{n,m}^\\pm(x)\\= \\sqrt{\\frac{4\\pi}B}~\\delta_{n,m} \\ .\n\\label{fpmnorm}\\end{equation}\nA better understanding of the physical meaning of these functions, and\nhow the formulation of the field theory in terms of them is related to\ntime-ordered perturbation theory, is currently lacking.\n\n\\subsubsection*{Unitarity and causality}\n\nEach set of functions $f_{n,m}^+$ and $f_{n,m}^-$ on its own generates\na complete basis for expansion of Schwartz fields $\\phi$. The domain\n$\\Phi$ is chosen such that both expansions can be taken\nsimultaneously. Both matrix bases together imply stability and CT-invariance during matrix regularization.\nBy stability we mean that expanding $\\phi$\nin the matrix bases $f_{n,m}^\\pm$ and imposing the matrix regularization by cutting off these sums at some finite $N$ yields action functionals $S_\\Lambda^\\pm[\\phi]$,\nwhose sum $S_\\Lambda[\\phi]=\\frac12\\, \\big(S_\\Lambda^+[\\phi]+S_\\Lambda^-[\\phi]\\big)$ is\nmanifestly real. The CT-symmetry $\\phi_{n,m}^\\mp\\=C\\,T\\phi_{n,m}^\\pm$ follows from\nthe behaviours of the electric Landau wavefunctions under $T$, $P\\,T$\nand $C$ transformations given by\n\\begin{eqnarray}\nf_{m,n}^\\pm(-t,x)&=&(-1)^{m-n}\\,f_{n,m}^\\pm(t,x) \\ , \\nonumber \\\\[4pt]\nf_{m,n}^\\pm(-t,-x)&=& (-1)^{m-n}\\, f_{m,n}^\\pm(t,x) \\ , \\nonumber \\\\[4pt]\nf_{m,n}^\\pm(t,x)^*&=&f_{n,m}^\\mp(t,x)\\=(-1)^{m-n}\\, f_{m,n}^\\mp(-t,x)\\,.\n\\end{eqnarray}\n\n\\subsubsection*{Quantum duality}\n\nThe proof of classical duality follows exactly the same route as in\nthe euclidean case -- it does not depend on the signature of the inner\nproducts used in Fourier transformation. At the quantum level, \nanalogously to the euclidean case the regulated propagators in Minkowski\n  space are obtained by replacing\n\\begin{equation}\nC^\\pm(n,m)\\=\\big\\langle\\,{\\phi_{m,n}^\\pm}^*\\,\n\\phi_{m,n}^\\mp\\big\\rangle\n\\label{Cpmnm}\\end{equation}\nwith\n\\begin{equation}\nC_\\Lambda^\\pm(n,m)\\=\n2{\\,{\\rm i}\\,}\\,\\big(\\pm{\\,{\\rm i}\\,} E_n+\\mu^2\\big)^{-1}\\,F\\left(\\Lambda^{-2}\\,\n|E_n+E_m|\\right) \\ .\n\\end{equation}\nThen the proof of quantum duality presented before carries through verbatim\nusing these two sets of two-point functions. By\nmultiplying (\\ref{Cpmnm}) with $f_{m,n}^\\pm(x)^*\\,f_{m,n}^\\mp(y)$ and summing over all\n$n,m\\in{\\mathbb{N}}_0$ one obtains the\nposition space representation of the propagator~\\cite{Fischer08}. For\nthe free Klein-Gordon field without electric field, after Fourier\ntransformation this representation can be shown to possess the standard physical\nmass-shell poles~\\cite{Fischer10}.\n\n\\subsubsection*{Coupled complex two-matrix model representation} \n\nA non-trivial interacting two-matrix model now describes the\nminkowskian theory, whose action is generically rather\ninvolved~\\cite{Fischer08}. With the same notation as before, it simplifies at the self-dual point to\n\\begin{equation}\nS[\\phi]\\=\\frac12\\,\\sum_{\\pm}\\,{\\rm Tr}\\Big(\\pm\\,\\phi_\\pm^\\dag{\\,{\\rm i}\\,}\\mathcal{B}\\,\n\\phi_{\\mp}+\\mu^2\\,\\phi_\\pm^\\dag\\,\\phi_{\\mp}+\\tilde g^2\\,\\big(\n\\phi_\\pm^\\dag\\,\\phi_{\\mp}\\big)^2\\Big) \\ .\n\\label{Sphi2matrix}\\end{equation}\nThis action possesses a much larger\n{$GL(\\infty)\\times GL(\\infty)$} symmetry\n{$\\phi_\\pm~\\longmapsto~\\phi_\\pm\\,U_\\pm, \\\n\\phi_\\pm^\\dag~\\longmapsto~U_{\\mp}^{-1}\\,\\phi_\\pm^\\dag$}, and also the\ndiscrete CT-symmetry\n\\begin{equation}\n\\big(\\phi_\\pm\\,,\\,\\phi_\\pm^\\dag\\,\\big)~\\longmapsto~\n\\big(\\phi_{\\mp}\\,,\\,\\phi_{\\mp}^\\dag\\big) \\quad , \\quad\n\\theta~\\longmapsto~-\\theta \\ .\n\\end{equation}\nThis two-matrix model representation clearly demonstrates that the\nMinkowski theory is not simply a Wick rotation of the euclidean theory.\n\n\\subsection{Derivations\\label{deriv}}\n\nWe will now sketch how these results are obtained, including a description of the configuration space $\\Phi$. Just like the analysis\nof the standard Landau problem, and hence the duality covariant field theory\nin euclidean signature, is related to the harmonic oscillator, the\nmodel in Minkowski signature is related to the \\emph{inverted\nharmonic oscillator} whose hamiltonian is given by\n\\begin{equation}\nH\\=\\mbox{$\\frac12$}\\,\\big(p^2-\\omega^2\\,q^2\\big) \\ ,\n\\label{invham}\\end{equation}\nwhere $\\omega\\in{\\mathbb{R}}$ and $(p,q)$ are canonically conjugate variables. The functional analytic properties of the corresponding\nquantum hamiltonian are described in~\\cite{Chruscinski03}. It is related to\nthe usual harmonic oscillator hamiltonian by a \\emph{complex scaling}, which is a\nnon-unitary similarity transformation which sends\n{$\\omega~\\longrightarrow~\\pm{\\,{\\rm i}\\,}\\omega$}. The first quantized operator\n{$\\hat H$} corresponding to (\\ref{invham}) is\nsymmetric on a suitable domain in {$L^2(\\mathbb{R})$} with\nspectrum {${\\rm Spec}(\\hat H)\\=\\mathbb{R}$. In fact it \n  admits a one-parameter family of self-adjoint extensions. This parameter should have some significance within the\n  context of our duality covariant quantum field theory, but for\n  simplicity we fix a self-adjoint extension and simply work with\n  that. The relation between the classical hamiltonian $H$, the\n  quantum hamiltonian $\\hat{H}$, and the differential operator $D_\\mu^2$ is established via the star\nproduct and the Wigner transform, defined for a rank one operator\n$\\vech{\\phi}=\\kb{\\psi}{\\varphi}\\in L^2({\\mathbb{R}})\\otimes L^2({\\mathbb{R}})^\\vee$ by\n\\begin{equation}\n{\\sf W}(\\vech{\\phi}\\,)(t,x) \\=\\frac1{2\\pi}\\,\n\\int_{{\\mathbb{R}}}\\, {\\rm d} k~ \\epsilon^{{\\,{\\rm i}\\,} k\\,x}\\,\\bk{t-\\theta\n   \\,k\/2}{\\psi}\\,\\bk{\\varphi}{t+\\theta\\, k\/2} \\ .\n\\label{Wigenfn}\\end{equation}\nFor every function $f(x)\\={\\sf W}(\\hat{f}\\,)(x)$\none has\n\\begin{equation}\nD_\\mu^2 f(x)\\=H\\star f(x)\\={\\sf W}(\\hat{H}\\,\\hat{f}\\,)(x) \\ .\n\\end{equation}\nThus instead of working with\n$D_\\mu^2$ operating on a set of fields, we will work with $\\hat{H}$ acting on a\nsuitable quantum mechanical Hilbert space.\n\nThe hamiltonian $\\hat H$ also has a set of {generalized eigenfunctions} with {\\it imaginary}\neigenvalues. They occur as residues of the original eigenfunctions analytically\n  continued to the complex energy plane. By closing the contour of\n  integration in the eigenfunction expansion of a wavefunction appropriately, we pick up\n  these states and obtain the analog of the {\\it\n    discrete} expansion in Landau wavefunctions. These techniques are\n  analogous to those of the Bohm--Gadella theory of resonant\n  states in quantum mechanics, wherein the instabilities\n  mentioned above describe nuclear decay phenomena. The inverted\n  harmonic oscillator potential and its resonance expansion also\n  defines the analytic continuation of the Grosse--Wulkenhaar model to Minkowski\n  signature. To explore the renormalization in this case, one needs to\n  establish suitable decay properties of the free\n  propagators in the matrix basis analogous to the euclidean\n  case~\\cite{Grosse04}. The properties of these Green's functions are\n  currently under investigation~\\cite{Fischer10}.\n\nLet us now explain the concepts introduced above. The mathematical setting we need\nis the extension of the notion of Hilbert space to that of a \\emph{rigged Hilbert space} (also known in the literature\nas a Gel'fand triple), which is a triple of spaces\n\\begin{equation}\n\\Phi~\\subset~{\\mathcal H}~\\subset~\\Phi'\n\\end{equation}\nwhere $\\Phi$ is a dense nuclear subspace of a Hilbert space {${\\mathcal H}$} with dual\n{$\\Phi'$}, the space of continuous linear functionals $\\Phi\\to{\\mathbb{C}}$. If\n$A\\in{\\rm End}({\\mathcal H})$ is self-adjoint on $\\Phi$, then we can define its\naction on $\\Phi'$ using the dual pairing. A vector $F_\\lambda\\in\\Phi'$\nis then said to be a generalized eigenvector of $A$ with generalized\neigenvalue $\\lambda\\in\\mathbb{C}$ if\n\\begin{equation}\n\\langle AF_\\lambda|\\phi\\rangle~:=~ \\langle\n   F_\\lambda |A\\phi\\rangle\\= \\lambda\\,\\langle F_\\lambda|\\phi\\rangle\n\\end{equation}\nholds for all $\\phi\\in\\Phi$. The Gel'fand--Maurin theorem asserts\n  that for any\n  {$\\phi\\in\\Phi$}, there exists\n  {$F_\\lambda\\in\\Phi'$} such that there is an expansion\n\\begin{equation}\n\\phi\\=\\int_{{\\rm Spec}(A)}\\,\n    {\\rm d}\\mu(\\lambda)~F_\\lambda\\,\\langle F_\\lambda|\\phi\\rangle \\\n    ,\n\\end{equation}\nwhere ${\\rm d}\\mu$ is discrete measure on the discrete part of the\nspectrum of $A$ and Lebesgue measure on the continuous part. \n\nFor the example of the inverted harmonic\noscillator, the rigged Hilbert space is\n\\begin{equation}\n\\mathcal{S}(\\mathbb{R})~\\subset~L^2(\\mathbb{R})~\\subset\n    ~\\mathcal{S}'(\\mathbb{R}) \\ ,\n\\end{equation}\nwhere $\\mathcal{S}(\\mathbb{R})$ is the\n  topological vector space of Schwartz functions on ${\\mathbb{R}}$ (with the\n  usual semi-norm topology) and $\\mathcal{S}'(\\mathbb{R})$ is the space of\n  tempered distributions on ${\\mathbb{R}}$.\nBy parity invariance, each eigenvalue {${\\mathcal E}\\in{\\rm Spec}(\\hat H)={\\mathbb{R}}$}\ncorresponds to \ntwo-fold degenerate eigenfunctions {$\\chi_\\pm^{\\mathcal E} \\ , \\\n    \\eta_\\pm^{\\mathcal E}\\in \\mathcal{S}'(\\mathbb{R})$} which after rescaling $B\\to\n  B\/2$ are given explicitly by\n\\begin{eqnarray}\n\\chi_\\pm^{\\mathcal E}(q)&=&\\frac{C}{\\sqrt{2\\pi \\,B}}\\, {\\,{\\rm i}\\,}^{\\frac{\\nu}{2}+\\frac{1}{4}}\\, \\Gamma(\\nu+1)\\, D_{-\\nu-1}\\big(\\mp\\,\\sqrt{-2{\\,{\\rm i}\\,} B}\\,q\\big) \\ , \\nonumber\\\\[4pt]\n\\eta_\\pm^{\\mathcal E}(q)&=&\\frac{C}{\\sqrt{2\\pi \\,B}}\\, {\\,{\\rm i}\\,}^{\\frac{\\nu}{2}+\\frac{1}{4}}\\, \\Gamma(-\\nu)\\,D_\\nu\\big(\\mp\\,\\sqrt{2{\\,{\\rm i}\\,} B}\\,q\\big) \\ ,\n\\end{eqnarray}\nwhere $C$ is a numerical constant, $\\nu=-{\\,{\\rm i}\\,}\\frac{{\\mathcal E}}{B}-\\frac12$,\nand $D_\\nu(z)$ are parabolic cylinder functions. Only two \nof them are linearly independent, so for any {$\\phi\\in\\mathcal{S}(\\mathbb{R})$}\nthe Gel'fand--Maurin theorem gives a pair of expansions\n\\begin{equation}\n\\phi(q)\\=\\sum_{\\pm}~\\int_{\\mathbb{R}}\\,{\\rm d}{\\mathcal E}~\\chi^{\\mathcal E}_\\pm(q)~\\big\\langle\n\\chi^{\\mathcal E}_\\pm\\,\\big|\\,\\phi\\big\\rangle \\=\n\\sum_{\\pm}~\\int_{\\mathbb{R}}\\,{\\rm d}{\\mathcal E}~\\eta^{\\mathcal E}_\\pm(q)~\\big\\langle \n\\eta^{\\mathcal E}_\\pm\\,\\big|\\,\\phi\\big\\rangle \\ .\n\\label{phichieta}\\end{equation}\nThe oscillator hamiltonian {$\\hat H$} also has generalized eigenfunctions {$f_n^\\pm$}\n  with discrete eigenvalues\n  {$\\pm\\,{\\,{\\rm i}\\,} B\\,\\big(n+\\frac12\\big) \\ , \\ n\\=0,1,\\dots$},\n  occuring as residues of {$\\chi_\\pm^{\\mathcal E} \\ \/ \\ \\eta_\\pm^{\\mathcal E}$} \n  in the lower \/ upper complex half-plane. Then in a suitable domain\n  {$\\phi\\in\\Phi\\subset\\mathcal{S}(\\mathbb{R})$}, an application of the\n  residue theorem to the energy integrals in (\\ref{phichieta}) gives\n  the respective \\emph{resonance expansions}\n\\begin{equation}\n\\phi(q)\\= \\sum_{n=0}^\\infty\\,f_n^-(q)~\n\\big\\langle f_n^+\\big|\\phi\\big\\rangle \\= \\sum_{n=0}^\\infty\\,f_n^+(q)~\n\\big\\langle f_n^-\\big|\\phi\\big\\rangle \\ .\n\\label{phiq}\\end{equation}\n\nThe choice of configuration space $\\Phi$ must ensure that both integrals over ${\\mathcal E}$\nin (\\ref{phichieta}) can be extended to closed contour integrals for\nwhich the residue theorem applies in the usual way and such that the\nresonance expansions in (\\ref{phiq}) converge. In particular, it consists\nof fields $\\phi$ such that $\\chi^{\\mathcal E}_\\pm(q)\\,\\big\\langle\n\\chi^{\\mathcal E}_\\pm\\,\\big|\\,\\phi\\big\\rangle$ vanishes uniformly almost everywhere as ${\\mathcal E}$ tends to\ninfinity in the lower complex half-plane, and $\\eta^{\\mathcal E}_\\pm(q)\\,\\big\\langle\n\\eta^{\\mathcal E}_\\pm\\,\\big|\\,\\phi\\big\\rangle$ vanishes uniformly almost everywhere as ${\\mathcal E}$ tends to\ninfinity in the upper complex half-plane, together with the analogous\nvanishing requirements on the scalar products\n$\\big\\langle\\psi\\,\\big|\\,\\chi^{\\mathcal E}_\\pm\\,\\big\\rangle \\,\\big\\langle\n\\chi^{\\mathcal E}_\\pm\\,\\big|\\,\\phi\\big\\rangle$ and $\\big\\langle\\psi\\,\\big|\\,\n\\eta^{\\mathcal E}_\\pm\\big\\rangle \\,\\big\\langle\n\\eta^{\\mathcal E}_\\pm\\,\\big|\\,\\phi\\big\\rangle$ for all $\\phi,\\psi\\in\\Phi$. For this, consider the rigged Hilbert space\n\\begin{equation}\n\\mathcal{S}^\\alpha_\\alpha(\\mathbb{R})~\\subset~L^2(\\mathbb{R})~\\subset~\n\\mathcal{S}^\\alpha_\\alpha(\\mathbb{R})' \\ ,\n\\end{equation}\nwhere {$\\mathcal{S}^\\alpha_\\alpha(\\mathbb{R})$} is a Gel'fand--Shilov space with\n{$\\alpha\\geq\\frac12$}, and its dual\n{$\\mathcal{S}^\\alpha_\\alpha(\\mathbb{R})'$} is a space of tempered\n  ultra-distributions of Roumieu type. These Gel'fand--Shilov spaces\n  contain entire functions ${\\phi(q)}$\non {$\\mathbb{C}$} restricted to {$\\mathbb{R}$}, with $L^\\infty$-norms obeying\n{$\\big\\|q^m\\,\\partial_q^n\\phi\\big\\|_\\infty\\leq\n  C\\,M^{n+m}\\,n^{\\alpha\\,n}\\,m^{\\alpha\\,m}$} for all $n,m\\in{\\mathbb{N}}_0$ with\nsome constant $C$ and given $M$. They form dense subspaces of Schwartz\nspace {$\\mathcal{S}^\\alpha_\\alpha(\\mathbb{R})\\subset \\mathcal{S}(\\mathbb{R})\\=\n  \\mathcal{S}^\\infty_\\infty(\\mathbb{R})$} which are closed under Fourier\ntransformation and the star-product~\\cite{Soloviev07,Chaichian07}, and\nwhich are generated by the basis of harmonic oscillator\nwavefunctions~\\cite{Lozanov07}. They are thus natural configuration\nspaces for duality covariant noncommutative field theories. The\nboundedness properties of these functions, together with the\nasymptotic behaviours of the parabolic cylinder functions and the\ngamma-functions, appear to be sufficient to ensure that the integrands in (\\ref{phichieta})\nand all pertinent pairing factors vanish appropriately~\\cite{Fischer08,Fischer10}.\n\nThe mapping from functions on ${\\mathbb{R}}$ in (\\ref{phiq}) to fields on ${\\mathbb{R}}^2$\nin (\\ref{phix}) is given by applying the Wigner transformation ${\\sf\n  W}\\,:\\,\\mathcal{S}_\\alpha^\\alpha({\\mathbb{R}})\\otimes\\mathcal{S}_\\alpha^\\alpha({\\mathbb{R}})^\\vee~\\longrightarrow~ \n\\mathcal{S}_\\alpha^\\alpha({\\mathbb{R}}^2)$ to a rank one operator\n$\\vech{\\phi}=\\kb{\\psi}{\\varphi}$ using the integral formula (\\ref{Wigenfn}).\nExpanding $\\vech{\\phi}$ in parabolic cylinder functions,\n\\begin{eqnarray}\n \\vech{\\phi} &=& \\sum_{s,s'=\\pm}~\\int_{{\\mathbb{R}}}\\,\n {\\rm d}\\mathcal{E}~\\int_{{\\mathbb{R}}}\\,{\\rm d}\\mathcal{E}'~\\ket{\\chi_s^\\mathcal{E}}\\,\\bk{\\chi_s^\\mathcal{E}}{\\psi}\\,\n\\bk{\\varphi}{\\eta_{s'}^{\\mathcal{E}'}}\\,\\bra{\\eta_{s'}^{\\mathcal{E}'}}\n\\nonumber\\\\[4pt] &=& \\sum_{s,s'=\\pm}~\\int_{{\\mathbb{R}}}\\,\n {\\rm d}\\mathcal{E}~\\int_{{\\mathbb{R}}}\\,{\\rm d}\\mathcal{E}'~\\ket{\\eta_s^\\mathcal{E}}\\,\\bk{\\eta_s^\\mathcal{E}}{\\psi}\\,\n\\bk{\\varphi}{\\chi_{s'}^{\\mathcal{E}'}}\\,\\bra{\\chi_{s'}^{\\mathcal{E}'}} \\ , \\label{chichi}\n\\end{eqnarray}\none can read off the respective resonance expansions (\\ref{phix}) from\n(\\ref{phichieta}) and (\\ref{phiq}). Using this mapping one can also explicitly compute the electric Landau wavefunctions~\\cite{Fischer08}\n\\begin{eqnarray}\nf_{m,n}^\\pm(t,x)&=&\n(-1)^{\\min(m,n)}~\\sqrt{\\frac{B}{\\pi}}~\\sqrt{\\frac{\\min(m,n)!}{\\max(m,n)!}}~(\\pm{\\,{\\rm i}\\,}\nB)^{|m-n|\/2} \\nonumber\\\\ && \\times ~ \\epsilon^{\\mp{\\,{\\rm i}\\,}\n  B\\,x_+\\,x_-\/2}\\,x_{\\mp\\,{\\rm sgn}(m-n)}^{|m-n|}\\,\nL_{\\min(m,n)}^{|m-n|}(\\pm{\\,{\\rm i}\\,} B\\,x_+\\,x_-) \\ , \\label{fmn}\n\\end{eqnarray}\nwhere $x_\\pm=t\\pm x$ and\n\\begin{equation}\nL_n^\\alpha(z)\\= \\sum_{q=0}^n\\, {\\alpha+n\\choose n-q} \\, \\frac{(-z)^q}{q!}\n\\end{equation}\nare the generalized Laguerre polynomials. Using these explicit forms\none can straightforwardly derive all properties of $f_{m,n}^\\pm$\nstated above.\n\nEssential for the proof of orthogonality is the\noccurance of the phase factors $\\epsilon^{\\mp{\\,{\\rm i}\\,} B\\,x_+\\, x_-\/2}$ in (\\ref{fmn}), which\ngenerate derivatives via the identity $\\int_{\\mathbb{R}}\\,{\\rm d} x_-\n~(x_-)^p~\\epsilon^{\\mp {\\,{\\rm i}\\,} x_+\\, x_-} \\= 2\\pi\\,(\\pm{\\,{\\rm i}\\,}\\partial_{+})^p\\delta(x_+)$ and ensure that the integrals\nover $(x_+,x_-)\\in{\\mathbb{R}}^2$ converge. This is the reason why only the\n$L^2$-inner products $\\big({f_{m,n}^\\mp}\\,,\\,{f_{k,l}^\\pm}\\big)_{L^2}$ are permitted,\nsince this exponential factor is absent for the other\ncombinations. For the same reason only terms with equal powers of\n$x_+$ and $x_-$ survive the integration. After some\nalgebra one then readily verifies the\northogonality relations\n\\begin{eqnarray}\n\\big({f_{m,n}^\\mp}\\,,\\,{f_{k,l}^\\pm}\\big)_{L^2} \\= \\int\\,{\\rm d} x~{\\rm d}\nt~f_{n,m}^\\pm(t,x)\\,f_{k,l}^\\pm(t,x) \\= \n\\delta_{n,l}\\,\\delta_{m,k} \\ .\n\\end{eqnarray}\nThe normalization relation (\\ref{fpmnorm}) is computed in an analogous way.\n\n\\bigskip\n\n\\section*{Note added}\n\n\\noindent\nAfter this paper was submitted for publication, the\npreprint~\\cite{Zahn10} appeared with some critiques of our approach\nin~\\cite{Fischer08}. In particular, a counterexample to our\nTheorem~4.2 is found, casting doubt on our choice of domain $\\Phi$. While this critique is fully justified, we have\namended our calculations, and found that both our\nusage of and conclusions infered from the matrix basis are still in\nfact valid. Briefly, one splits the Minkowski space action functional\n$S_{\\rm M}$ at $g=0$ into two parts as\n\\begin{eqnarray}\nS_{\\rm M}\\= \\mbox{$\\frac{1}2$}\\,\\big(S^{(+\\epsilon)}+S^{(-\\epsilon)}\\big) \\\n, \\qquad S^{(\\pm\\,\\epsilon)}\\= S_{\\rm M}\\pm{\\,{\\rm i}\\,}\\tan(\\epsilon)\\, S_{\\rm E} \\ ,\n\\end{eqnarray}\nwhere $0<\\epsilon<\\frac\\pi2$ and $S_{\\rm E}$ is the euclidean\naction at $g=0$. In the same manner that $S_{\\rm E}$ and $S_{\\rm M}$ can be\nrelated to the harmonic and inverted harmonic oscillators,\nrespectively, the actions $S^{(\\pm\\,\\epsilon)}$ are related to the\n\\emph{complex harmonic oscillator} with hamiltonian\n$\\frac12\\,\\big(p^2-\\epsilon^{\\mp\\, 2{\\,{\\rm i}\\,}\\epsilon}\\, q^2\\big)$. In contrast to\nthe inverted harmonic oscillator, this family of hamiltonians have\ndiscrete eigenvalues given by the harmonic oscillator spectrum scaled\nby $\\pm{\\,{\\rm i}\\,}\\epsilon^{\\mp{\\,{\\rm i}\\,}\\epsilon}$, while its eigenfunctions span\n$L^2({\\mathbb{R}})$ and have the usual star-product projection property. Path integral quantisation can now be easily carried out,\nleading to the results summarised above in the limit $\\epsilon\\rightarrow0$.\nThe details of our modified analysis, together with various\napplications to the computation of propagators in our model, will\nappear in a forthcoming paper~\\cite{Fischer10}. As stressed\nin~\\cite{Langmann02,Fischer08} and reviewed above, the existence of the matrix basis is the crux\nof the proof of duality covariance at the \\emph{quantum} level. It also led to the\noriginal proof~\\cite{Grosse04} of renormalizability of the\nGrosse--Wulkenhaar model. From our point of view,\nthe matrix basis is taken as part of the definition of the\n(regularized) duality covariant quantum field theory. We do not know\nhow to establish quantum duality using the basis of continuum\neigenfunctions, on which the analysis of~\\cite{Zahn10} is (partly) based.\n\n\\bigskip\n\n\\section*{Acknowledgments}\n\n\\noindent\nWe thank A.P.~Balachandran, E.B.~Davies, K.~Fredenhagen,\nH.~Grosse, H.~Steinacker and J.~Zahn for helpful comments, discussions\nand correspondence. The work of RJS is supported in part by grant ST\/G000514\/1 ``String Theory\nScotland'' from the UK Science and Technology Facilities Council.\n\n\\bigskip\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Data for Topological Orders:\\\\ Fusion and Braiding}\nImagine a renormalization-group-fixed-point \ntopologically ordered quantum system\non a spacetime manifold $\\cM$. The\nmanifold can be viewed as a long-wavelength continuous limit of certain lattice regularization of the system. \nWe aim to compute the quantum amplitude from ``gluing'' one ket-state $| R\\rangle$ with another bra-state $\\langle L |$,\nsuch as $\\langle L | R\\rangle$. \nA quantum amplitude also defines a path integral or a partition function $Z$ \nwith the linking of worldlines\/worldsheets on \na $d$-manifold $\\cM^d$,\nread as\n\\begin{equation} \\label{eq:qaZ}\n\\langle L | R\\rangle=Z(\\cM^d; \\text{Link}[\\text{worldline,worldsheet}, \\dots]).\n\\end{equation}\nFor example, the $|R\\rangle$ state can\nrepresent a ground state of 2-torus $T^2_{xy}$ \nif we put the system on a solid torus $D^2_{xt} \\times S^1_y$ \\cite{TQFTGSD} \n(see Fig.\\ref{lnklp}(a) as the product space of 2-dimensional disk $D^2$ and 1-dimensional circle $S^1$). \nNote that its boundary\nis $\\partial (D^2 \\times S^1)=T^2$, and we can view the time $t$ along the radial direction.\nWe label the trivial vacuum sector without any operator insertions\nas $| 0_{D^2_{xt} \\times S^1_y} \\rangle$, which is trivial\nrespect to the measurement of any contractible line operator along $S^1_x$.\nA worldline operator creates a pair of\nanyon and anti-anyon at its end points, if it forms a closed loop then it can be viewed as creating \nthen annihilating a pair of anyons\nin a closed trajectory \\cite{beyond_gauge}. \nInserting a line operator $W^{S^1_y}_{\\si}$ in the interior of $D^2_{xt} \\times S^1_y$ gives a new state \n$W^{S^1_y}_{\\si} | 0_{D^2_{xt} \\times S^1_y} \\rangle \\equiv | \\si_{D^2_{xt} \\times S^1_y} \\rangle$.\nHere ${\\si}$ denotes the anyon\ntype \\cite{Representation} along the oriented line, see Fig.\\ref{lnklp}.\nInsert all possible line operators of all $\\si$ can completely span the ground state sectors for 2+1D topological order.\nThe gluing of\n$\n\\< 0_{D^2_{} \\times S^1_{}} |0_{D^2_{} \\times S^1_{}}\\>\n$\ncomputes the path integral\n$Z(S^2 \\times S^1)$.\nIf we view the $S^1$ as a compact time,\nthis counts the ground state degeneracy (GSD) on \na 2D\nspatial sphere $S^2$ without quasiparticle insertions,  \nthus it is a 1-dimensional Hilbert space with $Z(S^2 \\times S^1)=1$.\nSimilar relations hold for other dimensions, e.g. 3+1D topological orders on a $S^3$ without quasi-excitation\nyields\n$\n\\< 0_{D^3 \\times S^1} |0_{D^3 \\times S^1}\\>=Z(S^3 \\times S^1)=1\n$.\n\n\n\n\n\n\\se{2+1D Data}--\nIn 2+1D, we\nconsider the worldline operators creating particles.\nWe define the \\emph{fusion data} via fusing worldline operators:\n\\begin{eqnarray}\n\\label{GW}\nW^{S^1_y}_{\\si_1} W^{S^1_y}_{\\si_2}= \\mathcal{F}^\\si_{\\si_1\\si_2} W^{S^1_y}_{\\si}, \\;\\;\\text{and }\nG^\\al_\\si \\equiv \\<\\al|  \\si_{D^2_{xt} \\times S^1_y} \\rangle. \\;\\;\n\\end{eqnarray}\nHere $G^\\al_\\si$ is read from the projection to a\ncomplete basis\n$ \\<\\al|$.\nIndeed the $W^{S^1_y}_{\\si}$\ngenerates all the canonical bases from $|0_{D^2_{xt} \\times S^1_y}\\>$.\nThus the canonical projection can be \n$\n\\<\\al|=\\<0_{D^2_{xt} \\times S^1_y}| (W^{S^1_y}_{\\al})^\\dagger=\\<0_{D^2_{xt} \\times S^1_y}| (W^{S^1_y}_{\\bar{\\al}})\n=\\< \\al_{D^2_{xt} \\times S^1_y}| ,\n$\nthen we have\n$\nG^\\al_\\si \n =\\< 0_{D^2_{xt} \\times S^1_y} | (W^{S^1_y}_{\\bar{\\al}}) W^{S^1_y}_{\\si} |0_{D^2_{xt} \\times S^1_y}\\>\n=Z(S^2 \\times S^1; {\\bar{\\al}},\\si)=\\del_{\\al\\si}$,\nwhere a pair of particle-antiparticle $\\si$ and ${\\bar{\\si}}$ can fuse to the vacuum.\nWe derive\n\\begin{eqnarray}\n&&\\mathcal{F}^\\al_{\\si_1\\si_2}\n=\\< 0_{D^2_{xt} \\times S^1_y} | (W^{S^1_y}_{\\bar{\\al}}) W^{S^1_y}_{\\si_1} W^{S^1_y}_{\\si_2}|0_{D^2_{xt} \\times S^1_y}\\> \\nonumber \\\\\n&&=Z(S^2 \\times S^1; {\\bar{\\al}},\\si_1,\\si_2) \\equiv \\cN^\\al_{\\si_1\\si_2},\n\\end{eqnarray}\n\\cblue{where this path integral\ncounts the dimension of the Hilbert space (namely the GSD or the number of channels\n$\\si_1$ and $\\si_2$ can fuse to $\\al$) on the spatial $S^2$.\nThis shows the fusion data $\\mathcal{F}^\\al_{\\si_1\\si_2}$ is \nequivalent to the fusion rule $\\cN^\\al_{\\si_1\\si_2}$, symmetric under exchanging $\\si_1$ and ${\\si_2}$.} \n\\begin{comment}\nSince this path integral\ncounts the dimension of the Hilbert space on the spatial $S^2$ (namely the number of channels\n$\\si_1$ and $\\si_2$ can fuse to $\\al$),\nthis shows the fusion data $\\mathcal{F}^\\al_{\\si_1\\si_2}$ is \nequivalent to the fusion rule $\\cN^\\al_{\\si_1\\si_2}$, symmetric under exchanging $\\si_1$ and ${\\si_2}$.\n\\end{comment}\n\\begin{comment}\nFrom\n$W^y_{\\si_1} W^y_{\\si_2}|0_{D^2_{xt} \\times S^1_y}\\> = \\mathcal{F}^\\al_{\\si_1\\si_2} W^y_{\\al}|0_{D^2_{xt} \\times S^1_y}\\>$, \nwe have \n{\n\\begin{eqnarray}\n&&W^y_{\\si_1} W^y_{\\si_2} = \\mathcal{F}^\\al_{\\si_1\\si_2} W^y_{\\al}\n=\\cN^\\al_{\\si_1\\si_2} W^y_{\\al}.\n\\end{eqnarray}\nSo far, we also have (use $0$ to denote the vacuum sector):\n\\begin{eqnarray}\n&& G^\\al_\\si =Z(S^2 \\times S^1; {\\bar{\\al}},\\si)=\\del_{\\al\\si}=\\cN^{0}_{\\bar{\\al} \\si}=\\cN^{{\\al}}_{0 \\si}\\\\\n&&\\mathcal{F}^\\al_{\\si_1\\si_2}=Z(S^2 \\times S^1; {\\bar{\\al}},\\si_1,\\si_2)=\\cN^\\al_{\\si_1\\si_2} \\\\\n&& G^0_0= \\mathcal{F}^0_{00}=Z(S^2 \\times S^1)=1=\\cN^0_{0 0}\n\\end{eqnarray}\n}\n\\end{comment}\n\nMore generally we can glue the $T^2_{xy}$-boundary of $D^2_{xt} \\times S^1_y$ via its mapping class group (MCG), namely $\\MCG(T^2)=\\SL(2,\\Z)$\ngenerated by\n\\begin{eqnarray}\n\\hat{\\cS}=\\bpm 0&-1\\\\1&0\\epm,\\;\\; \\; \\hat{\\cT}=\\bpm 1&1\\\\0&1\\epm.\n\\end{eqnarray}\nThe $\\hat{\\cS}$ identifies $(x,y) \\to (-y, x)$ while $\\hat{\\cT}$ identifies $(x,y) \\to (x+y, y)$ of $T^2_{xy}$. \n\\begin{comment}\nThen we have the gluing surgery\n$D^2_{} \\times S^1_{} \\cup_{T^2; \\hat \\cS} D^2_{} \\times S^1_{} =S^3$\nand\n$D^2_{} \\times S^1_{} \\cup_{T^2; \\hat \\cT} D^2_{} \\times S^1_{}=S^2 \\times S^1$.\n\\end{comment}\nBased on Eq.(\\ref{eq:qaZ}), we write down\nthe quantum amplitudes of the two $\\SL(2,\\Z)$ generators $\\hat{\\cS}$ and $\\hat{\\cT}$\nprojecting to degenerate ground states.\nWe denote gluing two open-manifolds $\\cM_1$ and $\\cM_2$ along their boundaries $\\cB$ under the MCG-transformation $\\hat{\\cal U}$ to a new manifold\nas $\\cM_1 \\cup_{\\cB; \\hat{\\cal U}} \\cM_2$ \\cite{gluing}.\nThen it is amusing \nto visualize\nthe gluing\n$D^2_{} \\times S^1_{} \\cup_{T^2; \\hat \\cS} D^2_{} \\times S^1_{} =S^3$\nshows \nthat \nthe $\\cS_{\\bar\\si_1\\si_2}$\nrepresents the Hopf link of two $S^1$ worldlines $\\si_1$ and $\\si_2$ (e.g. Fig.\\ref{lnklp}(b)) in $S^3$ \nwith the given orientation (in the canonical basis \n$\\cS_{\\bar\\si_1\\si_2}=\\<{\\si_1}_{} | \\hat\\cS | {\\si_2}_{} \\>$):\n\\begin{eqnarray}\n\\label{Sgaga}\n\\cS_{\\bar\\si_1\\si_2}  &\n\\equiv \\<{\\si_1}_{D^2_{xt}\\times S^1_y} | \\hat\\cS | {\\si_2}_{D^2_{xt}\\times S^1_y} \\>\n=Z \\bpm \\includegraphics[scale=0.3]{S3ll12_uncut_l.pdf} \\includegraphics[scale=0.3]{S3_top.pdf}\n \\epm    \n.  \\;\\;\\;\\;\\;\n\\end{eqnarray}\nUse the gluing $D^2_{} \\times S^1_{} \\cup_{T^2; \\hat \\cT} D^2_{} \\times S^1_{}=S^2 \\times S^1$,\nwe can derive a well known result written in the canonical bases, \n\\begin{eqnarray}\n\\cT_{\\si_1\\si_2} \\equiv  \\<{\\si_1}_{D^2_{xt}\\times S^1_y} | \\hat\\cT | {\\si_2}_{D^2_{xt}\\times S^1_y} \\>\n=\\del_{\\si_1\\si_2}\\mathrm{e}^{\\ii \\th_{\\si_2}}.\\;\\;\\;\\;\\;\n\\end{eqnarray}\nIts spacetime configuration is that two unlinked closed worldlines $\\si_1$ and $\\si_2$,\nwith the worldline $\\si_2$ twisted by $2\\pi$.\nThe amplitude of a twisted worldline\nis given by the amplitude of untwisted worldline multiplied by  \n$\\mathrm{e}^{\\ii \\th_{\\si_2}}$, where \n$\\th_\\si\/2\\pi$ is the spin of the $\\si$ excitation.\nIt means that $\\cS^\\text{}_{\\bar\\si_1\\si_2}$ measures\nthe \\emph{mutual braiding statistics} of $\\si_1$-and-$\\si_2$,\nwhile $\\cT_{\\si \\si}$ measures the \\emph{spin} and \\emph{self-statistics} of $\\si$.\n\nWe can introduce additional data, the Borromean rings (BR) linking between three $S^1$ circles in $S^3$, \nwritten as\n$Z \\bpm \\includegraphics[scale=0.3]{2+1D_Borromean_mid_123_l.pdf} \\epm$$\\cblue{\\equiv}Z[S^3; \\text{BR}[\\sigma_1,\\sigma_2,\\sigma_3]]$.\nAlthough we do not know a bra-ket expression for this amplitude,\nwe can reduce this configuration to an easier one \n$Z[T^3_{xyt};\\sigma'_{1x},\\sigma'_{2y},\\sigma'_{3t}]$,\na path integral\nof 3-torus\nwith three orthogonal line operators each inserting along a non-contractible $S^1$ direction.\nThe later is a simpler expression because \nwe can uniquely \ndefine the three line insertions exactly along the homology group generators of $T^3$, namely\n$H_1(T^3,\\Z)=\\Z^3$. \nThe two path integrals\nare related by three consecutive modular $\\cS$ surgeries done along the $T^2$-boundary of $D^2 \\times S^1$ tubular neighborhood around three $S^1$ rings \\cite{S3T3}.\nNamely,\n$Z[T^3_{xyt};\\sigma'_{1x},\\sigma'_{2y},\\sigma'_{3t}]=\n\\underset{\\small{\\sigma_{1},\\sigma_{2},\\sigma_{3}}}{\\sum}\n\\cS_{ \\sigma'_{1x} \\sigma_1}\n\\cS_{ \\sigma'_{2y} \\sigma_2}\n\\cS_{ \\sigma'_{3z} \\sigma_3} Z[S^3; \\text{BR}[\\sigma_1,\\sigma_2,\\sigma_3]].$\n\n\n\n\n\n\\se{3+1D Data}--\nIn {3+1D}, there are intrinsic meanings of braidings of string-like excitations.\nWe need to consider both the worldline and the worldsheet operators which create particles and strings.\nIn addition to the $S^1$-worldline operator $W^{S^1}_{\\si}$,\nwe introduce $S^2$- and $T^2$-worldsheet operators as $V_{\\mu}^{S^2}$ and $V_{\\mu'}^{T^2}$ which create closed-strings (or loops) at their spatial cross sections. \nWe consider the vacuum sector ground state on open 4-manifolds:\n$| 0_{D^3 \\times S^1} \\rangle $,\n$| 0_{D^2 \\times S^2} \\rangle$, $| 0_{D^2 \\times T^2} \\rangle$ and $| 0_{S^4 \\- D^2 \\times T^2} \\rangle$,\nwhile their boundaries are $\\partial({D^3 \\times S^1})=\\partial({D^2 \\times S^2})=S^2 \\times S^1$\nand $\\partial({D^2 \\times T^2})=\\partial({S^4 \\- D^2 \\times T^2})=T^3$.\nHere ${\\cM_1 \\- \\cM_2}$ means the complement space of $\\cM_2$ out of $\\cM_1$. \nSimilar to 2+1D, we define the \\emph{fusion data} $\\mathcal{F}^{M}$ by fusing operators:\n\\begin{eqnarray}\n\\label{F3+1DS1}\n&&W^{S^1}_{\\si_1} W^{S^1}_{\\si_2}= (\\mathcal{F}^{S^1})^\\si_{\\si_1\\si_2} W^{S^1}_{\\si}, \\\\\n&&V^{S^2_{}}_{\\mu_1} V^{S^2_{}}_{\\mu_2}=  (\\mathcal{F}^{S^2})_{{\\mu_1}{\\mu_2}}^{\\mu_3} V^{S^2_{}}_{\\mu_3}, \\label{F3+1DS2} \\\\\n&&V^{T^2_{}}_{\\mu_1} V^{T^2_{}}_{\\mu_2}=  (\\mathcal{F}^{T^2})_{{\\mu_1}{\\mu_2}}^{\\mu_3} V^{T^2_{}}_{\\mu_3}. \\label{F3+1DT2}\n\\end{eqnarray}\nNotice that we introduce additional upper indices in the fusion algebra $\\mathcal{F}^{M}$ to specify the topology of $M$ for the fused operators \\cite{FM}.\nWe require normalizing worldline\/sheet operators for a proper basis, \nso that the $\\mathcal{F}^{M}$ is also properly normalized in order for $Z(  Y^{d-1} \\times S^1; \\dots )$ as the GSD on\na spatial closed manifold $Y^{d-1}$ always be a positive integer.\nIn principle,\nwe can derive the fusion rule\nof excitations in any closed spacetime 4-manifold.\nFor instance, the fusion rule for fusing three particles on a spatial $S^3$ is\n$Z(S^3 \\times S^1; {\\bar{\\al}},\\si_1,\\si_2)\n=\\langle 0_{D^3 \\times S^1} | W^{S^1}_{\\bar{\\al}}  W^{S^1}_{\\sigma_1} W^{S^1}_{\\sigma_2} | 0_{D^3 \\times S^1} \\rangle=\n(\\mathcal{F}^{S^1})^\\alpha_{\\si_1  \\sigma_2 }$.\nMany more examples of fusion rules can be derived from\ncomputing  $Z(\\cM^4; {\\si}, {\\mu}, \\dots)$ \\cite{index}\nby using $\\mathcal{F}^{M}$ and Eq.(\\ref{eq:qaZ}),\nhere the worldline and worldsheet are submanifolds \\emph{parallel not linked}\n with each other.\n\\begin{comment}\nOverall we choose the operators $W^{S^1}_{\\sigma}$ and $V^{M^2}_{\\mu}$\nso that\nthey generate linear-independent states when acting on the $|0_{M'} \\rangle$ state.\n\\end{comment}\n\n\nIf the worldline and worldsheet are linked as Eq.(\\ref{eq:qaZ}), then the path integral\nencodes \nthe \\emph{braiding data}. Below we discuss the important braiding processes in 3+1D.\nFirst, the Aharonov-Bohm particle-loop braiding  \ncan be represented as a $S^1$-worldline of particle and a $S^2$-worldsheet of loop linked in $S^4$ spacetime,\n\\begin{align}\n\\label{S2S1glue}\n{\\tL}^{(S^2,S^1)}_{ \\mu \\sigma}\n\\equiv \\langle 0_{D^2 \\times S^2} | V_{\\mu}^{S^2_{}\\dagger} W^{S^1}_{\\si} | 0_{D^3 \\times S^1} \\rangle\n=Z \\bpm \\includegraphics[scale=0.32]{Link_S2_S1_in_S4.pdf} \\epm,   \n\\end{align}\nif we design the worldline and worldsheet along the generators of the first and the second homology group $H_1(D^3 \\times S^1,\\Z)=H_2(D^2 \\times S^2,\\Z)=\\Z$ respectively \nvia Alexander duality.\nWe also use the fact $ S^2_{} \\times D^2_{}  \\cup_{S^2 \\times S^1}  D^3_{} \\times S^1_{} =S^4$,\nthus $\\langle 0_{D^2 \\times S^2 } | 0_{D^3 \\times S^1} \\rangle= Z(S^4)$.\nSecond, we can also consider\nparticle-loop braiding  \nas a $S^1$-worldline of particle and a $T^2$-worldsheet \n(below $T^2$ drawn as a $S^2$ with a handle) \nof loop linked in $S^4$,\n\\begin{eqnarray}\n\\label{T2S1glue}\n \\langle  0_{D^2 \\times T^2}  | V_{\\mu}^{T^2_{}\\dagger} W^{S^1}_{\\si}   | 0_{S^4 \\- D^2 \\times T^2} \\rangle\n=Z \\bpm \\includegraphics[scale=0.35]{3+1D_T2_S1_mid.pdf} \\epm,\\;\\;\\;\\;\\;\n\\end{eqnarray}\nif we design the worldline and worldsheet along the generators of $H_1({S^4 \\- D^2 \\times T^2},\\Z)=H_2(D^2 \\times T^2,\\Z)=\\Z$ respectively. \nCompare Eqs.(\\ref{S2S1glue}) and (\\ref{T2S1glue}),\n\n the loop excitation of $S^2$-worldsheet\n is shrinkable \\cite{nocharge},\nwhile the loop of $T^2$-worldsheet needs not to be shrinkable. \n\nThird, \nwe can \nrepresent \na three-loop braiding process \\cite{Wang:2014xba, Jiang:2014ksa, Moradi:2014cfa, Wang:2014oya,Jian:2014vfa, Bi:2014vaa}\nas three $T^2$-worldsheets\n \\emph{triple-linking} \\cite{carter2004surfaces} in the spacetime $S^4$ (as the first figure in Eq.(\\ref{T2T2T2glue})).\nWe find that\n\\begin{eqnarray} \\label{T2T2T2glue}\n&&\n{\\tL^{\\text{Tri}}_{{\\mu_3}, {\\mu_2}, {\\mu_1}}}\n\\equiv\n\\langle 0_{S^4 \\- {D^2_{wx} \\times T^2_{yz} }}| V^{T^2_{zx} \\dagger}_{\\mu_3} V^{T^2_{xy} \\dagger}_{\\mu_2} V^{T^2_{yz}}_{\\mu_1} | 0_{{D^2_{wx} \\times T^2_{yz} }} \\rangle \\nonumber\\\\\n&&=Z \\bpm \\includegraphics[scale=0.35]{3+1D_Triple_link_middle_123.pdf} \\epm\n=Z \\bpm \\includegraphics[scale=0.35]{3+1D_Triple_link_mid_SpinHopfLink_Large_S2_123_l.pdf} \\epm,\n\\end{eqnarray}\nwhere we design the worldsheets $V^{T^2_{yz}}_{\\mu_1}$ along the generator of homology group $H_2(D^2_{wx} \\times T^2_{yz},\\Z)=\\Z$\nwhile we design  $V^{T^2_{xy} \\dagger}_{\\mu_2}$ and $V^{T^2_{zx} \\dagger}_{\\mu_3}$  along the two generators of $H_2({S^4\\- {D^2_{wx} \\times T^2_{yz}}},\\Z)=\\Z^2$ respectively. \n\\cblue{We find that Eq.(\\ref{T2T2T2glue})\nis also equivalent to the\nspun surgery construction\nof a Hopf link (denoted as $\\mu_2$ and $\\mu_3$) \nlinked by a third $T^2$-torus (denoted as $\\mu_1$) \\cite{Jian:2014vfa, Bi:2014vaa}. \nNamely, we can view the above figure as \na Hopf link of two loops\nspinning along the dotted path of a $S^1$ circle, which\n becomes a pair of $T^2$-worldsheets $\\mu_2$ and $\\mu_3$.\nAdditionally the $T^2$-worldsheet $\\mu_1$ \n(drawn in gray as a $S^2$ added a thin handle), \ntogether with $\\mu_2$ and $\\mu_3$, the three worldsheets have a \ntriple-linking topological invariance \\cite{carter2004surfaces}.\n}\n\\begin{comment}\nWe find that Eq.(\\ref{T2T2T2glue})\nis also equivalent to the\nspun surgery construction \\cite{Spunsurgery} of a Hopf link linked by a third $T^2$-torus,\nshown as the second figure in Eq.(\\ref{T2T2T2glue}).\n\\end{comment}\n\n\nFourth, the four-loop braiding process,\nwhere three loops dancing in the Borromean ring trajectory while linked by a fourth loop \\cite{PhysRevB.91.165119},\ncan characterize certain 3+1D non-Abelian topological orders \\cite{Wang:2014oya}. \nWe find it is also the spun surgery\nconstruction of Borromean rings of three loops linked by a fourth torus in the spacetime picture,\nand\nits path integral\n$Z[S^4; \\text{Link[Spun[BR}[\\mu_4 ,\\mu_3, \\mu_2]],\\mu_1]]$ can be transformed:\n\\begin{eqnarray} \\label{ZSpinBRS4}\n&& \n Z \\bpm \\includegraphics[scale=0.37]{3+1D_SpinBorromean_mid_S2_1234.pdf} \\epm\n  \\xrightarrow{\\;\\text{surgery}\\;}\nZ[T^4 \\# S^2  \\times S^2; \\mu_4',\\mu_3',\\mu_2',\\mu_1' ] \\nonumber \\\\\n&&=\\langle 0_{T^4 \\# S^2 \\times S^2 \\- {D^2_{wx} \\times T^2_{yz}}} |  V^{T^2\\dagger}_{\\mu_4' } V^{T^2\\dagger}_{\\mu_3' }  V^{T^2\\dagger}_{\\mu_2' } V^{T^2_{yz}}_{\\mu_1' } |0_{D^2_{wx} \\times T^2_{yz}} \\rangle,\\;\\;\\;\\;\\;\\;\\;\\;\n\\end{eqnarray}\nwhere the surgery \ncontains four consecutive modular $\\cS$-transformations done along the $T^3$-boundary of $D^2 \\times T^2$ tubular neighborhood around four $T^2$-worldsheets \\cite{S4T4}.\nThe final spacetime manifold is $T^4 \\# S^2  \\times S^2$, where $\\#$ stands for the connected sum.\n\nWe can glue the $T^3$-boundary of 4-submanifolds (e.g. $D^2 \\times T^2$ and ${S^4 \\- D^2 \\times T^2}$) via $\\MCG(T^3)=\\SL(3,\\Z)$ \ngenerated by  \\cite{3+1DST}\n\\begin{eqnarray}\n\\hat{\\cS}^{xyz}=\\bpm 0& 0&1 \\\\1& 0&0 \\\\0& 1&0\\epm, \\;\\;\\; \\hat{\\cT}^{xy}=\\bpm 1&1 &0\\\\0&1 & 0 \\\\0 &0 & 1\\epm.\n\\end{eqnarray}\n\\begin{comment}\nIn this work, we define  \\cite{3+1DST}  ${\\cS^{xyz}_{\\mu_2, \\mu_1}} \\equiv \n\\< 0_{D^2_{xw} \\times T^2_{yz}} | V^{T^2_{yz} \\dagger}_{\\mu_2}  \\hat{\\cS}^{xyz}  V^{T^2_{yz}}_{\\mu_1}  | 0_{D^2_{xw} \\times T^2_{yz}} \\rangle$\nas a spun-Hopf link in $S^3 \\times S^1$, and\n$ {\\cT^{xy}_{\\mu_2, \\mu_1}} \n=\\< 0_{D^2_{xw} \\times T^2_{yz}} | V^{T^2_{yz} \\dagger}_{\\mu_2}  \\hat{\\cT}^{xy}  V^{T^2_{yz}}_{\\mu_1}  | 0_{D^2_{xw} \\times T^2_{yz}} \\rangle$\nis related to the \\emph{topological spin} and \\emph{self-statistics} of closed strings \\cite{Wang:2014oya}.\n\\end{comment}\nIn this work, we define their representations as \\cite{3+1DST} \n\\begin{eqnarray}\n&& \\label{eq:Sxyz}\n{\\cS^{xyz}_{\\mu_2, \\mu_1}} \\equiv \n\\< 0_{D^2_{xw} \\times T^2_{yz}} | V^{T^2_{yz} \\dagger}_{\\mu_2}  \\hat{\\cS}^{xyz}  V^{T^2_{yz}}_{\\mu_1}  | 0_{D^2_{xw} \\times T^2_{yz}} \\rangle, \\\\\n&& {\\cT^{xy}_{\\mu_2, \\mu_1}} \n\\equiv \\< 0_{D^2_{xw} \\times T^2_{yz}} | V^{T^2_{yz} \\dagger}_{\\mu_2}  \\hat{\\cT}^{xy}  V^{T^2_{yz}}_{\\mu_1}  | 0_{D^2_{xw} \\times T^2_{yz}} \\rangle,\n\\end{eqnarray}\nwhile $\\cS^{xyz}$ is a spun-Hopf link in $S^3 \\times S^1$, and ${\\cT}^{xy}$ is related to the \\emph{topological spin} and \\emph{self-statistics} of closed strings \\cite{Wang:2014oya}.\n\n\n\\se{{Quantum surgery and general Verlinde formulas}} --\nNow we like to derive a powerful identity for fixed-point path integrals\nof topological orders. \nIf the path integral\nformed by disconnected manifolds\n$M$ and $N$, denoted as $M\\sqcup N$,\nwe have\n$\nZ(M\\sqcup N)=Z(M)Z(N)\n$.\nAssume that\n(1) we divide both $M$ and $N$ into two pieces such that\n$M=M_U\\cup_{B} M_D$, $N=N_U\\cup_{B} N_D$,\nand their cut\ntopology (dashed $B$)\nis equivalent $B={\\prt M_D}={\\prt M_U}={\\prt N_D}={\\prt N_U}$,\nand (2) the Hilbert space  on the spatial slice is 1-dimensional (namely the GSD=1)\\cite{GSD1}, \nthen we obtain\n\\begin{eqnarray}\n\\label{ZMN}\n&& Z \\bpm \\includegraphics[scale=0.33]{MNB} \\epm \n=\n Z \\bpm \\includegraphics[scale=0.33]{MNB1} \\epm \n\\\\\n\\Rightarrow\n\n&&Z(M_U \\cup_B M_D) \nZ(N_U \\cup_B N_D) \n\\nonumber\\\\\n&&\n =\nZ( N_U\\cup_B  M_D) \nZ(M_U \\cup_B N_D ). \\nonumber\n\\end{eqnarray}\nIn 2+1D, we can derive the\nrenowned Verlinde formula \\cite{Witten:1988hf,  Verlinde:1988sn, Moore:1988qv}\nby one version of Eq.(\\ref{ZMN}):\n\\begin{eqnarray}\n\\label{Z2Dcut}\n Z \\bpm \\includegraphics[scale=0.26]{S3l_lb} \\epm \n Z \\bpm \\includegraphics[scale=0.26]{S3lll_lb} \\epm \n&=&\n Z \\bpm \\includegraphics[scale=0.26]{S3ll12_lb} \\epm \n Z \\bpm \\includegraphics[scale=0.26]{S3ll13_lb} \\epm \\nonumber\\\\\n \\Rightarrow \\cS^\\text{}_{\\bar\\si_10}\\sum_{\\si_4} \\cS^\\text{}_{\\bar\\si_1\\si_4}   \\cN^{\\si_4}_{\\si_2\\si_3} \n&=&\n\\cS^\\text{}_{\\bar\\si_1\\si_2}\n\\cS^\\text{}_{\\bar\\si_1\\si_3}, \n\\end{eqnarray}\nwhere each spacetime manifold is $S^3$, with the line operator insertions such as an unlink and Hopf links.\nEach $S^3$ is cut into two $D^3$ pieces,\nso $D^3 \\cup_{S^2} D^3={S^3}$, while the boundary dashed cut is $B=S^2$.\nThe GSD for this spatial section $S^2$ with a pair of particle-antiparticle must be 1, so our surgery satisfies the assumptions for Eq.(\\ref{ZMN}).\nThe second line is derived from rewriting path integrals\nin terms of our data introduced before -- the fusion rule $\\cN^{\\si_4}_{\\si_2\\si_3}$ comes from \nfusing ${\\si_2\\si_3}$ into ${\\si_4}$ which Hopf-linked with ${\\si_1}$, while Hopf links render $\\cS$ matrices \\cite{Verlinde}.\n\\cblue{\nThe label $0$, in $\\cS^\\text{}_{\\bar\\si_10}$ and hereafter, denotes a vacuum sector without operator insertions in a submanifold.\n}\n\nIn 3+1D, the particle-string braiding in terms of $S^4$-spacetime path integral Eq.(\\ref{S2S1glue}) has constraint formulas:\n\\begin{widetext}\n\\begin{eqnarray}\n&&\\label{eq:S2S1S1inS4}\n Z \\bpm \\includegraphics[scale=0.35]{Link_S2_in_S4_lb.pdf}\\epm \n  Z \\bpm \\includegraphics[scale=0.35]{Link_S2_S1_S1_in_S4_lb.pdf} \\epm\n=\n Z \\bpm \\includegraphics[scale=0.35]{Link_S2S1_a_S4_lb.pdf} \\epm\n Z \\bpm \\includegraphics[scale=0.35]{Link_S2S1_b_S4_lb.pdf} \\epm\n \\Rightarrow\n {\n{\\tL}^\\text{($S^2$,$S^1$)}_{\\mu_1 0}\n\\sum_{\\si_4}  {\\tL}^\\text{($S^2$,$S^1$)}_{\\mu_1\\si_4}  (\\mathcal{F}^{S^1})^{\\si_4}_{\\si_2\\si_3}\n=\n{\\tL}^\\text{($S^2$,$S^1$)}_{\\mu_1\\si_2}\n{\\tL}^\\text{($S^2$,$S^1$)}_{\\mu_1\\si_3}} \n .\\\\\n&&\\label{eq:S1S2S2inS4}\n  Z \\bpm \\includegraphics[scale=0.35]{Link_S1_in_S4_lb.pdf} \\epm \n Z \\bpm \\includegraphics[scale=0.35]{Link_S1_S2_S2_in_S4_lb.pdf} \\epm \n=\n Z \\bpm \\includegraphics[scale=0.35]{Link_S1S2_a_S4_lb.pdf} \\epm \n Z \\bpm \\includegraphics[scale=0.35]{Link_S1S2_b_S4_lb.pdf} \\epm \n \\Rightarrow \n  {\n{\\tL}^\\text{($S^2$,$S^1$)}_{0 \\si_1}\n\\sum_{\\mu_4}  {\\tL}^\\text{($S^2$,$S^1$)}_{\\mu_4 \\si_1}  \n(\\mathcal{F}^{S^2})^{\\mu_4}_{\\mu_2\\mu_3}\n=\n{\\tL}^\\text{($S^2$,$S^1$)}_{\\mu_2 \\si_1}\n{\\tL}^\\text{($S^2$,$S^1$)}_{\\mu_3 \\si_1}}. \\;\\;\\;\\;\\;\\;\\;\\;\n \\end{eqnarray}\n\\cblue{Here the gray areas mean $S^2$-spheres.\nAll the data are well-defined in Eqs.(\\ref{F3+1DS1}),(\\ref{F3+1DS2}),(\\ref{S2S1glue})}. \nNotice that Eqs.(\\ref{eq:S2S1S1inS4}) and (\\ref{eq:S1S2S2inS4}) are symmetric by exchanging worldsheet\/worldline indices: $\\mu \\leftrightarrow \\sigma$,\nexcept that the fusion data is different: $\\mathcal{F}^{S^1}$ fuses worldlines, while $\\mathcal{F}^{S^2}$ fuses worldsheets.\n\nWe also derive a quantum surgery constraint formula \\cite{Supple}\nfor the three-loop braiding in terms of $S^4$-spacetime path integral Eq.(\\ref{T2T2T2glue}) via the ${\\cS}^{xyz}$-surgery and its matrix representation:\n\\begin{eqnarray} \n\\label{eq:Spin[HopfLink]S4}\n&&\\ \\ \\ \\\n  Z \\bpm \\includegraphics[scale=0.45]{3+1D_T2_S1_0_lb.pdf} \\epm \n Z \\bpm \\includegraphics[scale=0.45]{3+1D_Triple_link_double_SpinHopfLink_S2_double_12345_lb.pdf} \\epm \n_1=\n Z \\bpm \\includegraphics[scale=0.45]{3+1D_Triple_link_double_SpinHopfLink_S2_Up_123_lb.pdf} \\epm \n Z \\bpm \\includegraphics[scale=0.45]{3+1D_Triple_link_double_SpinHopfLink_S2_Down_145_lb.pdf} \\epm \\nonumber \\\\\n&& \\Rightarrow\n {{\\tL^{\\text{Tri}}_{0, 0, {\\mu_1}}} \\cdot\n \\sum_{{\\Gamma, \\Gamma'},{\\Gamma_1, \\Gamma_1'}}\n (\\mathcal{F}^{T^2})_{{\\zeta_2},{\\zeta_4}}^{\\Gamma}\n{(\\cS^{xyz})^{-1}_{\\Gamma', \\Gamma}}  (\\mathcal{F}^{T^2})_{{\\mu_1} {\\Gamma'}}^{\\Gamma_1}  {\\cS^{xyz}_{\\Gamma_1', \\Gamma_1}}  \\; {\\tL^{\\text{Tri}}_{0, 0, {\\Gamma_1'}}} } \\nonumber\n\\\\\n&&\n{=\n{\\sum_{{\\zeta_2'}, {\\eta_2}, {\\eta_2'}} \n{(\\cS^{xyz})^{-1}_{\\zeta_2', {\\zeta_2}}}  (\\mathcal{F}^{T^2})_{{\\mu_1} {\\zeta_2'}}^{\\eta_2}  {\\cS^{xyz}_{\\eta_2', \\eta_2}}  \\; {\\tL^{\\text{Tri}}_{0, 0, {\\eta_2'}}} } \n\\cdot\n{\\sum_{{\\zeta_4'}, {\\eta_4}, {\\eta_4'}}  \n{(\\cS^{xyz})^{-1}_{\\zeta_4', {\\zeta_4}}}  (\\mathcal{F}^{T^2})_{{\\mu_1} {\\zeta_4'}}^{\\eta_4}  {\\cS^{xyz}_{\\eta_4', \\eta_4}}  \\; {\\tL^{\\text{Tri}}_{0, 0, {\\eta_4'}}} }}, \n\\end{eqnarray}\n\\end{widetext}\n\\cblue{here the ${\\mu_1}$-worldsheet in gray represents a $T^2$ torus,\nwhile ${\\mu_2}$-${\\mu_3}$-worldsheets and ${\\mu_4}$-${\\mu_5}$-worldsheets are both a pair of\ntwo $T^2$ tori obtained by spinning the Hopf link.}\n\\cblue{All our data\nare well-defined in Eqs.(\\ref{F3+1DT2}),(\\ref{T2T2T2glue}),(\\ref{eq:Sxyz}) introduced earlier.\nFor example, the ${\\tL^{\\text{Tri}}_{0, 0, {\\mu_1}}}$ is defined in Eq.(\\ref{T2T2T2glue})\nwith 0 as a vacuum without insertion, so ${\\tL^{\\text{Tri}}_{0, 0, {\\mu_1}}}$ is a\npath integral of a $T^2$ worldsheet ${\\mu_1}$ in $S^4$.\n}\n\\cblue{The index ${\\zeta_2}$ is obtained from} fusing ${\\mu_2}$-${\\mu_3}$-worldsheets,\nand \\cblue{${\\zeta_4}$  is obtained from} fusing ${\\mu_4}$-${\\mu_5}$-worldsheets.\nOnly ${\\mu_1},{\\zeta_2},{\\zeta_4}$ are the fixed indices, other indices are summed over.\n\nFor all path integrals of $S^4$ in Eqs.(\\ref{eq:S2S1S1inS4}), (\\ref{eq:S1S2S2inS4}) and (\\ref{eq:Spin[HopfLink]S4}), each $S^4$ is cut into two $D^4$ pieces, \nso $D^4 \\cup_{S^3} D^4={S^4}$. \\cblue{We choose all the dashed cuts for 3+1D path integral\nrepresenting $B=S^3$, while\nwe can view the $S^3$ as a spatial slice,\nwith the following excitation configurations:\nA loop in Eq.(\\ref{eq:S2S1S1inS4}), a pair of particle-antiparticle in Eq.(\\ref{eq:S1S2S2inS4}), and a pair of loop-antiloop in Eq.(\\ref{eq:Spin[HopfLink]S4}).}\nHere we require a stronger criterion that all loop excitations are gapped without zero modes, \nthen the GSD is 1 for all above spatial section $S^3$. \nThus all our surgeries satisfy the assumptions for Eq.(\\ref{ZMN}).\n\n\\begin{comment}\nhere ${\\zeta_2}$ relates to fusing ${\\mu_2}$-${\\mu_3}$-worldsheets,\nand ${\\zeta_4}$ relates to fusing ${\\mu_4}$-${\\mu_5}$-worldsheets.\nOnly ${\\mu_1},{\\zeta_2},{\\zeta_4}$ are the fixed indices, other indices are summed over.\nFor all path integrals of $S^4$ in Eqs.(\\ref{eq:S2S1S1inS4}), (\\ref{eq:S1S2S2inS4}) and (\\ref{eq:Spin[HopfLink]S4}), each $S^4$ is cut into two $D^4$ pieces, \nso $D^4 \\cup_{S^3} D^4={S^4}$, while the dashed cut is $B=S^3$.\nHere we require a stronger criterion that all loop excitations are gapped without zero modes, \nthen the GSD is 1 for all above spatial section $S^3$ (with a loop in Eq.(\\ref{eq:S2S1S1inS4}), a pair of particle-antiparticle in Eq.(\\ref{eq:S1S2S2inS4}), and a pair of loop-antiloop in Eq.(\\ref{eq:Spin[HopfLink]S4})). \nThus all our surgeries satisfy the assumptions for Eq.(\\ref{ZMN}).\n\\end{comment}\n\nThe above Verlinde-like formulas\nconstrain the fusion data (e.g. $\\cN$, $\\mathcal{F}^{S^1}$, $\\mathcal{F}^{S^2}$, $\\mathcal{F}^{T^2}$, etc.) \nand braiding data (e.g. $\\cS$, $\\cT$, ${\\tL}^\\text{($S^2$,$S^1$)}$, ${\\tL^{\\text{Tri}}}$, $\\cS^{xyz}$, etc.). \nMoreover, we can derive constraints between the fusion data itself.\nSince a $T^2$-worldsheet contains two non-contractible $S^1$-worldlines along its two homology group generators in $H_1(T^2,\\Z)=\\Z^2$,\nthe $T^2$-worldsheet operator $V^{T^2}_{\\mu}$ contains the data of $S^1$-worldline operator $W^{S^1}_{\\si}$. \nMore explicitly, we can compute the state $W_{\\sigma_1}^{S^1_y}  W_{\\sigma_2}^{S^1_y} V_{\\mu_2}^{T^2_{yz}} | 0_{D^2_{wx} \\times T^2_{yz}} \\rangle$\nby fusing two $W^{S^1}_{\\si}$ operators and one $V^{T^2}_{\\mu}$ operator in different orders, then we obtain a consistency formula \\cite{Supple}:\n\\begin{eqnarray} \\label{eq:FS1FT2}\n\\sum_{\\sigma_3} (\\mathcal{F}^{S^1})_{{\\sigma_1}{\\sigma_2}}^{\\sigma_3}   (\\mathcal{F}^{T^2})_{{\\sigma_3}{\\mu_2}}^{\\mu_3}\n=\\sum_{\\mu_1} (\\mathcal{F}^{T^2})_{{\\sigma_2}{\\mu_2}}^{\\mu_1} (\\mathcal{F}^{T^2})_{{\\sigma_1}{\\mu_1}}^{\\mu_3}. \\;\\;\\;\\;\\;\n\\end{eqnarray}\nWe organize\nour quantum statistics data of fusion and braiding, and some explicit examples of \ntopological orders and their topological invariances in terms of our data in the Supplemental Material.\n\n\n\n\\section{Conclusion}\n\n\n\nIt is known that the quantum statistics of particles in 2+1D begets\nanyons, beyond the familiar statistics of bosons and fermions, while Verlinde formula \\cite{Verlinde:1988sn}\nplays a key role to dictate the consistent anyon statistics.\nIn this work, we derive a set of quantum surgery formulas analogous to Verlinde's constraining the fusion and braiding quantum statistics of\nanyon excitations of particle and string in 3+1D.\n\n\n\nA further advancement of our work, comparing to the pioneer work Ref.\\cite{Witten:1988hf} on 2+1D Chern-Simons gauge theory, \nis that we apply the surgery idea to generic 2+1D and 3+1D topological orders\nwithout assuming quantum field theory (QFT) or gauge theory description. \nAlthough many lattice-regularized topological orders happen to have TQFT descriptions at low energy, \nwe may not know which topological order derives which TQFT easily.\nInstead we simply use\nquantum amplitudes written in the bra and ket (over-)complete bases, obtained\nfrom inserting worldline\/sheet operators along the cycles of non-trivial homology group generators of a spacetime submanifold, \nto\ncut and glue to the desired path integrals.\nConsequently our approach, without the necessity of any QFT description, can be powerful to describe more generic quantum systems.\nWhile our result is originally based on\nstudying specific examples of TQFT in Dijkgraaf-Witten gauge theory \\cite{Dijkgraaf:1989pz,{Supple}}, \nwe formulate the\ndata without using QFT.\nWe have incorporated the necessary generic quantum statistic data and new constraints \nto characterize some 3+1D topological orders (including Dijkgraaf-Witten's),\nwe will leave the issue of their sufficiency and completeness\nfor future work. \nFormally, our approach can be applied to any spacetime dimensions.\n\n\n\nIt will be interesting to study the analogous Verlinde formula constraints for 2+1D boundary states, such as highly-entangled gapless modes,\nconformal field theories (CFT) and anomalies, for example through the bulk-boundary correspondence\n\\cite{{Witten:1988hf}, 2015PhST164a4009R, 2015arXiv150904266C, 2015arXiv151209111W}.\nThe set of consistent quantum surgery formulas\nwe derive may lead to an alternative effective way to \\emph{bootstrap} \\cite{Polyakov:1974gs,Ferrara:1973yt} 3+1D topological states of matter and 2+1D CFT.\\\\\n\n\n\\noindent\n{\\bf Note added}:\nThe formalism and some results discussed in this work have been partially reported in the first author's Ph.D. thesis \\cite{JWangthesis}.\nReaders may refer to Ref.\\cite{JWangthesis} for other discussions.\n\n\n\\section{Acknowledgements}\n\nWe are indebted to Clifford Taubes for many generous helps on the development of this work.\nJW is grateful to\nRonald Fintushel, \nRobert Gompf, Allen Hatcher, Shenghan Jiang, Greg Moore,\nNathan Seiberg, Ronald Stern, Andras Stipsicz, Brian Willet, Edward Witten and Yunqin Zheng for helpful comments, \nand to colleagues at\nHarvard University for discussions.\nJW gratefully acknowledges the Schmidt Fellowship at IAS supported by Eric and Wendy Schmidt and the NSF Grant PHY-1314311.\nThis work is supported by the NSF Grant PHY-1306313, PHY-0937443, DMS-1308244, DMS-0804454,  DMS-1159412 and\nCenter for Mathematical Sciences and Applications at Harvard University.\nThis work is also supported by NSF Grant\nDMR-1506475 and NSFC 11274192, the\nBMO Financial Group and the John Templeton Foundation No.\\  39901.  Research at\nPerimeter Institute is supported by the Government of Canada through Industry\nCanada and by the Province of Ontario through the Ministry of Research.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec1}\nIn a finite, asexually reproducing population with mutations, it is\nwell known that competition among multiple individuals that get\nbeneficial mutations can slow the rate of adaptation. This phenomenon\nis known as the Hill--Robertson effect, named for the authors of \\cite\n{HR}. One may wish to consider the effect on the rate of adaptation of\na population when there are many beneficial mutations present\nsimultaneously. It is easily observed that when such a population is\nfinite and all mutations are either neutral or deleterious, the fitness\nof the population will decrease over time. This scenario is known as\nMuller's ratchet. The first rigorous results regarding Muller's ratchet\nwere due to Haigh~\\cite{H}. In an asexually reproducing population,\nbeneficial mutations are necessary to overcome Muller's ratchet. Yu,\nEtheridge and Cuthbertson~\\cite{YEC} proposed a model that gives\ninsight into both the Hill--Robertson effect and Muller's ratchet in\nlarge populations with fast mutation rates.\n\nThe model introduced in~\\cite{YEC} is a Moran model with mutations and\nselection. There are $N$ individuals in this model, each with an\ninteger valued fitness. The dynamics of the model are determined by\nthree parameters, $\\mu$, $q$ and $\\gamma$, which are independent of\n$N$. The parameters must satisfy $\\mu> 0$, $0 < q \\leq1$ and $\\gamma\n> 0$. Let $X_t^i$ be the fitness of individual $i$ at time $t$. Then $X\n= (X^1,X^2,\\ldots, X^N)$ is a stochastic process with state space\n$\\Z^N$. The system has the following dynamics:\n\\begin{longlist}[(3)]\n\\item[(1)] Mutation: Each individual acquires mutations at rate $\\mu\n$. When individual $i$ gets a mutation, it is beneficial with\nprobability $q$ and $X^i$ increases by 1. With probability $1-q$ the\nmutation is deleterious and $X^i$ decreases by 1.\n\\item[(2)] Selection: For each pair of individuals $(i,j)$, at rate\n$\\frac{\\gamma}{N}(X^i-X^j)^+$, we set $X^j$ equal to $X^i$.\n\\item[(3)] Resampling:\\vspace*{1pt} For each pair of individuals\n$(i,j)$, at rate $1\/N$, we set $X^j$ equal to $X^i$.\n\\end{longlist}\nNote that the upper bound we establish for the rate of adaptation still\nholds in the absence of deleterious mutations, which corresponds to the\ncase $q = 1$. Under the selection mechanism the event that $X^j$ is set\nto equal $X^i$ represents the more fit individual $i$ giving birth and\nthe less fit individual $j$ dying. Likewise, the resampling event that\ncauses $X^j$ to equal $X^i$ represents individual $i$ giving birth and\nindividual $j$ dying.\n\nWe give an equivalent description of the model involving Poisson\nprocesses that may make the coupling arguments more clear. The Poisson\nprocesses that determine the dynamics of $X$ are as follows:\n\\begin{itemize}\n\\item There are $N$ Poisson processes $\\mathcal{P}^{i\\uparrow}$, $1\n\\leq i \\leq N$, on $[0,\\infty)$ of rate $q \\mu$. If $\\mathcal\n{P}^{i\\uparrow}$ gets a mark at $t$ then the $i$th coordinate of $X$\nincreases by 1 at time $t$.\n\\item There are $N$ Poisson processes $\\mathcal{P}^{i\\downarrow}$, $1\n\\leq i \\leq N$, on $[0,\\infty)$ of rate $(1-q)\\mu$. If $\\mathcal\n{P}^{i\\downarrow}$ gets a mark at $t$ then the $i$th coordinate of $X$\ndecreases by 1 at time $t$.\n\\item For each ordered pair of coordinates $(i,j)$ with $i \\neq j$\nthere is a Poisson process on $[0,\\infty)$, $\\mathcal{P}^{i,j}$, of\nrate $1\/N$. If $\\mathcal{P}^{i,j}$ gets a mark at $t$ then the $j$th\ncoordinate changes its value to agree with the $i$th coordinate at time $t$.\n\\item For each ordered pair of coordinates $(i,j)$ with $i \\neq j$\nthere is a Poisson processes on $[0,\\infty) \\times[0,\\infty)$,\n$\\mathcal{P}^{i,j\\uparrow}$, which has intensity $\\frac{\\gamma\n}{N}\\lambda$ where $\\lambda$ is Lebesgue measure on $\\R^2$. If\n$\\mathcal{P}^{i,j\\uparrow}$ gets a mark in $\\{t\\} \\times\n[0,X_{t-}^i-X_{t-}^j]$ then the $j$th coordinate changes its value to\nagree with the $i$th coordinate at time $t$.\n\\end{itemize}\n\nA heuristic argument in~\\cite{YEC} shows that as $N$ tends to infinity\nthe mean rate of increase of the average fitness of the individuals in\n$X$ is $O(\\log N\/\\break(\\log\\log N)^2)$. Due to a typo on page 989 they\nstate that the rate is $O(\\log N\/\\break\\log\\log N)$. By equation (10) in\n\\cite{YEC},\n\\[\nK\\log(\\gamma K) = 2\\log N.\n\\]\nThis implies that\n\\[\nK \\approx\\frac{2\\log N}{\\log\\log N}.\n\\]\nPlugging $2\\log N\/\\log\\log N$ into each side of the consistency\ncondition that they derive gives a rate of adaption of $O(\\log N\/(\\log\n\\log N)^2)$.\n\nThe heuristic argument is difficult to extend to a rigorous argument. Let\n\\[\n\\overline{X} = \\frac{1}{N} \\sum_{i=1}^N\nX^i\n\\]\nbe the continuous-time process which represents the average fitness of\nthe individuals in $X$. The rigorous results established in~\\cite{YEC}\nare as follows:\n\\begin{itemize}\n\\item The centered process $X^C$, in which individual $i$ has fitness\n$X^{C,i} = X^i - \\overline{X}$, is ergodic and has a stationary\ndistribution $\\pi$.\n\\item If\n\\[\nc_2 = \\frac{1}{N}\\sum_{i=1}^N\n\\bigl(X^{C,i}\\bigr)^2\n\\]\nis the variance of the centered process under the stationary\ndistribution, then\n\\[\nE^\\pi[\\overline{X}_t] = \\bigl(\\mu(2q-1)+\\gamma\nE^\\pi[c_2]\\bigr)t,\n\\]\nwhere $E^\\pi$ means that the initial configuration of $X$ is chosen\naccording to the stationary distribution $\\pi$.\n\\item For any $\\delta> 0$ there exists $N_0$ large enough so that for\nall $N \\geq N_0$ we have $E\\pi[\\overline{X}_1] \\geq\\log^{1-\\delta} N$.\n\\end{itemize}\nIt is difficult to say anything rigorous about $E^\\pi[c_2]$ so other\nmethods are needed to compute $E[\\overline{X}_t]$. The third result of\n\\cite{YEC} shows that if there is a positive ratio of beneficial\nmutations then a large enough population will increase in fitness over\ntime. A paper by Etheridge and Yu~\\cite{EY} provides further results\npertaining to this model.\n\nOther similar models can be found in the biological literature. In\nthese models the density of the particles is assumed to act as a\ntraveling wave in time. The bulk of the wave behaves approximately\ndeterministically and the random noise comes from the most fit classes\nof individuals. One tries to determine how quickly the fittest classes\nadvance and pull the wave forward. This traveling wave approach is used\nin~\\cite{YE} and~\\cite{YEC} to approximate the rate of evolution as\n$O(\\log N\/(\\log\\log N)^2)$. For other work in this direction see\nRouzine, Brunet and Wilke~\\cite{RBW}, Brunet, Rouzine and Wilke \\cite\n{BRW}, Desai and Fisher~\\cite{DF} and Park, Simon and Krug \\cite\n{PSK}. Using nonrigorous arguments, these authors get estimates of\n$O(\\log N)$, $O(\\log N\/\\log\\log N)$ and $O(\\log N\/(\\log\\log N)^2)$,\nwhere the differences depend\\vadjust{\\goodbreak} on the details of the models that they\nanalyze. For more motivation and details concerning this model, please\nsee the Introduction in~\\cite{YEC}.\n\nMotivated by applications to cancer development, Durrett and Mayberry\nhave established rigorous results for a similar model in~\\cite{DM}.\nThey consider two models in which all mutations are beneficial and the\nmutation rate tends to 0 as the population size tends to infinity. In\none of their models the population size is fixed and in the other it is\nexponentially increasing. For the model with the fixed population size\nthey show that the rate at which the average fitness is expected to\nincrease is $O(\\log N)$. By considering the expected number of\nindividuals that have fitness $k$ at time $t$, they establish\nrigorously that the density of the particles in their model will act as\na traveling wave in time.\n\nOur result is the following theorem.\n\\begin{Theo} \\label{Theorem}\nLet $X_0^i = 0$ for $1 \\leq i \\leq N$. There exists a positive constant\n$C$ which may depend on $\\mu$, $q$ and $\\gamma$ such that for $N$\nlarge enough\n\\[\n\\frac{E[\\overline{X}]}{t} \\leq\\frac{C\\log N}{(\\log\\log N)^2}\n\\]\nfor all $t \\geq\\log\\log N$.\n\\end{Theo}\n\nA difference between the result in~\\cite{YEC} and our result is that\nin~\\cite{YEC} the initial state of the process is randomly chosen\naccording to the stationary distribution $\\pi$, while we make the\nassumption that all of the individuals initially have fitness 0.\n\nThe statements of the propositions needed to prove Theorem \\ref\n{Theorem} and the proof of Theorem~\\ref{Theorem} are included in\nSection~\\ref{sec2}. At the end of the paper there is a table which\nincludes the\nnotation that is used throughout the paper and the \\hyperref\n[app]{Appendix} that includes\nsome general results on branching processes.\n\n\\section{\\texorpdfstring{Proof of Theorem \\protect\\ref{Theorem}}{Proof of Theorem 1}}\\label{sec2}\nBefore stating the propositions we use to prove the theorem we need to\nestablish some notation. Let $X_t^+ = \\max\\{X_t^i\\dvtx1 \\leq i \\leq N\\}$\nbe the maximum fitness of any individual at time $t$ and $X_t^- = \\min\n\\{X_t^i\\dvtx1 \\leq i \\leq N\\}$ be the minimum fitness of any\nindividual at\ntime $t$. Define the width of the process to be $W_t = X_t^+-X_t^-$ and\ndefine $D_t = X_t^+ - X_0^+$ be the distance the front of the process\nhas traveled by time $t$. Theorem~\\ref{Theorem} states that all\nindividuals initially have fitness~0. Therefore, a bound on $D_t$\nimmediately yields a bound on $\\overline{X}_t$. The bounds we\nestablish on $D_t$ will depend on the width, $W_t$.\n\nLet $w = w(N)$ be any positive, increasing function that satisfies\n\\[\n\\lim_{N\\rightarrow\\infty}w(N) = \\infty\n\\quad\\mbox{and}\\quad\\lim_{N\\rightarrow\n\\infty}\\frac{w(N)}{\\log\\log N}\n= 0.\n\\]\nLet $\\mathcal{W} = \\lfloor w\\log N\/\\log\\log N \\rfloor$ and $\\mathcal\n{T} = w^{-1\/2}\\log\\log N$. Heuristically, we conjecture that $W_t$ is\ntypically of size $O(\\log N\/\\log\\log N)$ so $\\mathcal{W}$ is larger\nthan the typical width of $X$. With probability\\vadjust{\\goodbreak} tending to 1, selection\nshould cause any width larger than $\\mathcal{W}$ to shrink within\n$\\mathcal{T}$ time units. Because the width is a stochastic process,\nwe are motivated to make the following definitions:\n\\begin{eqnarray*}\nt_1 &=& 0,\n\\\\[-2pt]\ns_n &=& \\inf\\{t \\geq t_n\\dvtx W_t \\geq2\n\\mathcal{W}\\} \\qquad\\mbox{for } n \\geq1,\n\\\\[-2pt]\nt_n &=& \\inf\\{t \\geq s_{n-1}\\dvtx W_t <\n\\mathcal{W}\\} \\qquad\\mbox{for } n \\geq2,\n\\\\[-2pt]\nY_i &=& \\sup_{s_i \\leq t \\leq t_{i+1}}D_t - D_{s_i}\n\\qquad\\mbox{for }i \\geq1,\n\\\\[-2pt]\nN_t &=& \\max\\{i\\dvtx s_i \\leq t\\} \\qquad\\mbox{for }t \\geq0.\n\\end{eqnarray*}\nNote that $s_n$ and $t_n$ exist for all $n \\geq1$ with probability 1.\n\nWe define branching processes $Z^{k,\\uparrow}$ for $k \\geq0$ which\nhave the following dynamics:\n\\begin{itemize}\n\\item Initially there are $N$ particles of type $k$ in $Z_0^{k,\\uparrow}$.\n\\item Each particle changes from type $i$ to $i+1$ at rate $\\mu$.\n\\item A particle of type $i$ branches at rate $\\gamma i+1$ and, upon\nbranching, the new particle is also type $i$.\n\\end{itemize}\nLet $\\overline{M}{}^{k,\\uparrow}_t$ be the maximum type of any particle\nin $Z_t^{k,\\uparrow}$ and let \\mbox{$M_t^{k,\\uparrow} = \\overline\n{M}{}^{k,\\uparrow}_t-k$}, so that $M_0^{k,\\uparrow} = 0$. Note that we\nrefer to individuals in branching processes as particles to distinguish\nthem from the individuals in $X$. This will make the coupling arguments\nin the next section more clear.\n\nWe define a stochastic process $X'$ that will be coupled with $X$ as\ndescribed in the proof of Proposition~\\ref{TheoProp1} for reasons that\nwill become clear shortly. Let $\\{\\mathcal{Z}^n\\}_{n=0}^\\infty$ be an\ni.i.d. sequence of continuous-time stochastic processes which each have\nthe same distribution as $Z^{\\mathcal{W},\\uparrow}$. Let\n$\\overline{\\mathcal{M}}{}^n_t$ be the maximum type of any particle in $\\mathcal\n{Z}_t^n$ and let $\\mathcal{M}_t^n = \\overline{\\mathcal\n{M}}{}^n_t-\\mathcal{W}$ so that $\\mathcal{M}_0^n = 0$ for all $n$. Define\n\\[\nX_t' = \\cases{\nX_0^+ +\n\\mathcal{M}_t^0, &\\quad if $t \\in[0,\\mathcal{T}]$,\n\\vspace*{1pt}\\cr\nX_{i\\mathcal{T}}' + \\mathcal{M}_{t-i\\mathcal{T}}^i, &\\quad\nif $t \\in\\bigl(i\\mathcal{T}, (i+1)\\mathcal{T}\\bigr]$\\qquad for any integer $i \\geq1$,}\n\\]\nand $D_t' = X_t' - X_0^+$. The idea is that $D_t'$ is the maximum type\nof any particle in a branching process $X'$ that has the same\ndistribution as $Z^{\\mathcal{W},\\uparrow}$ except that at each time\n$i\\mathcal{T}$ we restart the branching process so that there are once\nagain $N$ particles of type $\\mathcal{W}$. For each integer $i \\geq0$\nat time $i\\mathcal{T}$, the $N$ particles initially have type $D_t'$\nwhich is the maximum type achieved by any particle in $X_t'$ up to time $t$.\n\nNow we are able to state the four propositions used to prove Theorem\n\\ref{Theorem}. Proposition~\\ref{TheoProp1} is a result of the\ncoupling of $X$ and $X'$.\n\\begin{Prop} \\label{TheoProp1}\nLet $X_0^i = 0$ for $1 \\leq i \\leq N$. Then\n\\[\nD_t \\leq D_t'+\\sum\n_{i=1}^{N_t}Y_i\n\\]\nfor all times $t \\geq0$.\\vadjust{\\goodbreak}\n\\end{Prop}\n\\begin{Prop} \\label{TheoProp4}\nLet $X_0^i = 0$ for $1 \\leq i \\leq N$. For $N$ large enough we have\n\\[\n\\sup_{t \\in[\\mathcal{T},\\infty)} \\frac{E[D_t']}{t} \\leq\\frac\n{6\\mathcal{W}}{\\mathcal{T}}.\n\\]\n\\end{Prop}\n\nWith the initial condition $X_0^i = 0$ for $1 \\leq i \\leq N$, we let\n$\\mathcal{F} = \\{\\mathcal{F}_t\\}_{t \\geq0}$ be the natural\nfiltration associated with $X$.\n\\begin{Prop} \\label{TheoProp2}\nLet $X_0^i = 0$ for $1 \\leq i \\leq N$. For $N$ large enough we have\n$E[Y_i|\\mathcal{F}_{s_i}] \\leq5\\mathcal{W}$ for all $i \\geq1$.\n\\end{Prop}\n\\begin{Prop} \\label{TheoProp3}\nLet $X_0^i = 0$ for $1 \\leq i \\leq N$. For $N$ large enough,\n\\[\n\\sup_{s \\in[0,\\infty)}\\frac{1}{s}E[N_s] \\leq\n\\frac{1}{\\mathcal{T}}.\n\\]\n\\end{Prop}\n\\begin{pf*}{Proof of Theorem~\\ref{Theorem}}\nFix $t \\geq\\log\\log N$. It follows by definition of $\\mathcal{T}$\nthat $t > \\mathcal{T}$ so that the hypotheses of the preceding four\npropositions are satisfied. There exists $N_0$ which does not depend on\n$t$ such that for any $N \\geq N_0$ we have\n\\begin{eqnarray*}\nE \\biggl[\\frac{D_t}{t} \\biggr] & \\leq & E \\biggl[\\frac{D_t'+\\sum\n_{i=1}^{N_t}Y_i}{t} \\biggr]\n\\qquad\\mbox{by Proposition~\\ref{TheoProp1}}\n\\\\\n& = &E \\biggl[\\frac{D_t'}{t} \\biggr] + E \\biggl[\\frac{\\sum\n_{i=1}^{N_t}Y_i}{t} \\biggr]\n\\\\\n& \\leq &\\frac{6\\mathcal{W}}{\\mathcal{T}} + \\frac{1}{t}E \\Biggl[\\sum\n_{i=1}^{N_t}Y_i \\Biggr] \\qquad\\mbox{by\nProposition~\\ref{TheoProp4}}\n\\\\\n& = &\\frac{6\\mathcal{W}}{\\mathcal{T}} + \\frac{1}{t}\\sum_{i=1}^\\infty\nE[Y_i 1_{\\{N_t \\geq i\\}}]\n\\\\\n& = &\\frac{6\\mathcal{W}}{\\mathcal{T}} + \\frac{1}{t}\\sum_{i=1}^\\infty\nE\\bigl[E[Y_i 1_{\\{N_t \\geq i\\}}|\\mathcal{F}_{s_i}]\\bigr]\n\\\\\n& = &\\frac{6\\mathcal{W}}{\\mathcal{T}} + \\frac{1}{t}\\sum_{i=1}^\\infty\nE\\bigl[1_{\\{N_t \\geq i\\}}E[Y_i |\\mathcal{F}_{s_i}]\\bigr]\n\\\\\n& \\leq &\\frac{6\\mathcal{W}}{\\mathcal{T}} + \\frac{5\\mathcal\n{W}}{t}\\sum_{i=1}^\\infty\nE[1_{\\{N_t \\geq i\\}}] \\qquad\\mbox{by Proposition~\\ref{TheoProp2}}\n\\\\\n& = &\\frac{6\\mathcal{W}}{\\mathcal{T}} + \\frac{5\\mathcal\n{W}}{t}E[N_t]\n\\\\\n& \\leq &\\frac{6\\mathcal{W}}{\\mathcal{T}} + \\frac{5\\mathcal\n{W}}{\\mathcal{T}} \\qquad\\mbox{by Proposition\n\\ref{TheoProp3}}\n\\\\\n& = &\\frac{11w^{1\/2}\\log N}{(\\log\\log N)^2}.\n\\end{eqnarray*}\nSince $w$ may go to infinity arbitrarily slowly with $N$ there must\nexist a constant $C$ such that\n\\[\n\\frac{E[D_t]}{t} \\leq\\frac{C\\log N}{(\\log\\log N)^2}\n\\]\nfor all $t \\geq\\log\\log N$. This immediately gives a bound on\n$E[\\overline{X}_t]\/t$.\n\\end{pf*}\n\n\\section{Bounding the rate when the width is small}\\label{sec3}\nThrough the use of branching processes we establish a bound on $D_t$\nthat depends on the width. We will make use of the strong Markov\nproperty of $X$ at the times $s_n$ and $t_n$ for $n \\geq1$. For this\nreason, many of the statements we prove below will include conditions\nfor which $W_0 > 0$ even though according to the conditions of Theorem\n\\ref{Theorem} we have $W_0 = 0$. In this section we establish a small\nupper bound for $D_t$ on the time intervals $[t_n,s_n)$.\n\nThe following proofs will involve coupling $X$ with various branching\nprocesses. While the individuals in $X$ each have an integer value that\nwe refer to as the fitness of the individual, the particles in a\nbranching process will each be given an integer value that we refer to\nas the type of the particle. Let $Z^C = \\{Z_t^C\\}_{t \\geq0}$ be a\nmulti-type Yule process in which there are initially $N$ particles of\ntype 0. Particles increase from type $i$ to type $i+1$ at rate $\\mu$\nand branch at rate $C$. When a particle of type $i$ branches, the new\nparticle is also type $i$. Let $M_t^C$ be the maximum type of any\nparticle at time $t$.\n\nThe next proposition will give a lower bound on the fitness of any\nindividual up to time $t$ given that we know the least fitness at time\n0 is $X_0^-$. We do this by establishing an upper bound on the amount\nthat any individual will decrease in fitness. Let\n\\[\nS_t = \\sup_{0 \\leq s \\leq t}\\bigl(X_0^- -\nX_s^-\\bigr).\n\\]\n\n\\begin{figure}\n\n\\includegraphics{873f01.eps}\n\n\\caption{Picture of the coupling of $X$ with $Z^1$ when $N = 3$.}\\label{fig1}\n\\end{figure}\n\n\\begin{Prop} \\label{BackSpeed}\nFor any population size $N$, initial configuration $X_0$, time $t \\geq\n0$ and natural number $l$,\n\\[\nP(S_t \\geq l) \\leq\\frac{N (t\\mu)^l e^t}{l!}.\n\\]\n\\end{Prop}\n\\begin{pf}\nBy Lemma~\\ref{YuleLem} in the \\hyperref[app]{Appendix} we have\n\\[\nP\\bigl(M_t^1 \\geq l\\bigr) \\leq\\frac{N(t\\mu)^le^t}{l!}\n\\]\nfor any population size $N$, time $t \\geq0$ and natural number $l$.\nNote that from our notation above $Z^1$\\vadjust{\\goodbreak} is a Yule process with\nbranching rate 1. To complete the proof we establish a\ncoupling\\vspace*{1pt} between $X$ and $Z^1$ such that for any\npopulation size $N$ and time $t \\geq0$ we have $M_t^1 \\geq S_t$. See\nFigure~\\ref{fig1} for an illustration of the coupling. At all times\nevery individual in $X$ will be paired with one particle in $Z^1$. The\ncoupling is as follows:\n\\begin{itemize}\n\\item We initially have a one-to-one pairing of each individual $i$ in\n$X_0$ with each particle $i$ in $Z_0^1$.\n\\item The particle in $Z^1$ that is paired with individual $i$ will\nincrease in type by 1 only when individual $i$ gets a mutation.\n\\item For each individual $i$ in $X$ and each $j \\neq i$, individual\n$j$ is replaced by individual $i$ at rate $1\/N$ due to resampling\nevents. If individual $i$ replaces individual $j$ due to resampling,\nthen the particle labeled $i$ in $Z^1$ branches. If particle $i$ has a\nhigher type than particle $j$, then the new particle is paired with\nindividual $j$. The particle that was paired with individual $j$ before\nthe branching event is no longer paired with any individual in $X$. If\nparticle $i$ has a lower type than particle $j$ then the particle that\nwas paired with individual $j$ remains paired with individual $j$ and\nthe new particle is not paired with any individual in $X$.\n\\item The particle paired with individual $i$ in $Z^1$ branches at rate\n$1\/N$ and these branching events are independent of any of the events\nin $X$. When the particle paired with individual $i$ branches due to\nthese events, the new particle is not paired with any individual in $X$.\n\\item Any particles in $Z^1$ that are not paired with an individual in\n$X$ branch and acquire mutations independently of $X$. The selection\nevents in $X$ are independent of any events in $Z^1$.\n\\end{itemize}\n\n\n\nLet $R^i$ be the type of the particle in $Z^1$ that is paired with\nindividual $i$ and let\n\\[\nS_s^i = \\sup_{0\\leq r \\leq s}\\bigl(X_0^--X_r^i\n\\bigr).\n\\]\nTo show $M_t^1 \\geq S_t$ it is enough to show $R_t^i \\geq S_t^i$ for\nall $i$. Initially $S_0^i \\leq R_0^i = 0$ for all $i$. Note that both\n$s \\mapsto S_s^i$ and $s \\mapsto R_s^i$ are increasing functions and\nthat increases in these functions correspond to decreases in $X^i$.\n\nWhen individual $i$ gets a mutation, $R^i$ increases by 1. However, if\nindividual $i$ gets a mutation at time $s$, then $S^i$ will only\nincrease by 1 if $S_{s-}^i = X_0^- - X_{s-}^i$ and the mutation is\ndeleterious. Therefore, if individual $i$ gets a mutation at time $s$\nand $S_{s-}^i \\leq R_{s-}^i$, then\n\\[\nS_s^i \\leq S_{s-}^i+1 \\leq\nR_{s^-}^i+1 = R_s^i.\n\\]\n\nSuppose individual $j$ is replaced by individual $i$ due to a\nresampling event at time $s$ and that both $S_{s-}^j \\leq R_{s-}^j$ and\n$S_{s-}^i \\leq R_{s-}^i$ hold. With probability 1 we have $S_s^i =\nS_{s-}^i$ and $R_s^i = R_{s-}^i$. If $X_0^- - X_s^i \\leq S_{s-}^j$ then\n$S_{s-}^j = S_s^j$. From this it follows that $S_s^j \\leq R_s^j$. If\n$X_0^- - X_s^i > S_{s-}^j$ then $S_s^j = X_0^--X_s^i \\leq S_s^i \\leq\nR_s^i$. If $R_s^i \\geq R_{s-}^j$, then by the definition of the\ncoupling, $R_s^j = R_s^i$. If $R_s^i < R_{s-}^j$, then by definition of\nthe coupling, $R_s^j = R_{s-}^j$. Therefore, $R_s^j \\geq R_s^i$ which\ngives us $S_s^j \\leq R_s^j$.\n\nSelection events will never increase $S^i$ and since $S^i$ and $R^i$\nare increasing in time, a selection event at time $s$ will preserve the\ninequality $S_s^i \\leq R_s^i$. This shows that any event that occurs at\ntime $s$ which may change the fitness of an individual $i$ in $X$ will\npreserve the inequality $S_s^i \\leq R_s^i$. Since the result holds for\neach individual~$i$, we have $S_t \\leq M_t^1$.\n\\end{pf}\n\nWe now wish to bound the distance the front of the wave moves as a\nfunction of the initial width.\n\\begin{Prop} \\label{UpBound}\nFor any initial configuration $X_0$, fixed time $t \\geq0$ and any\ninteger $l \\geq0$, we have\n\\[\nP\\Bigl(\\sup_{0\\leq s \\leq t} D_s > l\\Bigr) \\leq\\frac{2N(t\\mu\n)^le^{(\\gamma\n(W_0+2l)+\\mu+1)t}}{(l-1)!}.\n\\]\n\\end{Prop}\n\\begin{pf}\nRecall that $W_0$ is the width of $X$ at time 0. We first establish a\ncoupling between $X$ and $Z^{W_0+k,\\uparrow}$ for each integer $k \\geq\n0$. See Figure~\\ref{fig2} for an illustration of the coupling.\n\\begin{figure}\n\n\\includegraphics{873f02.eps}\n\n\\caption{Picture of the coupling of $X$ with $Z^{k,\\uparrow}$ when $N\n= 3$.}\\label{fig2}\n\\end{figure}\nLet $T^k= \\inf\\{\nt\\dvtx S_t > k\\}$ for $k \\geq1$. Every individual in $X$ will be paired\nwith one particle in $Z^{W_0+k,\\uparrow}$ until time $T^k$. We couple\n$Z^{W_0+k,\\uparrow}$ with $X$ for all times $t \\in[0,T^k)$ as\nfollows:\n\\begin{itemize}\n\\item We initially have a one-to-one pairing of each individual $i$ in\n$X_0$ with each particle $i$ in $Z_0^{W_0+k,\\uparrow}$. When a\nparticle in $Z_t^{W_0+k,\\uparrow}$ is coupled with individual $i$, we\nrefer to the particle as particle $i$.\n\\item Particle $i$ increases in type by 1 only when individual $i$ gets\na mutation.\n\\item For each individual $i$ in $X$ and each $j \\neq i$, individual\n$j$ is replaced by individual $i$ at rate $1\/N$ due to resampling\nevents. If individual $i$ replaces individual $j$ due to resampling,\nthen particle $i$ branches. If particle $i$ has a higher type than\nparticle $j$, then the new particle is paired with individual $j$. The\nparticle that was paired with individual $j$ before the branching event\nis no longer paired with any individual in $X$. If particle $i$ has a\nlower type than particle $j$, then the particle that was paired with\nindividual $j$ remains paired with individual $j$ and the new particle\nis not paired with any individual in $X$.\n\\item Additionally, particle $i$ branches at rate $1\/N$ and these\nbranching events are independent of any of the events in $X$. When\nparticle $i$ branches due to these events the new particle is not\npaired with any individual in $X$.\n\\item In $X$ there is a time dependent rate $\\gamma U_s^i$ at which\nindividuals $j \\neq i$ are replaced by individual $i$ due to selection\nevents, namely,\n\\[\nU_s^i = \\frac{1}{N}\\sum\n_{j=1}^N \\bigl(X_s^i -\nX_s^j\\bigr)^+.\n\\]\nIf individual $j$ is replaced by individual $i$ in $X$ due to a\nselection event, then particle $i$ branches. If particle $i$ has a\nhigher type than particle $j$, then the new particle is paired with\nindividual $j$. The particle that was paired with individual $j$ before\nthe branching event is no longer paired with any individual in $X$. If\nparticle $i$ has a lower type than particle $j$, then the particle that\nwas paired with individual $j$ remains paired with individual $j$. The\nnew particle is not paired with any individual in $X$.\n\\item Additionally, particle $i$ branches at a time dependent rate\n$\\gamma(R_t^{i,k}-U_t^i)$ where $R_t^{i,k}$ is the type of particle\n$i$. These branching events are independent of any of the events in\n$X$. When such a branching event occurs, the new particle is not paired\nwith any individual in $X$.\n\\item Any particles in $Z^{W_0+k,\\uparrow}$ that are not paired with\nan individual in $X$ branch and change type independently of $X$.\n\\end{itemize}\nFix $k \\geq1$. For the above coupling between $X$ and\n$Z^{W_0+k,\\uparrow}$ to be well defined until time $T^k$, we need\n$R_t^{i,k}-U_t^i \\geq0$ for all $i \\in\\{1,\\ldots,N\\}$ and for all\ntimes $t \\in[0,T^k)$. Let $\\overline{T}{}^{k,i} = \\inf\\{t\\dvtx R_t^{i,k} -\nU_t^i < 0\\}$. The coupling between $X$ and $Z^{W_0+k,\\uparrow}$ is\nwell defined until time $\\overline{T}{}^k = \\min\\{\\overline{T}{}^{k,i}\\dvtx1\n\\leq i \\leq N\\}$. We will show that $T^k \\leq\\overline{T}{}^k$.\n\n\nLet\n\\[\n\\overline{S}{}^i_t = \\sup_{0\\leq s \\leq t}\n\\bigl(X_s^i-X_0^+\\bigr) \\quad\\mbox{and}\\quad\n\\overline{R}{}^{i,k}_t = R_0^{i,k}-W_0-k.\n\\]\nInitially $\\overline{S}{}^i_0 \\leq\\overline{R}{}^{i,k}_0 = 0$ for all\n$i$. Note that both $t \\mapsto\\overline{S}{}^i_t$ and $t \\mapsto\nR_t^{i,k}$ are increasing functions, from which it follows that $t\n\\mapsto\\overline{R}{}^{i,k}_t$ is also an increasing function.\n\nWhen individual $i$ gets a mutation, $\\overline{R}{}^{i,k}$ increases by\n1. However, if individual $i$ gets a mutation at time $s$ then\n$\\overline{S}{}^i$ will only increase by 1 if $\\overline{S}{}^i_{s-} =\nX_{s-}^i - X_0^+$ and the mutation is beneficial. Therefore, if\nindividual $i$ gets a mutation at time $s$ and $\\overline{S}{}^i_{s-}\n\\leq\\overline{R}{}^{i,k}_{s-}$, then\n\\[\n\\overline{S}{}^i_s \\leq\\overline{S}{}^i_{s-}+1\n\\leq\\overline{R}{}^{i,k}_{s^-}+1 = \\overline{R}{}^{i,k}_s.\n\\]\n\nSuppose individual $j$ is replaced by individual $i$ due to a\nresampling or selection event at time $s$ and that both $\\overline\n{S}{}^j_{s-} \\leq\\overline{R}{}^{j,k}_{s-}$ and $\\overline{S}{}^i_s =\n\\overline{S}{}^i_{s-} \\leq\\overline{R}{}^{i,k}_{s-} = \\overline\n{R}{}^{i,k}_s$ hold. If $X_s^i - X_0^+ \\leq\\overline{S}{}^j_{s-}$, then\n$\\overline{S}{}^j_{s-} = \\overline{S}{}^j_s$. It follows that $\\overline\n{S}{}^j_s \\leq\\overline{R}{}^{j,k}_s$. If $X_s^i-X_0^+ > \\overline\n{S}{}^j_{s-}$ then $\\overline{S}{}^j_s = X_0^--X_s^i \\leq\\overline\n{S}{}^i_s \\leq\\overline{R}{}^{i,k}_s$. If $\\overline{R}{}^{i,k}_s \\geq\n\\overline{R}{}^{j,k}_{s-}$, then by the definition of the coupling,\n$\\overline{R}{}^{j,k}_s = \\overline{R}{}^{i,k}_s$. If $\\overline\n{R}{}^{i,k}_s < \\overline{R}{}^{j,k}_{s-}$, then by definition of the\ncoupling, $\\overline{R}{}^{j,k}_s = \\overline{R}{}^{j,k}_{s-}$.\nTherefore, $\\overline{R}{}^{j,k}_s \\geq\\overline{R}{}^{i,k}_s$ which\ngives us $\\overline{S}{}^j_s \\leq\\overline{R}{}^{j,k}_s$.\n\nFor any time $s < T^k$ we have $R^{i,k}_s \\geq\\overline{S}{}^i_s+W_0+k\n\\geq X^i_s-X_0^++W_0+k = X^i_s-X_0^-+k$. If there were $N$ individuals\nwith fitness $X_0^--k$ at time $s \\in[0,\\overline{T}{}^{k,i})$, then\nthe rate at which individual $i$ replaces these $N$ individuals due to\nselection is $\\gamma(X^i_s-X_0^-+k)$. However, for any time $s < T^k$,\nthere are fewer than $N$ individuals being replaced by individual $i$\ndue to selection and they will all have fitnesses at least as large as\n$X_0^--k$. This gives us a bound on the rate at which resampling events\noccur on individual $i$ before time $T^k$, namely, $U^i_s \\leq\nX^i_s-X_0^-+k \\leq R^{i,k}_s$ for all $s \\in[0,T^k)$. This shows that\n$T^k \\leq\\overline{T}{}^{k,i}$ for all $i$. Hence, $T^k \\leq\\overline\n{T}{}^k$ and the coupling is well defined until time $T^k$.\n\nWe have shown that any event that occurs at time $s \\in[0,T^k)$ which\nmay change the fitness of an individual $i$ in $X$ will preserve the\ninequality $\\overline{S}{}^i_s \\leq\\overline{R}{}^{i,k}_s$. Since the\nresult holds for each individual $i$, for any $s \\in[0,T^k)$ we have\n\\[\n\\sup_{0 \\leq r \\leq s} D_r = \\sup_{1 \\leq i \\leq N}\n\\overline{S}{}^i_s \\leq\\sup_{1 \\leq i \\leq N}\n\\overline{R}{}^{i,k}_s \\leq M_s^{W_0+k,\\uparrow}.\n\\]\n\nNote that if $\\sup_{0\\leq s \\leq t}(X_0^- - X_s^-) \\leq k$ then $t <\nT^k$. On the event\\break $\\{\\sup_{0\\leq s \\leq t}(X_0^- - X_s^-) \\leq k\\}$\nwe have $M_t^{W_0+k,\\uparrow} \\geq\\sup_{0\\leq s \\leq t}D_s$. This\nallows us to do the following computation:\n\\begin{eqnarray}\n\\label{simpEq} P\\Bigl(\\sup_{0\\leq s \\leq t} D_s > l\\Bigr) & = & \\sum\n_{i=0}^\\infty P\\Bigl(\\Bigl\\{\n\\sup_{0\\leq s \\leq t} D_s > l\\Bigr\\} \\cap\\Bigl\\{\\sup_{0\\leq s \\leq t}\n\\bigl(X_0^- - X_s^-\\bigr) = i\\Bigr\\}\\Bigr)\n\\nonumber\n\\\\[-2pt]\n& \\leq &\\sum_{i=0}^\\infty P\\Bigl(\\bigl\n\\{M_t^{W_0+i,\\uparrow} > l\\bigr\\} \\cap\\Bigl\\{\\sup_{0\\leq s \\leq t}\n\\bigl(X_0^- - X_s^-\\bigr) = i\\Bigr\\}\\Bigr)\n\\nonumber\n\\\\[-2pt]\n& \\leq &\\sum_{i=0}^\\infty P\\Bigl(\\bigl\n\\{M_t^{W_0+i,\\uparrow} > l\\bigr\\} \\cap\\Bigl\\{\\sup_{0\\leq s \\leq t}\n\\bigl(X_0^- - X_s^-\\bigr) \\geq i\\Bigr\\}\\Bigr)\n\\nonumber\n\\\\[-2pt]\n& \\leq &\\sum_{i=0}^\\infty P\n\\bigl(M_t^{W_0+i,\\uparrow} > l\\bigr) \\wedge P\\Bigl(\\sup_{0\\leq s \\leq t}\n\\bigl(X_0^- - X_s^-\\bigr) \\geq i\\Bigr)\n\\nonumber\n\\\\[-2pt]\n& \\leq &\\sum_{i=0}^\\infty P\n\\bigl(M_t^{W_0+i,\\uparrow} > l\\bigr) \\wedge\\biggl(\\frac{N (t\\mu)^i e^t}{i!}\n\\biggr) \\qquad\\mbox{by Proposition~\\ref{BackSpeed}}\n\\nonumber\n\\\\[-2pt]\n& \\leq &\\sum_{i=0}^\\infty\\biggl(\n\\frac{N (t\\mu)^l e^{(\\gamma\n(W_0+i+l)+1)t}}{l!} \\biggr) \\wedge\\biggl(\\frac{N (t\\mu)^i\ne^t}{i!} \\biggr) \\nonumber\\\\[-2pt]\n&&\\mbox{by\nLemma~\\ref{ZUpBound} in the \\hyperref[app]{Appendix}}\n\\nonumber\n\\\\[-2pt]\n& \\leq &\\frac{N (t\\mu)^l e^{(\\gamma(W_0+l)+1)t}}{l!} \\sum_{i=0}^{l-1}\ne^{i\\gamma t} + Ne^t \\sum_{i=l}^\\infty\n\\frac{(t\\mu\n)^i}{i!}\n\\nonumber\n\\\\[-2pt]\n& \\leq &\\frac{N (t\\mu)^l e^{(\\gamma(W_0+l)+1)t}}{l!}\\cdot le^{l\\gamma\nt} + Ne^t \\sum\n_{i=l}^\\infty\\frac{(t\\mu)^i}{i!}\n\\\\[-2pt]\n& \\leq &\\frac{N (t\\mu)^l e^{(\\gamma(W_0+2l)+1)t}}{(l-1)!} + \\frac\n{N(t\\mu)^l e^{(\\mu+1)t}}{l!} \\nonumber\\\\[-2pt]\n&&\\mbox{by Lemma~\\ref{TRTBoundLem}\nin the \\hyperref[app]{Appendix}}\n\\nonumber\n\\\\[-2pt]\n& \\leq &\\frac{2N(t\\mu)^le^{(\\gamma(W_0+2l)+\\mu+1)t}}{(l-1)!}.\\nonumber\n\\end{eqnarray}\n\\upqed\\end{pf}\n\nWe now extend the bound we got on the least fit individuals in\nProposition~\\ref{BackSpeed} to a slightly stronger result.\n\\begin{Dfn} \\label{S}\nLet $x \\in\\Z$ and let $\\mathcal{S}_t^x \\subset\\{1,2,\\ldots, N\\}$\ncorrespond to a collection of individuals at time $t$ which is\ndetermined by the following dynamics:\n\\begin{itemize}\n\\item Initially, $\\mathcal{S}_0^x$ consists of all individuals whose\nfitness lies in the interval $(x,\\infty)$.\n\\item If a resampling or selection event occurs at time $t$ and an\nindividual not in $\\mathcal{S}_{t-}^x$ is replaced by a individual in\n$\\mathcal{S}_{t-}^x$, then it is added to $\\mathcal{S}_t^x$.\n\\item If a beneficial mutation occurs at time $t$ on an individual not\nin $\\mathcal{S}_{t-}^x$ that causes its fitness to increase from $x$\nto $x+1$, it is added to $\\mathcal{S}_t^x$.\n\\item If a resampling event occurs at time $t$ to an individual in\n$\\mathcal{S}_{t-}^x$ and it is replaced by a individual not in\n$\\mathcal{S}_{t-}^x$, then it is removed from $\\mathcal{S}_t^x$.\n\\end{itemize}\n\\end{Dfn}\nMutation and selection events do not cause individuals to be lost\nfrom~$\\mathcal{S}^x$. We now prove the following corollary to Proposition\n\\ref{UpBound}.\n\\begin{Cor} \\label{BackSpeedLabel}\nLet $A_t^{x,l}$ be the event that an individual in $\\mathcal{S}_s^x$\nhas fitness in $(-\\infty, x-l]$ for some time $s \\in[0,t]$. For any\ninitial configuration $X_0$, time $t \\geq0$ and any integer $l$,\n\\[\nP\\bigl(A_t^{x,l}\\bigr) \\leq\\frac{2N(t\\mu)^le^{(\\gamma(W_0+2l)+\\mu+1)t}}{(l-1)!}.\n\\]\n\\end{Cor}\n\nNote that we cannot use the bound found in Proposition~\\ref{BackSpeed}\nbecause individuals not in $\\mathcal{S}_t^x$ may move\\vadjust{\\goodbreak} to $\\mathcal\n{S}_t^x$ due to selection events. In the proof of Proposition \\ref\n{BackSpeed} the number of individuals with the least fitness cannot\nincrease due to selection events. However, the number of individuals\nwith the least fitness in $\\mathcal{S}_t^x$ may increase due to\nselection events involving individuals not in $\\mathcal{S}_t^x$.\n\\begin{pf*}{Proof of Corollary~\\ref{BackSpeedLabel}}\nFor $k \\geq1$ let $X$ be coupled with $Z^{W_0+k,\\uparrow}$ as in the\nproof of Proposition~\\ref{UpBound}. Let $T^k$, $R^{i,k}_t$ and\n$\\overline{R}{}^{i,k}_t$ be defined as they were in the proof of\nProposition~\\ref{UpBound}. Define $\\overline{T}{}^i_s = \\{r \\in[0,s]\\dvtx\ni \\in\\mathcal{S}_r^x\\}$ and let\n\\[\nS^i_s = \\cases{\n\\displaystyle \\sup_{r \\in\\overline{T}{}^i_s}\n\\bigl(x-X_r^i\\bigr), &\\quad if $\\overline{T}{}^i_s\n\\neq\\varnothing$,\n\\vspace*{2pt}\\cr\n-\\infty, &\\quad if $\\overline{T}{}^i_s =\n\\varnothing$.}\n\\]\n\nThe goal is to show that for all $s \\in[0,T^k)$ we have\n\\[\n\\sup_{1 \\leq i \\leq N} S^i_s \\leq\\sup_{1 \\leq i \\leq N}\n\\overline{R}{}^{i,k}_s \\leq M_s^{W_0+k,\\uparrow}.\n\\]\nNote that we can only consider the coupling of $X$ with\n$Z^{W_0+k,\\uparrow}$ until time $T^k$ because after this time the\ncoupling is not well defined.\n\nInitially all of the individuals in $\\mathcal{S}_0^x$ have fitness in\n$(x,\\infty)$. Therefore, if $i \\in\\mathcal{S}_0^x$ then $S_0^i \\leq\n0 = \\overline{R}{}^{i,k}_0$. If $i \\notin\\mathcal{S}_0^x$ then $S_0^i\n= -\\infty< \\overline{R}{}^{i,k}_0$.\n\nSuppose individual $i$ gets a mutation at time $s$ and that for any\ntime $s' \\in[0,s-)$ we have $S_{s'}^i \\leq\\overline{R}{}^{i,k}_{s'}$.\nThen $\\overline{R}{}^{i,k}$ increases by 1. If $i \\in\\overline\n{S}{}^x_{s-}$ then $S^i_s$ will only increase by 1 if $S_{s-}^i =\nx-X^i_s$ and the mutation is deleterious. If $i \\notin\\mathcal\n{S}_{s-}^x$ and the mutation does not cause the fitness of individual\n$i$ to change from $x$ to $x+1$, then $S^i_s = S_{s-}^i$. If $i \\notin\n\\mathcal{S}_{s-}^x$ and the mutation does cause the fitness of\nindividual $i$ to change from $x$ to $x+1$, then $S^i_s = S_{s-}^i \\vee\n0$. In any of these three cases, $S^i_s \\leq\\overline{R}{}^{i,k}_s$.\n\nSuppose individual $j$ is replaced by individual $i$ due to a\nresampling or selection event at time $s$ and that $S_{s-}^j \\leq\n\\overline{R}{}^{j,k}_{s-}$ and $S_{s-}^i \\leq\\overline{R}{}^{i,k}_{s-}$.\nIf $i \\notin\\overline{S}{}^x_{s-}$ then $S_{s-}^j = S_s^j \\leq\n\\overline{R}{}^{j,k}_{s-}$. Suppose $i \\in\\mathcal{S}_{s-}^x$. If $x -\nX^i_s \\leq S_{s-}^j$ then $S_{s-}^j = S_s^j$. From this it follows that\n$S_s^j \\leq\\overline{R}{}^j_s$. If $x - X^i_s > S_{s-}^j$, then $S_s^j\n= x-X^i_s \\leq S^i_s \\leq\\overline{R}{}^i_s$. If $\\overline{R}{}^i_s\n\\geq\\overline{R}{}^j_{s-}$, then by the definition of the coupling,\n$\\overline{R}{}^j_s = \\overline{R}{}^i_s$. If $\\overline{R}{}^i_s <\n\\overline{R}{}^j_{s-}$, then by definition of the coupling, $\\overline\n{R}{}^j_s = \\overline{R}{}^j_{s-}$. Therefore, $\\overline{R}{}^j_s \\geq\n\\overline{R}{}^i_s$ which gives us $S^j_s \\leq\\overline{R}{}^j_s$.\n\nNote that if $\\sup_{0\\leq s \\leq t}(X_0^- - X_s^-) \\leq k$ then $t <\nT^k$. Therefore, on the event $\\{\\sup_{0\\leq s \\leq t}(X_0^- - X_s^-)\n\\leq k\\}$ we have $M^{W_0+k,\\uparrow}_t \\geq\\sup_{1 \\leq i \\leq N}\nS^i_s$. This allows us to do the following computation:\n\\begin{eqnarray*}\n&&\nP\\Bigl(\\sup_{0\\leq s \\leq t} \\sup_{1 \\leq i \\leq N} S^i_s\n> l\\Bigr) \\\\\n&&\\qquad = \\sum_{i=0}^\\infty P\\Bigl(\\Bigl\n\\{\\sup_{0\\leq s \\leq t} \\sup_{1 \\leq i \\leq N} S^i_s > l\n\\Bigr\\} \\cap\\Bigl\\{\\sup_{0\\leq s \\leq t}\\bigl(X_0^- -\nX_s^-\\bigr) = i\\Bigr\\}\\Bigr)\n\\\\\n&&\\qquad \\leq \\sum_{i=0}^\\infty P\\Bigl(\\bigl\n\\{M_t^{W_0+i,\\uparrow} > l\\bigr\\} \\cap\\Bigl\\{\\sup_{0\\leq s \\leq t}\n\\bigl(X_0^- - X_s^-\\bigr) = i\\Bigr\\}\\Bigr).\n\\end{eqnarray*}\nThis is the same bound as equation (\\ref{simpEq}) in the proof of\nProposition~\\ref{UpBound}. Therefore, we have established the same bound.\n\\end{pf*}\n\\begin{pf*}{Proof of Proposition~\\ref{TheoProp4}}\nBy definition $D_\\mathcal{T}'$ has the same distribution as\n$M_\\mathcal{T}^{\\mathcal{W},\\uparrow}$ so by Lemma~\\ref{ZUpBound}\nin the \\hyperref[app]{Appendix} we have\n\\[\nP\\bigl(D_\\mathcal{T}' > l\\bigr) \\leq\\frac{N(\\mathcal{T}\\mu)^l e^{(\\gamma\n(\\mathcal{W}+l)+1)\\mathcal{T}}}{l!}.\n\\]\nThen\n\\begin{eqnarray}\n\\label{eq1} \\frac{E[D_\\mathcal{T}']}{2\\mathcal{W}} & = & \\frac\n{1}{2\\mathcal\n{W}}\\sum\n_{l=0}^\\infty P\\bigl(D_\\mathcal{T}'\n> l\\bigr)\n\\nonumber\\\\[-8pt]\\\\[-8pt]\n& \\leq &\\frac{1}{2\\mathcal{W}} \\Biggl[2\\mathcal{W} + \\sum\n_{l=2\\mathcal{W}}^\\infty\\frac{N(\\mathcal{T}\\mu)^l e^{(\\gamma\n(\\mathcal{W}+l)+1)\\mathcal{T}}}{l!} \\Biggr].\\nonumber\n\\end{eqnarray}\nBy Lemma~\\ref{TRTBoundLem} in the \\hyperref[app]{Appendix} we have\n\\begin{equation}\n\\label{eq2} \\sum_{l=2\\mathcal{W}}^\\infty\n\\frac{N(\\mathcal{T}\\mu)^l e^{(\\gamma\n(\\mathcal{W}+l)+1)\\mathcal{T}}}{l!} \\leq\\frac{N e^{(\\gamma\\mathcal\n{W} + 1)\\mathcal{T}}(\\mathcal{T}\\mu e^{\\gamma\\mathcal\n{T}})^{2\\mathcal{W}}e^{\\mathcal{T} \\mu e^{\\gamma\\mathcal\n{T}}}}{(2\\mathcal{W})!}.\n\\end{equation}\n\nNote that for any $k \\geq2$ both $D_{k\\mathcal{T}}'-D_{(k-1)\\mathcal\n{T}}'$ and $D_\\mathcal{T}'$ have the same distribution, namely, that\nof $M_\\mathcal{T}^{\\mathcal{W}}$. Choose $t \\in[k\\mathcal{T},\n(k+1)\\mathcal{T})$ for some $k \\geq1$. Because $D_t'$ is increasing\nin $t$ we have\n\\[\n\\frac{D_t'}{t} \\leq\\frac{1}{k\\mathcal{T}} \\bigl(D_{(k+1)\\mathcal\n{T}}'\n- D_{k\\mathcal{T}}'+D_{k\\mathcal{T}}'-\n\\cdots+D_{2\\mathcal\n{T}}'-D_\\mathcal{T}'+D_\\mathcal{T}'\n\\bigr).\n\\]\nTherefore,\n\\[\n\\frac{E[D_t']}{t} \\leq\\frac{(k+1)E[D_\\mathcal{T}']}{k\\mathcal{T}} \\leq\n\\frac{2E[D_\\mathcal{T}']}{\\mathcal{T}}.\n\\]\n\nLet $t > \\mathcal{T}$. Dividing both sides by $2\\mathcal{W}\/\\mathcal\n{T}$ and using the bounds found in equations (\\ref{eq1}) and (\\ref\n{eq2}) gives us\n\\[\n\\frac{\\mathcal{T}E[D_t']}{2t\\mathcal{W}} \\leq\\frac{2E[D_\\mathcal\n{T}']}{2\\mathcal{W}} \\leq2 + \\frac{N e^{(\\gamma\\mathcal{W} +\n1)\\mathcal{T}}(\\mathcal{T}\\mu e^{\\gamma\\mathcal{T}})^{2\\mathcal\n{W}}e^{\\mathcal{T} \\mu e^{\\gamma\\mathcal{T}}}}{2\\mathcal\n{W}(2\\mathcal{W})!}.\n\\]\nBy Stirling's formula we have\n\\[\n\\frac{N e^{(\\gamma\\mathcal{W} + 1)\\mathcal{T}}(\\mathcal{T}\\mu\ne^{\\gamma\\mathcal{T}})^{2\\mathcal{W}}e^{\\mathcal{T} \\mu e^{\\gamma\n\\mathcal{T}}}}{2\\mathcal{W}(2\\mathcal{W})!} \\sim\\frac{N e^{(\\gamma\n\\mathcal{W} + 1)\\mathcal{T}}(\\mathcal{T}\\mu e^{\\gamma\\mathcal\n{T}})^{2\\mathcal{W}}e^{\\mathcal{T} \\mu e^{\\gamma\\mathcal\n{T}}+2\\mathcal{W}}}{(2\\mathcal{W})^{2\\mathcal{W}+1}\\sqrt{4\\pi\n\\mathcal{W}}} = e^x,\n\\]\nwhere\n\\begin{eqnarray*}\nx &=& \\log N + \\mathcal{T}\\bigl(\\gamma\\mathcal{W}+1+\\mu e^{\\gamma\\mathcal\n{T}}\\bigr)+2\n\\mathcal{W}\\bigl(\\log\\bigl(\\mathcal{T}\\mu e^{\\gamma\\mathcal\n{T}}\\bigr)+1\\bigr)\\\\\n&&{}-(2\n\\mathcal{W}+1)\\log(2\\mathcal{W})-\\log(4\\pi\\mathcal{W})\/2.\n\\end{eqnarray*}\nAs $N \\rightarrow\\infty$ we have $x \\sim-(2\\mathcal{W}+1)\\log\n(2\\mathcal{W}) \\sim-2w\\log N$. Therefore,\n\\[\n\\frac{\\mathcal{T}E[D_t']}{2t\\mathcal{W}} \\leq3\n\\]\nfor $N$ large enough.\n\\end{pf*}\n\\begin{pf*}{Proof of Proposition~\\ref{TheoProp1}}\nWe now couple $X$ with $X'$ by coupling $X$ with the sequence of\nprocesses $\\{\\mathcal{Z}^m\\}_{m=0}^\\infty$. Let\n\\[\nI_m = \\bigl(m\\mathcal{T},(m+1)\\mathcal{T}\\bigr] \\cap\\bigcup\n_{n=1}^\\infty[t_n,s_n)\n\\quad\n\\mbox{and}\n\\quad\nJ_m = (0,\\mathcal{T}] \\cap\\bigcup\n_{n=1}^\\infty[t_n-m\\mathcal{T},s_n-m\n\\mathcal{T}).\n\\]\nFor any $m \\geq0$ we couple $X$ and $\\mathcal{Z}^m$ as follows:\n\\begin{itemize}\n\\item The particles in $\\mathcal{Z}_0^m$ are labeled\n$1,2,\\ldots,N$.\\vspace*{1pt}\n\\item For any time in $I_m^C$ the process $X$ behaves independently of\n$\\mathcal{Z}^m$. For any time in $J_m^C$ the process $\\mathcal{Z}^m$\nbehaves independently of the process $X$. During the time~$J_m^C$, if a\nparticle labeled $i$ in $\\mathcal{Z}^m$ branches, the particle remains\nlabeled $i$ and the new particle is unlabeled.\n\\item The particle in $\\mathcal{Z}^m$ that is paired with individual\n$i$ will increase in type by 1 at time $t \\in J_m$ only when individual\n$i$ gets a mutation at time \\mbox{$t+m\\mathcal{T} \\in I_m$}.\n\\item For each individual $i$ in $X$ and each $j \\neq i$, individual\n$j$ is replaced by individual $i$ at rate $1\/N$ due to resampling\nevents. If individual $i$ replaces individual $j$ due to resampling at\ntime $t \\in I_m$, then the particle labeled $i$ in $\\mathcal{Z}^m$\nbranches at time $t-m\\mathcal{T} \\in J_m$. If particle $i$ has a\nhigher type than particle $j$, then the new particle is paired with\nindividual $j$. The particle that was paired with individual $j$ before\nthe branching event is no longer paired with any individual in $X$. If\nparticle $i$ has a lower type than particle~$j$, then the particle that\nwas paired with individual $j$ remains paired with individual $j$ and\nthe new particle is not paired with any individual in $X$.\n\\item The particle paired with individual $i$ in $\\mathcal{Z}^m$\nbranches at rate $1\/N$ for all times $t \\in J_m$ and these branching\nevents are independent of any of the events in $X$. When the particle\npaired with individual $i$ branches due to these events the new\nparticle is not paired with any individual in $X$.\n\\item In $X$ there is a time dependent rate $\\gamma U^i_s$ at which\nindividuals $j \\neq i$ are replaced by individual $i$ due to selection\nevents. If individual $j$ is replaced by individual $i$ in $X$ due to a\nselection event at time $t \\in I_m$, then the particle labeled\\vadjust{\\goodbreak} $i$ in\n$\\mathcal{Z}^m$ splits at time $t-m\\mathcal{T} \\in J_m$. If particle\n$i$ has a higher type than particle $j$, then the new particle is\npaired with individual~$j$. The particle that was paired with\nindividual $j$ before the branching event is no longer paired with any\nindividual in $X$. If particle $i$ has a lower type than particle $j$,\nthen the particle that was paired with individual $j$ remains paired\nwith individual $j$. The new particle is not paired with any individual\nin $X$.\n\\item A particle labeled $i$ in $\\mathcal{Z}^m$ splits at a\ntime-dependent rate $\\gamma(R^{i,k}_t-U_t^i)$ for all times $t \\in\nJ_m$ where $R^{i,k}_t$ is the type of particle $i$. These branching\nevents are independent of any of the events in $X$. When such a\nbranching event occurs, the new particle is not paired with any\nindividual in $X$.\n\\item Any particles in $\\mathcal{Z}^m$ that are not paired with an\nindividual in $X$ branch and acquire mutations independently of $X$.\n\\end{itemize}\n\nObserve the following bound for $D_t$:\n\\begin{eqnarray*}\nD_t &\\leq&\\sum_{i=1}^{N_t-1}(D_{t_{i+1}}-D_{s_i})\n+ \\sum_{i=1}^{N_t}(D_{s_i}-D_{t_i})\n+ \\sup_{s_{N_t} \\leq s \\leq t_{N_t+1}} (D_s - D_{s_{N_t}})\\\\\n&&{} +\n\\sup_{t_{N_t+1} \\leq s \\leq t} (D_s - D_{t_{N_t+1}}),\n\\end{eqnarray*}\nwhere we consider the supremum over the empty set to be 0. By\ndefinition we have\n\\[\n\\sum_{i=1}^{N_t-1}(D_{t_{i+1}}-D_{s_i})\n+ \\sup_{s_{N_t} \\leq s \\leq\nt_{N_t+1}} (D_s - D_{s_{N_t}}) \\leq\\sum\n_{i=1}^{N_t} Y_i.\n\\]\n\nTo finish the proof we will show\n\\[\n\\sum_{i=1}^{N_t}\\sup_{t_i \\leq s \\leq s_i}(D_s-D_{t_i})\n+ \\sup_{t_{N_t+1} \\leq s \\leq t} (D_s - D_{t_{N_t+1}}) \\leq\nD_t'.\n\\]\nTo do this we define\n\\[\nM_t = \\sum_{i=1}^{N_t}\n\\sup_{t_i \\leq s \\leq s_i}(D_s-D_{t_i}) + \\sup_{t_{N_t+1} \\leq s \\leq t}\n(D_s - D_{t_{N_t+1}})\n\\]\nfor all times $t \\geq0$. Suppose $M_s \\leq D_s'$ for all $s \\in[0,t)$\nand a mutation, resampling or selection event occurs in $X$ at time\n$t$. If $t \\in(s_i, t_{i+1})$ for some $i \\geq0$, then $M_{t-} = M_t$\nbecause the process $M$ does not change on these time intervals. It is\npossible that $D_t'$ changes, but $D_t'$ can only increase. Therefore,\n$D_t' \\geq M_t$. If $t \\in[t_i, s_i] \\cap(m\\mathcal{T},(m+1)\\mathcal\n{T}]$ for some $i \\geq0$ and $m \\geq0$, then at time $t$ the\nprocesses $X$ and $X'$ are coupled. More precisely, $X$ and $\\mathcal\n{Z}^m$ are coupled and the coupling has the same dynamics as the\ncoupling in Proposition~\\ref{UpBound} except the time shift. The same\nargument used in Proposition~\\ref{UpBound} shows that $D_t' \\geq M_t$\nwhether the individual changed fitness due to mutation, resampling or\nselection. Since this inequality is preserved on any event that may\nchange $M_t$, it is true for all times $t$.\n\\end{pf*}\n\n\\section{Bounding the rate when the width is large}\\label{sec4}\nWe consider what happens when the width is large in this section. By\nlarge width we mean $W_t \\geq\\mathcal{W}$. The statements in this\nsection are easier to make when we consider an initial configuration of\n$X$ such that $W_0 \\geq\\mathcal{W}$. Although the conditions of\nTheorem~\\ref{Theorem} state that $W_0 = 0$, we can wait for a random\ntime $\\tau$ so that $W_\\tau\\geq\\mathcal{W}$ and apply the strong\nMarkov property.\n\nWe begin this section by showing that when the width is large enough\nthe selection mechanism will cause the width to decrease quickly. We\ngive a labeling to the individuals that will help\nus in this regard. Define the following subsets of $\\R$:\n\\begin{eqnarray*}\nI_1 &=& \\bigl(-\\infty, X_0^+-\\tfrac{3}{16}W_0\\bigr],\n\\\\\nI_2 &=& \\bigl(X_0^+-\\tfrac{3}{16}W_0,\nX_0^+-\\tfrac{2}{16}W_0\\bigr],\n\\\\\nI_3 &=& \\bigl(X_0^+-\\tfrac{2}{16}W_0,\nX_0^+-\\tfrac{1}{16}W_0\\bigr],\n\\\\\nI_4 &=& \\bigl(X_0^+-\\tfrac{1}{16}W_0,\n\\infty\\bigr).\n\\end{eqnarray*}\n\nWe will label each individual in $X_0$ with two labels. For the first\nlabeling, we use $\\mathfrak{a}$ to label the individuals in $I_1 \\cup\nI_2$, we use $\\mathfrak{b}$ to label the individuals in $I_3$ and we\nuse $\\mathfrak{c}$ to label the individuals in $I_4$. For the second\nlabeling we use $\\mathfrak{a}'$ to label the individuals in $I_1$, we\nuse $\\mathfrak{b}'$ to label the individuals in $I_2$ and we use\n$\\mathfrak{c}'$ to label the individuals in $I_3 \\cup I_4$.\n\nLet $\\mathfrak{A}_t$, $\\mathfrak{B}_t$ and $\\mathfrak{C}_t$ denote\nthe number of individuals labeled $\\mathfrak{a}$, $\\mathfrak{b}$ and\n$\\mathfrak{c}$ at time~$t$, respectively. Let $\\mathfrak{A}_t'$,\n$\\mathfrak{B}_t'$ and $\\mathfrak{C}_t'$ denote the number of\nindividuals labeled $\\mathfrak{a}'$, $\\mathfrak{b}'$ and $\\mathfrak\n{c}'$ at time $t$, respectively.\n\nThe individuals change labels over time according to the following dynamics:\n\\begin{itemize}\n\\item Mutations: If the fitness of an individual labeled $\\mathfrak\n{a}$ increases so that it is in~$I_3$, then the individual is relabeled\n$\\mathfrak{b}$. If the fitness of a individual labeled $\\mathfrak\n{a}'$ increases so that it is in $I_2$, then the individual is\nrelabeled $\\mathfrak{b}'$. Likewise, if the fitness of a individual\nlabeled $\\mathfrak{b}$ increases so that it is in $I_4$, then it is\nrelabeled $\\mathfrak{c}$ and if the fitness of a individual labeled\n$\\mathfrak{b}'$ increases so that it is in $I_3$, then it is relabeled\n$\\mathfrak{c}'$. Deleterious mutations do not cause individuals to be\nrelabeled.\n\\item Resampling: Any resampling event in which individual $i$ is\nreplaced by individual $j$ causes individual $i$ to inherit the labels\nof individual $j$.\n\\item Selection: If an individual labeled $\\mathfrak{a}$ is replaced\ndue to a selection event, it inherits the corresponding label of the\nindividual that replaced it. If an individual labeled $\\mathfrak{a}'$\nis replaced due to a selection event, it inherits the corresponding\nlabel of the individual that replaced it. If an individual labeled\n$\\mathfrak{b}$ is replaced by an individual labeled $\\mathfrak{c}$\ndue to a selection event, then the individual that was labeled\n$\\mathfrak{b}$ is relabeled $\\mathfrak{c}$. If an individual labeled\n$\\mathfrak{b}'$ is replaced by\\vadjust{\\goodbreak} an individual labeled $\\mathfrak{c}'$\ndue to a selection event, then the individual that was labeled\n$\\mathfrak{b}'$ is relabeled~$\\mathfrak{c}'$. Any other selection\nevents do not cause the labels of the individuals to be changed.\n\\end{itemize}\n\nLet $A_1$ be the event that there is an individual labeled $\\mathfrak\n{b}$ with fitness in $(-\\infty, X_0^+-\\frac{5}{32}W_0)$ for some time\n$t \\in[0,\\mathcal{T}]$. Let $A_2$ be the event that there is an\nindividual labeled $\\mathfrak{c}$ with fitness in $(-\\infty,X_0^+ -\n\\frac{3}{32}W_0)$ for some time $t \\in[0,\\mathcal{T}]$. Let $A_1'$\nbe the event that there is an individual labeled $\\mathfrak{b}'$ with\nfitness in $(-\\infty, X_0^+-\\frac{7}{32}W_0)$ for some time $t \\in\n[0,\\mathcal{T}]$. Let $A_2'$ be the event that there is an individual\nlabeled $\\mathfrak{c}'$ with fitness in $(-\\infty,X_0^+ - \\frac\n{5}{32}W_0)$ for some time $t \\in[0,\\mathcal{T}]$.\n\\begin{Lemma} \\label{AALem}\nSuppose $W_0 \\geq\\mathcal{W}$ for all $N$. Then\n\\[\nP\\bigl(A_1 \\cup A_2 \\cup A_1'\n\\cup A_2'\\bigr) \\rightarrow0 \\qquad\\mbox{as } N \\rightarrow\n\\infty.\n\\]\n\\end{Lemma}\n\\begin{pf}\nFirst we show the result for $A_1$. We apply Corollary \\ref\n{BackSpeedLabel} with $x = X_0^+-2W_0\/16$, $t = t_0$ and $l = W_0\/32$.\nRecall that we had defined $\\mathcal{S}_t^x$ in Definition~\\ref{S}.\nBecause $x = X_0^+-2W_0\/16$, we have that $\\mathcal{S}_0^x$ consists\nof all the individuals labeled $\\mathfrak{b}$ or $\\mathfrak{c}$.\nSetting $t = \\mathcal{T}$ and $l = W_0\/32$ will make $A_t^{x,l}$ the\nevent that an individual labeled $\\mathfrak{b}$ or $\\mathfrak{c}$ has\nfitness less than $X_0^+-\\frac{5}{32}W_0$ by time $\\mathcal{T}$. Note\nthat according to the relabeling dynamics, individual $i$ being labeled\n$\\mathfrak{b}$ or $\\mathfrak{c}$ is equivalent to $i \\in\\mathcal\n{S}^x$. Therefore, $A_1 \\subset A_t^{x,l}$ and we get\n\\[\nP(A_1) \\leq P\\bigl(A_t^l\\bigr) \\leq\n\\frac{2N(t\\mu)^le^{(\\gamma(W_0+2l)+\\mu\n+1)t}}{\\lfloor l-1 \\rfloor!}.\n\\]\nApplying Stirling's formula we have\n\\[\n\\frac{2N(t\\mu)^le^{(\\gamma(W_0+2l)+\\mu+1)t}}{\\lfloor l-1 \\rfloor!} \\sim\n\\frac{2N(t\\mu)^le^{(\\gamma(W_0+2l)+\\mu+1)t+\\lfloor l-1 \\rfloor\n}}{\\lfloor l-1 \\rfloor^{\\lfloor l-1 \\rfloor}\\sqrt{2\\pi\\lfloor l-1\n\\rfloor}} = e^x,\n\\]\nwhere\n\\begin{eqnarray*}\nx &=& \\log(2N)+l\\log(t\\mu)+\\bigl(\\gamma(W_0+2l)+\\mu+1\\bigr)t+\\lfloor\nl-1 \\rfloor\\\\\n&&{}-\\lfloor l-1 \\rfloor\\log\\bigl(\\lfloor l-1 \\rfloor\\bigr)-\\log\n\\bigl(2\\pi\n\\lfloor l-1 \\rfloor\\bigr)\/2.\n\\end{eqnarray*}\nAs $N \\rightarrow\\infty$ we have $x \\sim-\\lfloor l-1 \\rfloor\\log\n(\\lfloor l-1 \\rfloor) \\sim-w\\log N\/32$. Therefore,\n\\[\nP(A_1) \\rightarrow0 \\qquad\\mbox{as }N \\rightarrow\\infty.\n\\]\n\nWe can apply Corollary~\\ref{BackSpeedLabel} with $x = X_0^+-W_0\/16$,\n$t = \\mathcal{T}$ and $l = W_0\/32$ to get the same bound for $P(A_2)$.\nBy choosing $x$, $t$ and $l$ in this way, the event $A_t^{x,l}$ is the\nevent that an individual labeled $\\mathfrak{c}$ has fitness less than\n$X^+(0)-\\frac{3}{32}W_0$ by time~$\\mathcal{T}$. This shows that\n$P(A_2)$ also tends to 0 as $N$ tends to infinity.\n\nLikewise, to show $P(A_1')$ tends to 0 as $N$ goes to infinity we can\napply Corollary~\\ref{BackSpeedLabel} with $x = X_0^+-\\frac\n{3}{16}W_0$, $t = \\mathcal{T}$ and $l = W_0\/32$,\\vadjust{\\goodbreak} and to show $P(A_2')$\ntends to 0 as $N$ goes to infinity we can apply Corollary \\ref\n{BackSpeedLabel} with $x = X_0^+-\\frac{2}{16}W_0$, $t = \\mathcal{T}$\nand $l = W_0\/32$.\n\\end{pf}\n\\begin{Lemma} \\label{CLem}\nSuppose $W_0 \\geq\\mathcal{W}$ for all $N$. Let $T$ be a stopping time\nwhose definition may depend on $N$ such that $\\mathfrak{C}_T' \\geq\nN\/4$ for all $N$. Let $B_T = \\inf\\{t \\geq T\\dvtx X_t^- > X_0^+ - W_0\/4\\}\n$. Then\n\\[\nP\\bigl(B_T1_{\\{T<\\mathcal{T}\/2\\}} > \\tfrac{1}{2}\\mathcal\n{T}\\bigr)\n\\rightarrow0 \\qquad\\mbox{as } N \\rightarrow\\infty.\n\\]\n\\end{Lemma}\n\\begin{pf}\nLet $A_3'$ be the event that $\\mathfrak{C}_t' \\geq N\/5$ for all times\n$t \\in[T,T+\\frac{1}{2}\\mathcal{T})$. The only way for an individual\nlabeled $\\mathfrak{c}'$ to change its label is for it to be replaced\nby an individual labeled $\\mathfrak{a}'$ or $\\mathfrak{b}'$ via a\nresampling event. The rate at which individuals marked $\\mathfrak{c}'$\nundergo resampling events with individuals marked $\\mathfrak{a}'$ or\n$\\mathfrak{b}'$ at time $t$ is\n\\[\n\\frac{\\mathfrak{C}_t'(N-\\mathfrak{C}_t')}{N} \\leq\\frac{N}{4}.\n\\]\n\nLet $\\{U_n\\}_{n=0}^\\infty$ be a simple random walk with $U_0 = N\/4\n\\leq\\mathfrak{C}_T'$. Let $T \\leq t_1 < t_2 < \\cdots$ be the times at\nwhich individuals labeled $\\mathfrak{c}'$ are involved in resampling\nevents with individuals that are not labeled $\\mathfrak{c}'$ after\ntime $T$. We couple $\\{U_n\\}_{n=0}^\\infty$ with $X$ so that if at time\n$t_n$ an individual is labeled $\\mathfrak{c}'$ due to a resampling\nevent, then $U_n = U_{n-1}+1$. If at time $t_n$ an individual loses the\nlabel $\\mathfrak{c}'$ due to a resampling event, then $U_n =\nU_{n-1}-1$. To have $U_m < N\/5$ for some $m$ satisfying $0 \\leq m \\leq\nn$ we will need ${\\max_{0 \\leq m \\leq n}}|U_m-U_0| \\geq N\/20$. It\nfollows from the reflection principle that there exists a constant $C$\nsuch that $E[{\\max_{0 \\leq m \\leq n}}|U_m-U_0|] \\leq C\\sqrt{n}$ for all\n$n \\geq0$. By Markov's inequality,\n\\[\nP \\Bigl(\\max_{0 \\leq m \\leq n}|U_m-U_0| \\geq N\/20 \\Bigr)\n\\leq C\\sqrt{n}\/N\n\\]\nfor some constant $C$.\n\nLet $R$ be the number of resampling events that occur in the time\ninterval $[T,T+\\frac{1}{2}\\mathcal{T})$ that involve pairs of\nindividuals such that one is labeled $\\mathfrak{c}'$ and the other is\nnot. Using Lemma~\\ref{TRTBoundLem} in the \\hyperref[app]{Appendix} and\nthe fact that the\nrate at which resampling events occur is bounded above by $N\/4$, we have\n\\[\nP(R > k) \\leq\\sum_{i=k+1}^\\infty\n\\frac{(N\\mathcal{T})^i\ne^{-N\\mathcal{T}\/8}}{8^i i!} \\leq\\frac{(N\\mathcal{T})^k}{8^k k!}.\n\\]\nThen\n\\begin{eqnarray*}\nP\\bigl(\\bigl(A_3'\\bigr)^C\\bigr) & \\leq & P\n\\Bigl(\\Bigl\\{\\max_{0 \\leq m \\leq R}|U_m-U_0| \\geq N\/20\\Bigr\n\\} \\cap\\bigl\\{R \\leq N^{3\/2}\\bigr\\}\\Bigr)\n\\\\\n&&{} + P\\Bigl(\\Bigl\\{\\max_{0 \\leq m \\leq R}|U_m-U_0| \\geq\nN\/20\\Bigr\\} \\cap\\bigl\\{R > N^{3\/2}\\bigr\\}\\Bigr)\n\\\\\n& \\leq & P\\Bigl(\\Bigl\\{\\max_{0 \\leq m \\leq N^{3\/2}}|U_m-U_0| \\geq\nN\/20\\Bigr\\}\\Bigr) + P\\bigl(R > N^{3\/2}\\bigr)\n\\\\[-2pt]\n& \\leq &\\frac{C}{N^{1\/4}} + \\frac{(N\\mathcal\n{T})^{N^{3\/2}}}{8^{N^{3\/2}} \\lceil N^{3\/2} \\rceil!}\n\\\\[-2pt]\n& \\rightarrow & 0 \\qquad\\mbox{as } N \\rightarrow\\infty.\n\\end{eqnarray*}\n\nLet $A_4'$ be the event that $\\mathfrak{A}_t' = 0$ for some time $t\n\\in[T,T+\\frac{1}{2}\\mathcal{T})$. Notice that if $\\mathfrak{A}_t' =\n0$, then $\\mathfrak{A}_s' = 0$ for $s \\geq t$. Therefore, $A_4'$ is\nthe event that the label $\\mathfrak{a}'$ is eliminated by time\n$T+\\frac{1}{2}\\mathcal{T}$. By the given dynamics, $\\mathfrak{A}_t'$\ncan only increase when individuals marked $\\mathfrak{a}'$ replace\nindividuals marked $\\mathfrak{b}'$ or $\\mathfrak{c}'$ via resampling\nevents. At time $t$ the rate at which this happens is\n\\begin{equation}\n\\label{AUpper} \\frac{1}{2} \\cdot\\frac{\\mathfrak{A}_t'(N-\\mathfrak\n{A}_t')}{N} \\leq\n\\mathfrak{A}_t'.\n\\end{equation}\n\nWe define the event $\\mathcal{E}$ as\n\\[\n\\mathcal{E} = \\bigl(A_1'\\bigr)^C \\cap\n\\bigl(A_2'\\bigr)^C \\cap A_3'\n\\cap\\bigl\\{T < \\tfrac\n{1}{2}\\mathcal{T}\\bigr\\}.\n\\]\nSelection will cause $\\mathfrak{A}'$ to decrease. On the event\n$(A_2')^C$ all of the individuals marked $\\mathfrak{c}'$ will have\nfitness at least $\\frac{1}{32}W_0$ greater than any individual marked\n$\\mathfrak{a}$ until time $t_0$. Thus, on the event $(A_2')^C \\cap\\{T\n< \\frac{1}{2}t_0\\}$, all of the individuals marked $\\mathfrak{c}'$\nwill have fitness at least $\\frac{1}{32}W_0$ greater than any\nindividual marked $\\mathfrak{a}$ for all times $t \\in[T,T+\\frac\n{1}{2}\\mathcal{T})$. On the event $A_3'$ there are at least $N\/5$\nindividuals marked $\\mathfrak{c}$ for all times $t \\in[T,T+\\frac\n{1}{2}\\mathcal{T})$. Hence, on the event $\\mathcal{E}$ individuals\nmarked $\\mathfrak{a}'$ will become individuals marked $\\mathfrak{c}'$\nby a rate of at least\n\\begin{equation}\n\\label{ALower} \\frac{\\gamma\\mathfrak{A}_t'\\mathfrak{C}_t' W_0}{32N} \\geq\n\\frac\n{\\gamma}{160} W_0\n\\mathfrak{A}_t'\n\\end{equation}\nfor all times $t \\in[T,T+\\frac{1}{2}\\mathcal{T})$.\n\nLet $\\{U_n'\\}$ be a biased random walk which goes up with probability\n\\[\np' = \\frac{160}{160+\\gamma W_0}\n\\]\nand down with probability $1-p'$. Let $N$ be large enough so that $p' <\n1\/2$. Because the random walk is biased downward, the probability that\nthe random walk visits a state $j < U_0'$ is 1. Once the random walk is\nin state $j$, it goes up 1 with probability $p'$ and will eventually\nreturn to $j$ with probability~1. The random walk will go down 1 with\nprobability $1-p'$ and, from basic martingale arguments, the\nprobability that it never returns to $j$ again is $(1-2p')\/(1-p')$.\nTherefore, once $U'$ is in state $j$, the probability it never returns\nto state $j$ is\n\\[\n\\frac{(1-2p')}{1-p'} \\cdot\\bigl(1-p'\\bigr) = 1-2p'.\n\\]\nHence, the number of times $U'$ visits a state $j < U_0'$ has the\ngeometric distribution with mean $1\/(1-2p')$. For more details see\n\\cite{DNA}, pages 194--196.\\vadjust{\\goodbreak}\n\nBy equations (\\ref{AUpper}) and (\\ref{ALower}) we see that on the\nevent $\\mathcal{E}$, if $\\mathfrak{A}'$ changes during the time\ninterval $[T,T+\\frac{1}{2}\\mathcal{T})$, it decreases with\nprobability higher than $p'$. The expected number of times that\n$\\mathfrak{A}'$ will visit state $j$ is therefore less than or equal\nto $1\/(1-2p')$ for any $j \\in\\{1,2,\\ldots,N-1\\}$. Also, the rate at\nwhich $\\mathfrak{A}_t'$ changes state is at least\n\\[\n\\frac{\\gamma}{160}W_0\\mathfrak{A}_t'\n\\]\nfor all times $t \\in[T,T+\\frac{1}{2}\\mathcal{T})$ by equation (\\ref\n{ALower}). Let $\\overline{A} = \\{t \\geq T\\dvtx\\mathfrak{A}_t' > 0\\}$ and\nlet $\\lambda$ be Lebesgue measure. Then\n\\[\nE\\bigl[\\lambda(\\overline{A})1_\\mathcal{E}\\bigr] \\leq\\frac\n{160}{(1-2p')\\gamma\nW_0}\n\\sum_{j=1}^N \\frac{1}{j} \\sim\n\\frac{160\\log N}{\\gamma W_0}\n\\]\nas $N \\rightarrow\\infty$.\n\nObserve that\n\\begin{eqnarray*}\nP\\bigl(\\mathcal{E} \\cap\\bigl(A_4'\\bigr)^C\n\\bigr) & = & P \\biggl(\\mathcal{E} \\cap\\biggl\\{ \\lambda(\\overline{A}) \\geq\n\\frac{1}{2}\\mathcal{T} \\biggr\\} \\biggr)\n\\\\\n& = & P \\biggl(\\lambda(\\overline{A})1_\\mathcal{E} \\geq\\frac\n{1}{2}\n\\mathcal{T} \\biggr)\n\\\\\n& \\leq &\\frac{2 E[\\lambda(\\overline{A})1_\\mathcal{E}]}{\\mathcal{T}}\n\\qquad\\mbox{by Markov's inequality}\n\\\\\n& \\rightarrow &0 \\qquad\\mbox{as } N \\rightarrow\\infty.\n\\end{eqnarray*}\nTherefore,\n\\[\nP\\bigl(\\mathcal{E} \\cap A_4'\\bigr)-P \\bigl(T<\n\\tfrac{1}{2}\\mathcal{T} \\bigr) \\rightarrow0 \\qquad\\mbox{as } N \\rightarrow\n\\infty.\n\\]\n\nThis allows us to do the following computation:\n\\begin{eqnarray*}\n1 & = & \\lim_{N \\rightarrow\\infty} \\biggl(P \\biggl(T < \\frac\n{1}{2}\\mathcal{T}\n\\biggr)+P \\biggl(T \\geq\\frac{1}{2}\\mathcal{T} \\biggr) \\biggr)\n\\\\\n& = & \\lim_{N \\rightarrow\\infty} \\biggl(P\\bigl(\\mathcal{E} \\cap A_4'\n\\bigr)+P \\biggl(T \\geq\\frac{1}{2}\\mathcal{T} \\biggr) \\biggr)\n\\\\\n& = & \\lim_{N \\rightarrow\\infty} \\biggl(P \\biggl(\\bigl(A_1'\n\\bigr)^C \\cap\\bigl(A_2'\\bigr)^C\n\\cap A_3' \\cap A_4' \\cap\n\\biggl\\{T < \\frac{1}{2}\\mathcal{T} \\biggr\\} \\biggr)+P \\biggl(T \\geq\n\\frac{1}{2}\\mathcal{T} \\biggr) \\biggr)\n\\\\\n& \\leq &\\lim_{N \\rightarrow\\infty} \\biggl(P \\biggl( \\biggl\\{B_T \\leq\n\\frac{1}{2}\\mathcal{T} \\biggr\\} \\cap\\biggl\\{T < \\frac{1}{2}\\mathcal\n{T} \\biggr\\} \\biggr)+P \\biggl(T \\geq\\frac{1}{2}\\mathcal{T} \\biggr)\n\\biggr)\n\\\\\n& = &\\lim_{N \\rightarrow\\infty} P \\biggl(B_T1_{\\{T<\n\\mathcal{T}\/2\\}}\\leq\n\\frac{1}{2}\\mathcal{T} \\biggr).\n\\end{eqnarray*}\n\\upqed\n\\end{pf}\n\nLet $B = \\inf\\{t\\dvtx X_t^- > X_0^+ - W_0\/4\\}$.\n\\begin{Prop} \\label{BProp}\nSuppose $W_0 \\geq\\mathcal{W}$ for all $N$. As $N$ tends to infinity,\n\\[\nP(B > \\mathcal{T}) \\rightarrow0.\n\\]\n\\end{Prop}\n\\begin{pf}\nFirst note that if $\\mathfrak{B}_0+\\mathfrak{C}_0 \\geq N\/4$ then,\nbecause all of the individuals labeled $\\mathfrak{b}$ or $\\mathfrak\n{c}$ at time 0 are also labeled $\\mathfrak{c}'$, we have that\n$\\mathfrak{C}_0' \\geq N\/4$. The result then follows by Lemma \\ref\n{CLem} with $T = 0$. On the other hand, if $\\mathfrak{B}_0+\\mathfrak\n{C}_0 < N\/4$ then $\\mathfrak{A}_0 \\geq3N\/4$.\n\nLet $T = (\\inf\\{t\\dvtx\\mathfrak{A}_t < N\/4\\}) \\wedge(\\inf\\{t\\dvtx\n\\mathfrak{C}_t \\geq N\/4\\})$. Let $A_5$ be the event that $\\mathfrak\n{A}_t \\geq N\/4$ for all times $t \\in[0,\\frac{1}{2}\\mathcal{T})$. Let\n$A_6$ be the event that $\\mathfrak{C}_t < N\/4$ for all times $t \\in\n[0,\\frac{1}{2}\\mathcal{T})$. Define $\\zeta$ to be the infimum over\nall times such that an individual labeled $\\mathfrak{b}$ has fitness\nin $(-\\infty, X_0^+-\\frac{5}{32}W_0)$, an individual labeled\n$\\mathfrak{c}$ has fitness in $(-\\infty,X_0^+-\\frac{3}{32}W_0)$ or\n$\\mathfrak{A}_t < N\/4$. Note that $A_1^C \\cap A_2^C \\cap A_5 \\subset\\{\n\\zeta\\geq\\frac{1}{2}\\mathcal{T}\\}$.\n\nOn the event $\\{\\zeta\\geq\\frac{1}{2}\\mathcal{T}\\}$, the rate of\nincrease of $\\mathfrak{C}_t$ due to selection is at least\n\\begin{equation}\n\\label{CLower} \\frac{\\gamma\\mathfrak{A}_t \\mathfrak{C}_t W_0}{32N} \\geq\n\\frac\n{1}{128}\\gamma\n\\mathfrak{C}_t W_0\n\\end{equation}\nfor all $t \\in[0,\\frac{1}{2}\\mathcal{T})$. On the other hand,\nbecause $\\mathfrak{C}_t$ can only decrease due to resampling,\n$\\mathfrak{C}_t$ will decrease no faster than\n\\begin{equation}\n\\label{CUpper} \\frac{1}{2}\\cdot\\frac{\\mathfrak{C}_t(N-\\mathfrak\n{C}_t)}{N} \\leq\n\\mathfrak{C}_t.\n\\end{equation}\n\nLet $\\{U_n\\}_{n=0}^\\infty$ be a biased random walk with $U_0 = 1$\nwhich goes up with probability\n\\[\np = \\frac{\\gamma W_0}{128+\\gamma W_0}\n\\]\nand down with probability $1-p$. Let $N$ be large enough so that $p >\n1\/2$. By similar reasoning as was used in the proof of Lemma \\ref\n{CLem}, the number of times $U_n$ visits a state $j \\geq1$ has the\ngeometric distribution with mean $1\/(2p-1)$. Also, by basic martingale\narguments, the probability that $U_n$ ever reaches state 0 is\n\\[\n\\frac{1-p}{p} = \\frac{128}{\\gamma W_0}.\n\\]\n\nNote that $\\mathfrak{C}_0 \\geq U_0$ since the individual with the\nhighest fitness is initially labeled~$\\mathfrak{c}$. On the event $\\{\n\\zeta\\geq\\frac{1}{2}\\mathcal{T}\\}$, we see from equations (\\ref\n{CLower}) and (\\ref{CUpper}) that if $\\mathfrak{C}$ changes during\ntime $[0,\\frac{1}{2}\\mathcal{T})$, then it increases with a\nprobability of at least $p$. Therefore, the expected number of times\nthat $\\mathfrak{C}$ visits state $j$ is less than or equal to\n$1\/(2p-1)$ and the probability the $\\mathfrak{C}_t$ reaches state 0\nfor some time $t \\in[0,\\frac{1}{2}\\mathcal{T})$ is less than\n$128\/(\\gamma W_0)$. Let $A_7$ be the event that $\\mathfrak{C}_t$\nreaches state 0 for some time $t \\in[0,\\frac{1}{2}\\mathcal{T})$.\n\nBy equation (\\ref{CLower}), the rate at which $\\mathfrak{C}$ changes\nis at least\n\\[\n\\tfrac{1}{128}\\gamma\\mathfrak{C}_t W_0\\vadjust{\\goodbreak}\n\\]\nfor all times $t \\in[0,\\frac{1}{2}\\mathcal{T})$ on the event $\\{\n\\zeta> \\frac{1}{2}\\mathcal{T}\\}$. Let $\\overline{C} = \\{t \\in\n[0,\\frac{1}{2}\\mathcal{T})\\dvtx\\mathfrak{C} < \\frac{1}{4}N\\}$ and let\n$\\lambda$ be Lebesgue measure. Then\n\\begin{eqnarray*}\nE\\bigl[\\lambda(\\overline{C})1_{\\{\\zeta\\geq\\mathcal{T}\/2\\}}\\bigr] & = &\nE\\bigl[\\lambda(\n\\overline{C})1_{\\{\\zeta\\geq\\mathcal{T}\/2\\}}1_{A_7}\\bigr] + E\\bigl\n[\\lambda(\n\\overline{C})1_{\\{\\zeta\\geq\\mathcal{T}\/2\\}}1_{A_7^C}\\bigr]\n\\\\\n& \\leq &\\frac{1}{2}\\mathcal{T} P(A_7) + \\frac{128}{(2p-1)\\gamma W_0}\n\\sum_{j=1}^{\\lfloor N\/4 \\rfloor} \\frac{1}{j}\n\\\\\n& \\sim &\\frac{128\\log(N\/4)}{\\gamma W_0}.\n\\end{eqnarray*}\n\nBy Markov's inequality\n\\begin{eqnarray*}\nP\\bigl(A_1^C \\cap A_2^C \\cap\nA_5 \\cap A_6\\bigr) & \\leq & P\\biggl(A_1^C\n\\cap A_2^C \\cap A_5 \\cap\\biggl\\{\\lambda(\n\\overline{C}) \\geq\\frac{1}{2}\\mathcal{T}\\biggr\\}\\biggr)\n\\\\\n& \\leq & P\\biggl(\\biggl\\{\\zeta\\geq\\frac{1}{2}\\mathcal{T}\\biggr\\} \\cap\n\\biggl\n\\{\\lambda(\\overline{C}) \\geq\\frac{1}{2}\\mathcal{T}\\biggr\\}\\biggr)\n\\\\\n& = & P\\biggl(\\lambda(\\overline{C})1_{\\{\\zeta\\geq\\mathcal{T}\/2\\}} \\geq\n\\frac{1}{2}\n\\mathcal{T}\\biggr)\n\\\\\n& \\leq &\\frac{2E[\\lambda(\\overline{C})1_{\\{\\zeta\\geq\\mathcal{T}\/2\\}\n}]}{\\mathcal{T}}\n\\\\\n& \\leq &\\frac{256w^{1\/4}\\log(N\/4)}{\\mathcal{T}\\gamma W_0} \\qquad\\mbox{for\n}N\\mbox{ large enough}\n\\\\\n& \\rightarrow &0 \\qquad\\mbox{as } N \\rightarrow\\infty.\n\\end{eqnarray*}\nBecause $P(A_1^C \\cap A_2^C) \\rightarrow1$ we have $P(A_5^C \\cup\nA_6^C) \\rightarrow1$ as $N \\rightarrow\\infty$.\n\nNote that $A_5^C \\cup A_6^C \\subset\\{T < \\frac{1}{2}\\mathcal{T}\\}$.\nTherefore, $P(T < \\frac{1}{2}\\mathcal{T}) \\rightarrow1$ as $N\n\\rightarrow\\infty$. Let $E_2 = (A_1')^C \\cap(A_2')^C \\cap\\{T <\n\\frac{1}{2}\\mathcal{T}\\}$. Then $P(E_2) \\rightarrow1$ as $N\n\\rightarrow\\infty$. To show $P(B \\leq\\mathcal{T}) \\rightarrow1$ we\ncan show $P(\\{B \\leq\\mathcal{T}\\} \\cap E_2) \\rightarrow1$. At time\n$T$, at least $\\frac{1}{4}N$ individuals will be labeled either\n$\\mathfrak{b}$ or $\\mathfrak{c}$. According to the labeling, all of\nthese individuals are labeled $\\mathfrak{c}'$ so that at time $T$ we\nhave $\\mathfrak{C}_T \\geq\\frac{1}{4}N$. By Lemma~\\ref{CLem} we have\n\\[\nP \\bigl(B_T 1_{\\{T < \\mathcal{T}\/2\\}} \\leq\\tfrac\n{1}{2}\\mathcal{T}\n\\bigr) \\rightarrow1 \\qquad\\mbox{as } N \\rightarrow\\infty.\n\\]\nNote that\n\\[\n\\bigl\\{B_T 1_{\\{T < \\mathcal{T}\/2\\}} \\leq\\tfrac\n{1}{2}\\mathcal{T}\n\\bigr\\} = \\bigl\\{B_T \\leq\\tfrac{1}{2}\\mathcal{T} \\bigr\\}\n\\cup\\bigl\\{T \\geq\\tfrac{1}{2}\\mathcal{T} \\bigr\\}.\n\\]\nBecause $E_2 \\subset\\{T < \\frac{1}{2}\\mathcal{T}\\}$ we have\n\\[\n\\bigl\\{B_T 1_{\\{T < \\mathcal{T}\/2\\}} \\leq\\tfrac\n{1}{2}\\mathcal{T}\n\\bigr\\} \\cap E_2 = \\bigl\\{B_T \\leq\\tfrac\n{1}{2}\n\\mathcal{T} \\bigr\\} \\cap E_2.\n\\]\nIt then follows that\n\\[\nP \\bigl( \\bigl\\{B_T \\leq\\tfrac{1}{2}\\mathcal{T} \\bigr\\}\n\\cap E_2 \\bigr) \\rightarrow1 \\qquad\\mbox{as } N \\rightarrow\\infty.\n\\]\nHowever,\n\\[\n\\bigl\\{B_T \\leq\\tfrac{1}{2}\\mathcal{T} \\bigr\\} \\cap\nE_2 \\subset\\bigl\\{B_T \\leq\\tfrac{1}{2}\n\\mathcal{T} \\bigr\\} \\cap\\bigl\\{T < \\tfrac{1}{2}\\mathcal{T} \\bigr\\}\n\\subset\\{B \\leq\\mathcal{T}\\},\n\\]\nwhich gives the conclusion.\n\\end{pf}\n\nLet $V_t^1 = \\{i\\dvtx X_t^i > X_0^+ + W_0\/4\\}$ and $V_t^2 = \\{i\\dvtx\nX_t^i <\nX_0^- - W_0\/4\\}$. Let $F = \\inf\\{t\\dvtx V_t^1 \\cup V_t^2 \\neq\\varnothing\n\\}\n$. We now want to bound the time it takes for the width to increase.\n\\begin{Prop} \\label{FProp}\nSuppose $W_0 \\geq\\mathcal{W}$ for all $N$. Then\n\\[\n\\lim_{N \\rightarrow\\infty} P(F > \\mathcal{T}) = 1.\n\\]\n\\end{Prop}\n\\begin{pf}\nBy Proposition~\\ref{UpBound} with $l = W_0\/4$ and $t = \\mathcal{T}$\nwe have\n\\begin{eqnarray*}\nP\\bigl(\\inf\\bigl\\{s\\dvtx V_s^1 \\neq\\varnothing\\bigr\\} < t\n\\bigr) & = & P \\Bigl(\\sup_{0 \\leq s\n\\leq t} D_s \\geq l \\Bigr)\n\\\\\n& \\leq &\\frac{2N(t\\mu)^le^{(\\gamma(W_0+2l)+\\mu+1)t}}{(l-1)!}\n\\\\\n& \\rightarrow & 0 \\qquad\\mbox{as } N \\rightarrow\\infty.\n\\end{eqnarray*}\n\nBy Proposition~\\ref{BackSpeed} with $l = W_0\/4$ and $t = \\mathcal{T}$\nwe have\n\\begin{eqnarray*}\nP\\bigl(\\inf\\bigl\\{s\\dvtx V_s^2 \\neq\\varnothing\\bigr\\} < t\n\\bigr) & = & P \\Bigl(\\sup_{0 \\leq s\n\\leq t} \\bigl(X_0^- -\nX_s^-\\bigr) \\geq l \\Bigr)\n\\\\\n& \\leq &\\frac{N(t\\mu)^l e^t}{l!}\n\\\\\n& \\rightarrow & 0 \\qquad\\mbox{as } N \\rightarrow\\infty.\n\\end{eqnarray*}\n\\upqed\n\\end{pf}\n\nRecall that\\vspace*{1pt} $Y_i = \\sup_{s_i \\leq s \\leq t_{i+1}}D_{s} -\nD_{s_i}$ and that $\\{\\mathcal{F}_t\\}_{t \\geq0}$ is the natural\nfiltration associated with $X$. Note that if $W_0 < 2\\mathcal{W}$, then\nfor all $n \\geq1$ the width satisfies $W_{s_n} = \\lceil2\\mathcal{W}\n\\rceil$.\n\\begin{pf*}{Proof of Proposition~\\ref{TheoProp2}}\nWe consider a sequence of initial configurations $X_0$ depending on $N$\nsuch that $W_0 = \\lceil2\\mathcal{W} \\rceil$ for all $N$. Because\n$W_0 \\geq2\\mathcal{W}$ we have $s_1 = 0$ and $Y_1 = \\sup_{0 \\leq s\n\\leq t_2} D_s-D_0$. We will show that for $N$ large enough, $E[Y_1] <\n5\\mathcal{W}$. The result then follows because $X$ is a strong Markov process.\n\nWe make the following definitions:\n\\begin{eqnarray*}\nV_t^1(s) &=& \\bigl\\{i\\dvtx X_t^i >\nX_s^+ + W_s\/4\\bigr\\} \\qquad\\mbox{for } t \\geq s \\geq0,\n\\\\\nV_t^2(s) &=& \\bigl\\{i\\dvtx X_t^i <\nX_s^- - W_s\/4\\bigr\\} \\qquad\\mbox{for } t \\geq s \\geq0,\n\\\\\nF_0 &=& B_0 = r_0 = 0,\n\\\\\nF_n &=& \\inf\\bigl\\{t \\geq r_{n-1}\\dvtx V_t^1(r_{n-1})\n\\cup V_t^2(r_{n-1}) \\neq\\varnothing\\bigr\\}\n\\qquad\\mbox{for }n \\geq1,\n\\\\\nB_n &=& \\inf\\bigl\\{t \\geq r_{n-1}\\dvtx X_t^- >\nX_{r_{n-1}}^+-W_{r_{n-1}}\/4\\bigr\\} \\qquad\\mbox{for }n \\geq1,\n\\\\\nr_n &=& F_n \\wedge B_n \\qquad\\mbox{for }n \\geq1,\n\\\\\nn_* &=& \\inf\\{n \\geq1\\dvtx W_{r_n} < \\mathcal{W}\\}.\n\\end{eqnarray*}\nNote that $r_1$ is the first time that the event $F \\cup B$ occurs and\nthat, conceptually, $r_n$~acts like the first time that $F \\cup B$\noccurs when the process is started at time $r_{n-1}$ for $n \\geq2$.\nThe random variables $F_n$ and $B_n$ play the roles of the events $F$\nand $B$ when the processes are started at time $r_{n-1}$.\n\nOn the event $n-1 < n_*$, by Proposition~\\ref{BProp} and the strong\nMarkov property of $X$, we have $P(B_n \\leq r_{n-1}+\\mathcal\n{T}|\\mathcal{F}_{r_{n-1}}) \\rightarrow1$ uniformly on a set of\nprobability 1 as $N \\rightarrow\\infty$. Likewise, on the event $n-1 <\nn_*$, by Proposition~\\ref{FProp} and the strong Markov property, we\nhave $P(F_n > r_{n-1}+\\mathcal{T}|\\mathcal{F}_{r_{n-1}}) \\rightarrow\n1$ uniformly on a set of probability 1 as $N \\rightarrow\\infty$.\nTherefore, on the event $n-1 < n_*$, we have $P(B_n < F_n|\\mathcal\n{F}_{r_{n-1}}) \\rightarrow1$ uniformly on a set of probability 1.\n\nBecause the bounds in Propositions~\\ref{BProp} and~\\ref{FProp} do not\ndepend on $n$ we can choose a sequence $p = p_N$ such that $p\n\\rightarrow1$ as $N \\rightarrow\\infty$ and almost surely\n\\[\np 1_{\\{n-1 < n_*\\}} \\leq P(B_n < F_n|\n\\mathcal{F}_{r_{n-1}})1_{\\{n-1 <\nn_*\\}}\n\\]\nfor all $n \\geq0$. Let $\\{S_n\\}_{n=0}^\\infty$ be a random walk\nstarting at 1 which goes down 1 with probability $p$ and up 1 with\nprobability $1-p$ until it reaches 0. Once $S$ reaches 0 it is fixed.\nFor $n < n_*$ we couple $S$ with $X$ so that $2^{S_n-1}W_0 \\geq\nW_{r_n}$. The coupling is defined as follows:\n\\begin{itemize}\n\\item Each step of the process $S$ corresponds to a time $r_n$.\n\\item On the event $\\{F_n < B_n\\}$ we have $S_n - S_{n-1} = 1$.\n\\item On the event $\\{B_n \\leq F_n\\}$ we have $S_n - S_{n-1} = -1$ with\nprobability $p\/P(B_n \\leq F_n)$ and we have $S_n - S_{n-1} = 1$ with\nprobability $1-p\/P(B_n \\leq F_n)$.\n\\end{itemize}\n\nWe will\\vspace*{1pt} show that this coupling is well defined and gives\nthe necessary bound. Initially, $S_0 = 1$ and $2^{S_0-1}W_0 = W_0$. On\nthe event that $B_n \\leq F_n$, we have $W_{r_n} <\n\\frac{1}{2}W_{r_{n-1}}$ and $\\sup _{r_{n-1} \\leq t \\leq r_n}\nD_t-D_{r_{n-1}} \\leq\\frac{1}{4} W_{r_{n-1}}$. On the event that $F_n <\nB_n$, we have $W_{r_n} < 2W_{r_{n-1}}$ and $\\sup_{r_{n-1} \\leq t \\leq\nr_n}D_t-D_{r_{n-1}} \\leq \\frac{1}{4}W_{r_{n-1}}+1$. Therefore, if\n$2^{S_{n-1}-1}W_0 \\geq W_{r_{n-1}}$, then $2^{S_n-1}W_0 \\geq W_{r_n}$\nby the coupling. It follows that $2^{S_n-1}W_0 \\geq\\sup_{r_{n-1} \\leq t\n\\leq r_n} D_t-D_{r_{n-1}}$ as well. By induction,\\vspace*{1pt}\n$2^{S_n-1}W_0 \\geq W_{r_n}$ for all $n < n_* \\wedge\\inf\\{m\\dvtx S_m =\n0\\}$. If $n = \\inf\\{m\\dvtx S_m=0\\}$, then $W_{r_n} \\leq\\mathcal{W}$.\nTherefore, $n_* \\leq\\inf\\{m\\dvtx S_m=0\\} $ and the induction holds for\nall $n < n_*$.\n\nWe define a function $d$ on $(\\{0\\} \\cup\\N)^\\infty$ such that if $x\n= (x_0, x_1, \\ldots)$ then\n\\[\nd(x) = \\sum_{i=0}^\\infty1_{\\{x_i > 0\\}}\n2^{x_i-1} W_0.\n\\]\nConsider $S = (S_0, S_1, \\ldots)$ as a random element in $(\\{0\\} \\cup\n\\N)^\\infty$. Then\n\\[\nd\\bigl((S_0, S_1, \\ldots, S_n, 0, 0, \\ldots)\n\\bigr) \\geq\\sum_{i=1}^n \\Bigl(\n\\sup_{r_{i-1} \\leq t \\leq r_i} D_t-D_{r_{i-1}} \\Bigr) \\geq\n\\sup_{0 \\leq t\n\\leq r_n} D_t\n\\]\nfor all $n$ such that $n-1 < n_*$. By definition, $n_*$ is the first\n$n$ such that $W_{r_n} < \\mathcal{W}$. Hence, $d(S) \\geq\nY_1$.\n\nFor any $n \\geq0$ we have\n\\[\nP(S_{2n+1} = 0) = \\pmatrix{2n+1\n\\cr\nn}(1-p)^n\np^{n+1} \\leq4^n (1-p)^n p^{n+1}.\n\\]\nIf $S_{2n+1} = 0$ then\n\\[\nd(S) \\leq\\Biggl(2 + 2\\sum_{i=1}^n\n2^{i-1} \\Biggr) W_0 = 2^{n+1} W_0,\n\\]\nwhich is obtained by taking $n$ steps up followed by $n+1$ steps down.\n\nTherefore,\n\\[\nE[Y_1] \\leq E\\bigl[d(S)\\bigr] \\leq\\sum_{n=0}^\\infty\n\\bigl[4(1-p)\\bigr]^n p^{n+1} 2^{n+1}W_0\n= \\frac{2p W_0}{1-8(1-p)p} \\sim4\\mathcal{W},\n\\]\nbecause $W_0 = \\lceil2\\mathcal{W} \\rceil$ and $p \\rightarrow1$ as\n$N \\rightarrow\\infty$. This shows that for $N$ large enough we have\n$E[Y_1] < 5\\mathcal{W}$, which gives the conclusion.\n\\end{pf*}\n\nLet $l = \\lfloor\\mathcal{W}\/2 \\rfloor$. We make the following\ndefinitions for the rest of the section:\n\\begin{eqnarray*}\nK_1 &=& \\frac{2N(\\mathcal{T}\\mu)^l e^{(\\gamma(W_0+2l)+\\mu\n+1)\\mathcal{T}}}{(l-1)!},\n\\\\[-2pt]\nK_2 &=& \\frac{N(\\mathcal{T}\\mu)^l e^{\\mathcal{T}}}{l!},\n\\\\[-2pt]\np &=& 1-K_1-K_2.\n\\end{eqnarray*}\n\n\\begin{Lemma} \\label{WidthSpeed}\nSuppose $W_0 \\leq\\mathcal{W}$ for all $N$. Then\n\\[\nP \\Bigl(\\sup_{0 \\leq s \\leq\\mathcal{T}} W_s \\leq2\\mathcal{W} \\Bigr)\n\\geq1-K_1-K_2.\n\\]\n\\end{Lemma}\n\\begin{pf}\nBy Proposition~\\ref{UpBound} we have\n\\[\nP \\Bigl(\\sup_{0 \\leq s \\leq\\mathcal{T}} D_s \\geq l \\Bigr) \\leq\nK_1.\n\\]\nBy Proposition~\\ref{BackSpeed} we have\n\\[\nP \\Bigl(\\sup_{0 \\leq s \\leq\\mathcal{T}} \\bigl(X_0^- - X_s^-\\bigr)\n\\geq l \\Bigr) \\leq K_2.\n\\]\nOn the event that $\\sup_{0 \\leq s \\leq t} D_s \\leq\\mathcal{W}\/2$ and\n$\\sup_{0 \\leq s \\leq t}X_0^--X_s^- \\leq\\mathcal{W}\/2$, we have $\\sup_{0\n\\leq s \\leq t} W_t \\leq2\\mathcal{W}$. This gives the result.\\vadjust{\\goodbreak}\n\\end{pf}\n\\begin{pf*}{Proof of Proposition~\\ref{TheoProp3}}\nNotice that\n\\[\n\\{N_s \\geq i\\} = \\{s_i \\leq s\\} \\subset\\Biggl\\{\\sum\n_{j=1}^i (s_j-t_j)\n\\leq s \\Biggr\\}.\n\\]\nTherefore,\n\\[\nP(N_s \\geq i) \\leq P \\Biggl(\\sum_{j=1}^i\n(s_j-t_j) \\leq s \\Biggr).\n\\]\nApplying Lemma~\\ref{WidthSpeed} and the strong Markov property of $X$\nwe have\n\\[\n1-K_1-K_2 \\leq P (s_j-t_j \\geq\n\\mathcal{T}|\\mathcal{F}_{t_j} )\n\\]\nfor all $j$. Taking expectations of both sides yields\n\\[\n1-K_1-K_2 \\leq P(s_j-t_j \\geq\n\\mathcal{T})\n\\]\nfor all $j$, so\n\\[\n1-K_1-K_2 \\leq\\inf_j P(s_j-t_j\n\\geq\\mathcal{T}).\n\\]\n\nNote that $p \\rightarrow1$ as $N \\rightarrow\\infty$. Define an\ni.i.d. sequence $\\{V_i\\}_{i=1}^\\infty$ of random variables with\ndistribution $P(V_i = 0) = 1-p$ and $P(V_i = \\mathcal{T}) = p$. Then\n\\[\nP \\Biggl(\\sum_{j=1}^i\n(s_j-t_j) \\leq s \\Biggr) \\leq P \\Biggl(\\sum\n_{j=1}^i V_i \\leq s \\Biggr).\n\\]\nThis will allow us to define a new process $N_s'$ such that $N_s' = i$ if\n\\[\n\\sum_{j=1}^i V_i \\leq s <\n\\sum_{j=1}^{i+1} V_i.\n\\]\nNote that $P(N_s' = 0) = p$ for $s \\in[0,\\mathcal{T})$ and that\n$P(N_s' \\geq k) \\geq P(N_s \\geq k)$ for all~$k$. Therefore, it is\nenough to bound $E[N_s']\/s$.\n\nLet $V_0 = 0$. Jumps of the process $N_s'$ only occur at points\n$k\\mathcal{T}$ where $k$ is a positive integer. On the time interval\n$[0,\\mathcal{T})$ the process $N_s'$ is constant and has value $\\max\\{\ni \\geq0\\dvtx V_i = 0\\}$. Therefore, $N_s'$ has the shifted geometric\ndistribution for $s \\in[0,\\mathcal{T})$ with mean $(1-p)\/p$. We can\nnow make use of the fact that $N_s'$ is a Markov process. If we\nconsider values at $k\\mathcal{T}$ for $k \\geq0$, we have for $s \\in\n[(k-1)\\mathcal{T}, k\\mathcal{T})$ that $E[N_s'] = k(1-p)\/p$. For $k\n\\geq2$ we then have\n\\[\n\\frac{1}{s}E\\bigl[N_s'\\bigr] =\n\\frac{k(1-p)}{sp} \\leq\\frac{k(1-p)}{(k-1)p\n\\mathcal{T}}.\n\\]\nThis gives us\n\\[\n\\frac{\\mathcal{T}}{s}E\\bigl[N_s'\\bigr] \\leq\n\\frac{k(1-p)}{(k-1)p} \\rightarrow0 \\qquad\\mbox{as }N \\rightarrow\\infty.\n\\]\nOn the time interval $[0,\\mathcal{T})$ we have\n\\[\n\\frac{\\mathcal{T}}{s}E\\bigl[N_s'\\bigr] \\leq\n\\frac{(1-p)}{p} \\rightarrow0 \\qquad\\mbox{as } N \\rightarrow\\infty.\n\\]\n\\upqed\n\\end{pf*}\n\n\\begin{center}\n\\begin{tabular*}{\\tablewidth}{@{\\extracolsep{\\fill}}l p{0.85\\textwidth}@{}}\n\\multicolumn{2}{@{}c@{}}{NOTATION}\\\\[12pt]\n$N$ & The size of the population \\\\\n$\\mu$ & The rate at which individuals accumulate mutations \\\\\n$q$ & The probability that a mutation is beneficial \\\\\n$\\gamma$ & The selection coefficient \\\\\n$X^i$ & The stochastic process in $\\Z$ that represents the fitness of\nthe $i$th individual \\\\\n$X$ & The stochastic process in $\\Z^N$ that represents the fitnesses\nof the individuals \\\\[2.5pt]\n$\\overline{X}$ & $= \\frac{1}{N}\\sum_{i=1}^N X^i$ \\\\[2.5pt]\n$X_t^+$ & $= \\max\\{X_t^i\\dvtx1 \\leq i \\leq N\\}$ \\\\[2.5pt]\n$X_t^-$ & $= \\min\\{X_t^i\\dvtx1 \\leq i \\leq N\\}$ \\\\[2.5pt]\n$W_t$ & $= X_t^+-X_t^-$ \\\\[2.5pt]\n$D_t$ & $= X_t^+ - X_0^+$ \\\\[2.5pt]\n$w$ & is any positive, increasing function satisfying $\\lim_{N\n\\rightarrow\\infty} w(N) = \\infty$ \\\\[2.5pt]\n& and $\\lim_{N \\rightarrow\\infty} w(N)\/\\log\\log N = 0$\\\\[2.5pt]\n$\\mathcal{W}$ & $= \\lfloor w\\log N\/\\log\\log N \\rfloor$ \\\\[2.5pt]\n$\\mathcal{T}$ & $= w^{-1\/2}\\log\\log N$ \\\\[2.5pt]\n$t_1$ & $= 0$ \\\\[2.5pt]\n$s_n$ & $= \\inf\\{t \\geq t_n\\dvtx W_t \\geq2\\mathcal{W}\\} \\mbox{ for } n\n\\geq1$\\\\[2.5pt]\n$t_n$ & $= \\inf\\{t \\geq s_{n-1}\\dvtx W_t < \\mathcal{W}\\} \\mbox{ for } n\n\\geq2$ \\\\[2.5pt]\n$Y_i$ & $= \\sup_{s_i \\leq t \\leq t_{i+1}}D_t - D_{s_i} \\mbox{ for }i\n\\geq1$ \\\\[2.5pt]\n$N_t$ & $= \\max\\{i\\dvtx s_i \\leq t\\} \\mbox{ for }t \\geq0$ \\\\[2.5pt]\n$Z_t^{k,\\uparrow}$ & A multi-type Yule process in which there are\ninitially $N$ particles of type $k$.\nParticles increase from type $i$ to type $i+1$ at rate $\\mu$ and\nparticles of type $i$\nbranch at rate $\\gamma i + 1$ \\\\\n$\\overline{M}{}^{k,\\uparrow}_t$ & The maximum type of any particle in\n$Z_t^{k,\\uparrow}$ \\\\[2.5pt]\n$M_t^{k,\\uparrow}$ & $\\overline{M}{}^{k,\\uparrow}_t-k$\\\\[2.5pt]\n$X_t'$ & $X_0^++\\mathcal{M}_t^0$ if $t \\in[0,\\mathcal{T}]$ and\n$X_{i\\mathcal{T}}'+\\mathcal{M}_{t-i\\mathcal{T}}^i$ if $t \\in\n(i\\mathcal{T}, (i+1)\\mathcal{T}]$ for any\\\\[2pt]\n$\\{\\mathcal{Z}_t^n\\}_{n=0}^\\infty$ & An i.i.d.\\vspace*{2pt} sequence of stochastic\nprocesses each having the same distribution\nas $Z^{\\mathcal{W},\\uparrow}$ \\\\[2.5pt]\n$\\overline{\\mathcal{M}}{}^n_t$ & The maximum type of any particle in\n$Z^{\\mathcal{W},\\uparrow}\n\\end{tabular*}\n\\end{center}\n\n\\begin{center}\n\\begin{tabular*}{\\tablewidth}{@{\\extracolsep{\\fill}}l p{0.85\\textwidth}@{}}\n$\\mathcal{M}_t^n$ & $=\\overline{\\mathcal{M}}{}^n_t - \\mathcal{W}$\\\\[2.5pt]\n& integer $i \\geq1$ \\\\[2.5pt]\n$D_t'$ & $X_t' - X_0^+$ \\\\[2pt]\n$\\mathcal{F}$ & $=\\{\\mathcal{F}_t\\}_{t \\geq0}$ is the natural\nfiltration associated with $X$ under the initial condition\n$X_0^i = 0$ for $1 \\leq i \\leq N$ \\\\[2pt]\n$Z_t^C$ & A multi-type Yule process in which there are initially $N$\nparticles of type 0.\nParticles increase from type $i$ to type $i+1$ at rate $\\mu$ and\nbranch at rate~$C$ \\\\[2pt]\n$M_t^C$ & The maximum type of any particle in $Z_t^C$ \\\\[2pt]\n$S_t$ & $=\\sup_{0 \\leq s \\leq t}(X_0^- - X_s^-)$\\\\[2pt]\n$A_t^{x,l}$ & The event that an individual in $\\overline{S}{}^x_s$ has\nfitness in $(-\\infty, x-l]$\nfor some time $s \\in[0,t]$\\\\[2pt]\n$A_1$ & The event that there is an\nindividual labeled $\\mathfrak{b}$ with fitness in $(-\\infty,\nX_0^+-\\frac{5}{32}W_0)$\nfor some time $t \\in[0,\\mathcal{T}]$\\\\[2pt]\n$A_2$ & The event that there is an\nindividual labeled $\\mathfrak{c}$ with fitness in $(-\\infty,X_0^+ -\n\\frac{3}{32}W_0)$\nfor some time $t \\in[0,\\mathcal{T}]$\\\\[2pt]\n$A_1'$ & The event that there is an\nindividual labeled $\\mathfrak{b}'$ with fitness in $(-\\infty,\nX_0^+-\\frac{7}{32}W_0)$\nfor some time $t \\in[0,\\mathcal{T}]$\\\\[2pt]\n$A_2'$ & The event that there is an\nindividual labeled $\\mathfrak{c}'$ with fitness in $(-\\infty,X_0^+ -\n\\frac{5}{32}W_0)$\nfor some time $t \\in[0,\\mathcal{T}]$\\\\[2pt]\n$B$ & $= \\inf\\{t\\dvtx X_t^- > X_0^+ -\nW_0\/4\\}$\\\\[2pt]\n$V_t^1$ & $=\\{i\\dvtx X_t^i > X_0^++W_0\/4\\}\n$\\\\[2pt]\n$V_t^2$ & $=\\{i\\dvtx X_t^i < X_0^--W_0\/4\\}\n$\\\\[2pt]\n$F$ & $=\\inf\\{t\\dvtx V_t^1 \\cup V_t^2 \\neq\n\\varnothing\\}$\n\\end{tabular*}\n\\end{center}\n\n\\begin{appendix}\\label{app}\n\\section*{Appendix}\n\\begin{Lemma} \\label{TRTBoundLem}\nLet $x \\geq0$. The tail of the exponential series satisfies\n\\[\n\\sum_{i=k}^\\infty\\frac{x^i}{i!} \\leq\n\\frac{x^k e^x}{k!}.\\vspace*{-2pt}\n\\]\n\\end{Lemma}\n\\begin{pf}\nBy Taylor's remainder theorem we know that there exists a $\\xi\\in\n[0,x]$ such that\n\\[\ne^x = \\sum_{i=1}^{k-1}\n\\frac{x^i}{i!} + \\frac{x^k e^{\\xi}}{k!}.\n\\]\nUsing the series expansion of $e^x$ we have\n\\[\n\\sum_{i=k}^\\infty\\frac{x^i}{i!} =\n\\frac{x^k e^{\\xi}}{k!} \\leq\\frac{x^k e^x}{k!}.\\hspace*{125pt} \\qed\\hspace*{-125pt}\n\\]\n\\noqed\\end{pf}\n\nRecall that $M_t^C$ is the maximum type of any particle in the\nbranching process~$Z_t^C$.\n\\begin{Lemma} \\label{YuleLem}\nFor any population size $N$, time $t \\geq0$ and natural number~$l$,\n\\[\nP\\bigl(M_t^C \\geq l\\bigr) \\leq\\frac{N (t\\mu)^l e^{Ct}}{l!}.\\vspace*{-2pt}\n\\]\n\\end{Lemma}\n\\begin{pf}\nConsider a Yule process $Z$ which is the same as $Z^C$ except there is\nonly one particle at time 0. It is well known that the number of\nparticles in $Z_t$ has mean $e^{Ct}$. Let $M_t'$ be the maximum type of\nany particle at time $t$. When there are $k$ particles in the\npopulation, we let $B_1, \\ldots, B_k$ denote the types of the\nparticles, where the numbering is independent of the mutations. For any\n$l \\geq0$,\n\\begin{eqnarray*}\nP\\bigl(M_t' \\geq l\\bigr) & = & \\sum\n_{k=1}^\\infty P\\bigl(M_t'\n\\geq l|Z_t = k\\bigr)P(Z_t = k)\n\\\\[-2pt]\n& = & \\sum_{k=1}^\\infty P\\bigl(\n\\{B_1 \\geq l\\} \\cup\\cdots\\cup\\{B_k \\geq l\\}\n|Z_t = k\\bigr)P(Z_t = k)\n\\\\[-2pt]\n& \\leq &\\sum_{k=1}^\\infty k P(B_1\n\\geq l)P(Z_t = k)\n\\\\[-2pt]\n& = & E[Z_t]P(B_1 \\geq l)\n\\\\[-2pt]\n& = & e^{Ct} \\sum_{i=l}^\\infty\n\\frac{(t\\mu)^i}{i!}e^{-\\mu t}.\n\\end{eqnarray*}\nBy Lemma~\\ref{TRTBoundLem} it follows that\n\\[\nP\\bigl(M_t' \\geq l\\bigr) \\leq\\frac{(t\\mu)^l e^{Ct}}{l!}.\n\\]\n\nNow consider\\vspace*{1pt} $Z^C$. At time 0 label the particles $1,2,\\ldots,N$ and\nlet $M_{i,t}'$ be the maximum type of any particle among the progeny of\nparticle $i$ at time~$t$. Then\n\\begin{eqnarray*}\nP\\bigl(M_t^C \\geq l\\bigr) & = & P\\bigl(\\bigl\n\\{M_{1,t}' \\geq l\\bigr\\} \\cup\\cdots\\cup\\bigl\n\\{M_{N,t}' \\geq l\\bigr\\}\\bigr)\n\\\\[-2pt]\n& \\leq & NP\\bigl(M_{1,t}' \\geq l\\bigr)\n\\\\[-2pt]\n& \\leq &\\frac{N (t\\mu)^l e^{Ct}}{l!}.\n\\end{eqnarray*}\n\\upqed\\end{pf}\n\nRecall that $M_t^{k,\\uparrow} = \\overline{M}{}^{k,\\uparrow}_t-k$ where\n$\\overline{M}{}^{k,\\uparrow}_t$ is the maximum type of any individual\nin the branching process $Z_t^{k,\\uparrow}$.\n\\begin{Lemma} \\label{ZUpBound}\nFor any time $t \\geq0$ and any integers $k \\geq0$ and $l \\geq0$ we~have\n\\[\nP\\bigl(M_t^{k,\\uparrow} > l\\bigr) \\leq\\frac{N (t\\mu)^l e^{(\\gamma\n(k+l)+1)t}}{l!}.\n\\]\n\\end{Lemma}\n\\begin{pf}\nWhile all of the particles in $Z_t^{k,\\uparrow}$ have type less than\n$k+l$, they branch at a rate which is less than or equal to $\\gamma\n(k+l)+1$. Because of this, $P(M_t^{k,\\uparrow} > l) \\leq P(M_t^{\\gamma\n(k+l)+1} > l)$. By Lemma~\\ref{YuleLem} we have\n\\[\nP\\bigl(M_t^{\\gamma(k+l)+1} > l\\bigr) \\leq\\frac{N (t\\mu)^l e^{(\\gamma\n(k+l)+1)t}}{l!}.\n\\]\n\\upqed\\end{pf}\n\\end{appendix}\n\n\n\\section*{Acknowledgments}\n\nI would like to thank Jason Schweinsberg for suggesting the problem,\npatiently helping me work through various parts of the proof and for\nhelping me revise the first drafts of the paper. I would also like to\nthank the referee for helpful comments that led to an improved upper\nbound.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\refstepcounter{subsection} Let $X$ be an n-dimensional complex\nmanifold and let \n\\begin{equation}\n0\\longrightarrow \\OX{X} \\longrightarrow \\mathcal{E}_{X}^{0,0}\\overset%\n{\\overline{\\partial }}{\\longrightarrow }...\\overset{\\overline{\\partial }}{%\n\\longrightarrow }\\mathcal{E}_{X}^{0,n}\\longrightarrow 0  \\label{Rezol_Dolb1}\n\\end{equation}%\nbe the Dolbeault-Grothendieck resolution on $X$. Here, as usual, $\\OX{X}$ is \nthe sheaf of holomorphic functions and $\\mathcal{E}_{X}^{p,q}$ the\nsheaf of $(p,q)$ smooth differential forms on $X$. The problem is that in\nthe singular case the complex (\\ref{Rezol_Dolb1}) is no longer a resolution\nfor $\\OX{X}$. The purpose of this paper is to construct an analogue\nfor the Dolbeault-Grothendieck resolution on a complex space with\nsingularities.\n\nThere exist two recent constructions of analogues of Dolbeault-Grothendieck\nresolutions under suplimentary hypothesis on the singular space. Ancona and\nGaveau \\cite{Anc-Gav} considered analytic spaces with smooth singular locus;\ntheir solution is based on Hironaka desingularization. Andersson and\nSamuelsson \\cite{And-Samuel} considered the case of a reduced analytic\nspace; the resolution is obtained as a subcomplex of the complex of smooth\ncurrents on the space; their construction uses Koppelman representation\nformulas.\n\nOur construction is based on working in a category larger than that of\nanalytic spaces.--- the category of semi-simplicial analytic spaces. Recall\nthat a s.s.analytic space (throughout this paper s.s.is short for\nsemi-simplicial) is a contravariant functor from a simplicial complex (seen\nas a category) to the category of analytic spaces, or, equivalently, a\nfamily of analytic spaces indexed by the simplexes of a simplicial complex,\ntogether with a family of compatible connecting morphisms (see Section \\ref%\n{Sect_ss} for the definitions). S.s.analytic spaces and the corresponding\nanalytic modules proved a very flexible tool. They appeared implicitely or\nexplicitely, for instance, in Forster, Knorr \\cite{ForstK} for the proof of\nGrauert's direct image theorem, in Verdier \\cite{Verd}, Baran \\cite{B1} for\nthe introduction of natural topologies on the global (hyper)cohomological\ninvariants of analytic sheaves, in Ramis, Ruget \\cite{R-R} for the proof of\nrelative analytic duality, in Flenner \\cite{Flenner} and B\\u{a}nic\\u{a},\nPutinar, Schumacher \\cite{BPS} for computations linked to deformation theory.\n\nOur construction solves the problem for any analytic space. In fact for each\npair $(X,\\mathcal{A})$, where $\\mathcal{A}$\\ is an embedding atlas of the\nanalytic space $X$ (i.e. a family of local closed embeddings of $X$ in\ncomplex manifolds - see paragraph \\ref{paragr_Atlas}) we produce a\nresolution for $\\OX{X}$, denoted by $\\Dolb{\\mathcal{A}\\,}{\\OX{X}}$. The pair \n$(X,\\mathcal{A})$ will be called a \\textit{locally embedded analytic space}. \nIn particular, if $X$ is a complex manifold and $\\mathcal{A}$ the obvious atlas with \none chart, then one gets the usual resolution on $X$. For each morphism of locally embedded \nanalytic spaces: \n\\begin{equation*}\nf:(X,\\mathcal{A})\\rightarrow (Y,\\mathcal{B})\n\\end{equation*}%\none constructs a pullback morphism which extends the pullback morphism from\nthe smooth case: \n\\begin{equation}\nf^{\\ast }:\\Dolb{\\mathcal{B}\\,}{\\OX{Y}}\\rightarrow \nf_{\\ast }\\Dolb{\\mathcal{A}\\,}{\\OX{X}}\n\\label{pull_back_introd}\n\\end{equation}\n\nMoreover, the same construction produces a resolution for each $\\OX{X}$-module \n$\\mathcal{F}$, denoted $\\DolbA{A}{F}$.\n\nThe resolution $\\DolbA{A}{F}$ depends on the embedding atlas $\\mathcal{A}$. However \nthe resolution is unique up to unique isomorphism in the derived category of \n$\\OX{X}$-modules, $D(\\OX{X})$. More precisely, if $(X,\\mathcal{A}),(Y,\\mathcal{B})$ \nare locally embedded analytic spaces and $f:X\\rightarrow Y$ is a morphism of\nanalytic spaces (but not necessarily of locally embedded analytic spaces)\nthen there exists in $D(\\OX{Y})$\\ a unique pullback morphism similar to \n(\\ref{pull_back_introd}) (see Theorem \\ref{Theor_Dolb_cat_deriv}). In particular, \nif $\\mathcal{A}$ and $\\mathcal{B}$ are two embedding atlases on the same analytic space \n$X$ then there is a unique isomorphism between $\\DolbA{A}{F}$ and $\\DolbA{B}{F}$ \nin $D(\\OX{X})$.\n\nThe main result of the paper is:\n\n\\begin{theorem}\n\\label{Theor_Dolb}\n\n\\begin{enumerate}\n\\item Let $(X,\\mathcal{A})$ be a locally embedded analytic space and $%\n\\mathcal{F}\\in Mod(\\OX{X})$. Then there is a functor \n\\begin{equation}\n\\Dolb{\\mathcal{A}\\,}{\\bullet}:Mod(\\OX{X})\\rightarrow C^{+}(X)\n\\end{equation}\nsuch that:\n\n\\begin{enumerate}\n\\item $\\Dolb{\\mathcal{A}\\,}{\\bullet}$ is an exact functor\n\n\\item There is a functorial morphism $\\mathcal{F}\\rightarrow \\DolbA{A}{F}$ \nand $\\DolbA{A}{F}$ is a resolution of $\\mathcal{F}$.\n\n\\item $\\DolbA{A}{F}$ has soft components.\n\n\\item $\\Dolb{\\mathcal{A}\\,}{\\OX{X}}$ has $\\OX{X}$-flat components\n\n\\item One has a natural quasi-isomorphism: \n\\begin{equation}\n\\Dolb{\\mathcal{A}\\,}{\\OX{X}}\\otimes _{\\OX{X}}%\n\\mathcal{F}\\rightarrow \\DolbA{A}{F}\t\n\\label{ident_Dolb_A}\n\\end{equation}%\nMoreover, if $\\mathcal{F}\\in Coh(\\OX{X})$\\ then the above morphism\nis an isomorphism.\n\n\\item If $X$ is a complex manifold and $\\mathcal{A}$ consists of only one\nchart, namely $(X,id,X)$, then $\\Dolb{\\mathcal{A}\\,}{\\bullet}$ coincides \nwith the usual Dolbeault-Grothendieck resolution on $X$.\n\\end{enumerate}\n\n\\item Let $f:(X,\\mathcal{A})\\rightarrow (Y,\\mathcal{B})$ be a morphism of\nlocally embedded analytic spaces,\\linebreak $\\mathcal{F}\\in Mod(\\OX{X})$, $%\n\\mathcal{G}\\in Mod(\\OX{Y})$ and $u:\\mathcal{G}\\rightarrow f_{\\ast }%\n\\mathcal{F}$ a morphism of $\\OX{Y}$-modules. Then there exists a\nnatural pullback morphism: \n\\begin{equation}\nf^{\\ast }(u):\\DolbA{B}{G}\\rightarrow f_{\\ast }\\DolbA{A}{F}  \n\\label{def_f*_u}\n\\end{equation}%\nsuch that the following diagram commutes:%\n\\begin{equation}\n\\begin{CD} \\DolbA{B}{G} @>f^{\\ast }(u)>> f_{\\ast\n}\\DolbA{A}{F} \\\\ @AAb^{\\prime}A @AAf_{*}bA\\\\\n\\mathcal{G} @>{u}>> f_{\\ast }\\mathcal{F} \n\\end{CD}  \\label{diagr_f*_u}\n\\end{equation}%\nIn particular there is a natural morphism:%\n\\begin{equation}\nf^{\\ast }:\\Dolb{\\mathcal{B}\\,}{\\OX{Y}}\\rightarrow %\nf_{\\ast }\\Dolb{\\mathcal{A}\\,}{\\OX{X}}\n\\label{def_f*}\n\\end{equation}%\nover the mapping $f^{\\ast }:\\OX{Y} \\rightarrow f_{\\ast }\\OX{X}$.\n\n\\item Let $(X,\\mathcal{A})\\overset{f}{\\rightarrow }(Y,\\mathcal{B})\\overset{g}%\n{\\rightarrow }(Z,\\mathcal{C})$ be morphisms of locally embedded analytic\nspaces and $h=g\\circ f$. Let moreover $\\mathcal{F}\\in Mod(\\OX{X})$, \n$\\mathcal{G}\\in Mod(\\OX{Y})$, $\\mathcal{H}\\in Mod(\\OX{Z})$\nand morphisms $u:\\mathcal{G}\\rightarrow f_{\\ast }\\mathcal{F}$ \n$\\OX{Y}$-linear, $v:\\mathcal{H}\\rightarrow g_{\\ast }\\mathcal{G}$, \n$w:\\mathcal{H} \\rightarrow h_{\\ast }\\mathcal{F}$ $\\OX{Z}$-linear, such that \n$g_{\\ast }(u)\\circ v=w$ then one has the commutative diagram:\n\\begin{equation}\n\\begin{tikzcd}[column sep=-0.2cm] h_{\\ast}\\DolbA{A}{F} && g_{\\ast}\\DolbA{B}{G}\n\\arrow{ll}[swap]{g_{\\ast}(f^{\\ast}(u)} \\\\ & \\DolbA{C}{H} \n\\arrow{ul}{h^{\\ast}(w)} \\arrow{ur}[swap]{g^{\\ast}(v)}\n\\end{tikzcd}  \\label{diagr_3_A}\n\\end{equation}\n\\end{enumerate}\n\\end{theorem}\n\n\\medskip\n\nThe proof is based on two simple remarks:\n\n\\begin{enumerate}\n\\item \\label{rem_functorialit_pullback} Let $\\mathscr{X}=((X_{\\alpha\n})_{\\alpha \\in \\mathcal{S}})$ be a s.s.complex manifold relative to the\nsimplicial complex $(I,\\mathcal{S})$. The compatibility of the pullback of\ndifferential forms with the composition of mappings ensures that the\nDolbeault-Grothendieck resolutions on the manifolds $X_{\\alpha }$ form a\ncomplex of $\\mathscr{X}$-modules. We denote it $\\Dolb{\\mathscr{X}\\,}{\\OXss{X}}$. \nMoreover, if \n\\begin{equation}\nF:\\mathscr{X}\\rightarrow \\mathscr{Y}\n\\end{equation}%\nis a morphism of s.s.complex manifolds (see Definition \\ref{Def_morph_over_f})\nthen one defines a pullback morphism:%\n\\begin{equation}\nF^{\\sharp}:\\Dolb{\\mathscr{Y}\\,}{\\OXss{Y}} \\rightarrow\nF_{\\sharp} \\Dolb{\\mathscr{X}\\,}{\\OXss{X}}\n\\label{pullback_manif}\n\\end{equation}%\nwhere $F_{\\sharp}$ is a variant of the direct image functor which associates\nto each $\\mathscr{X}$-module a complex of $\\mathscr{Y}$-modules.\n\n\\item \\label{rem_emb_an_sp} Let $X\\overset{i}{\\hookrightarrow }D$ be an\nanalytic subspace of the complex manifold $D$, given by the coherent ideal $%\n\\mathcal{I}\\subset \\OX{D}$ (we say that $(X,i,D)$ is an embedding\ntriple - see paragraph \\ref{paragr_emb_triples}). The complex obtained by\ntensoring the Dolbeault-Grothendieck resolution on $D$ with $\\OX{D}\/%\n\\mathcal{I}$ (which comes to restricting the coefficients of the\ndifferential forms on $D$ to $X$) is a resolution of $\\OX{X}$,\nsince the $\\OX{D}$-modules $\\mathcal{E}_{D}^{p,q}$ are $\\OX{D}$-flat \n(see Malgrange \\cite{M}). We consider this complex as the analogue for the \nDolbeault-Grothendieck resolution on $X$. Note that the complex described here \nappears in the proof of the duality theorems of Serre-Malgrange (see Malgrange \n\\cite{M1}\\ or B\\u{a}nic\\u{a}, St\\u{a}n\\u{a}\\c{s}ila \\cite{B-S} Ch 7 \\S 4.b). \nIf $(\\mathscr{X},k,\\mathscr{D})$ is a s.s.embedding triple (see Remark \n\\ref{Rem_ss_embed_triple}) then the Dolbeault-Grothendieck resolutions on each component \nform a complex of $\\mathscr{X}$-modules that we denote by $\\Dolb{k\\,}{\\OX{X}|\\mathfrak{U}}$.\n\\end{enumerate}\n\nLet $(X,\\mathcal{A})$ be a locally embedded analytic space. By Lemma \\ref%\n{Lemma_assoc_ss_triples} and Example \\ref{Ex_incluz_acop} one associates to $%\n(X,\\mathcal{A})$ a s.s.embedding triple $(\\mathfrak{U},k,\\mathfrak{D})$ and\na natural morphism of s.s.analytic spaces $b:\\mathfrak{U}\\rightarrow X$.\nAccording to \\textbf{2.} there is a $\\Dolbn$-resolution on $\\mathfrak{U}$. \nWe need to define a $\\Dolbn$-resolution on $X$, \n$\\Dolb{\\mathcal{A}\\,}{\\OX{X}}$, such that a pullback mapping\nsimilar to (\\ref{pullback_manif}) exist for $b$, i.e. a mapping :%\n\\begin{equation*}\nb^{\\sharp}:\\Dolb{\\mathcal{A}\\,}{\\OX{X}}\\rightarrow b_{\\sharp }%\n\\Dolb{k\\,}{\\OX{X}|\\mathfrak{U}}\n\\end{equation*}%\nFor this we simply set:%\n\\begin{equation*}\n\\Dolb{\\mathcal{A}\\,}{\\OX{X}}=b_{\\sharp }\\Dolb{k\\,}\n{\\OX{X}|\\mathfrak{U}}\\text{ and }b^{\\sharp}=id\n\\end{equation*}%\nand check that all the properties are verified.\n\nHere are some applications of Theorem \\ref{Theor_Dolb}.\n\nSince the terms of the Dolbeault-Grothendieck resolution are soft sheaves\none can use it to define representatives for derived functors and morphisms.\nIn particular the complex $\\Gamma (X,\\DolbA{A}{F})$ computes the cohomology of \n$X$ with coefficients in $\\mathcal{F}$; furthermore, if $\\mathcal{F}$ is a coherent \nsheaf then the terms of $\\Gamma(X,\\DolbA{A}{F})$ are endowed with Fr\\'{e}%\nchet-Schwarz topologies which induce the natural topologies on the\ncohomology groups of $\\mathcal{F}$(see Corollary \\ref{Corol_topol_FS}). Note\nthat since for each open covering of the analytic space one produces a\nresolution and the construction has good functorial properties, it follows\nthat the resolution is suitable to produce good representatives for derived\nfunctors and morphisms.\n\nIf $X$ is a reduced analytic space then, by using a direct limit argument,\none can construct on $X$ an analogue of the Dolbeault-Grothendieck\nresolution which coincides with the classical one on $Reg(X)$, the regular\nlocus of $X$. However, in this case the topologies on the global sections of\nthe resolution are more complicated.\n\nOne can link the complex $\\Dolb{\\mathcal{A}\\,}{\\OX{X}}$ to the complex of smooth \ndifferential forms on $X$, namely there is a natural surjective morphism between \n$\\Dolb{\\mathcal{A}\\,}{\\OX{X}}$ and a suitable \\v{C}ech complex of the complex of \nsmooth differential forms on $X$(see theorem \\ref{Theor_Dolb_diff_forms}).\n\nAs in the smooth case, by using the $\\Dolb{\\mathcal{A}\\,}{\\bullet }$\n-functor one can construct a resolution with soft sheaves for the de Rham\ncomplex on an analytic space $X$(see Theorem \\ref{Theor_Dolb_deRham}).\n\nThe main result of this note was announced in \\cite{B2}. In the same paper\nthe functor $F_{\\sharp}$ (denoted there by $F_{\\ast }$) is defined.\n\nIn a future paper, using roughly the same technique as here, but replacing\nthe functor $F_{\\sharp }$ with the direct image with proper supports we\nshall give a construction of the dualizing complex of an analytic space.\n                    \n\n\\section{Preliminaries \\label{Sect_prelim}}\n\n\\textbf{Review and notations.} Throughout this paper analytic space will\nmean complex analytic space.\n\nLet $(X,\\OX{X})$ be an analytic space. We use the following notations:\n\n\\begin{itemize}\n\\item[-] $Mod(\\OX{X})$ - the abelian category of $\\OX{X}$-modules; $Coh(\\OX{X})$ - the subcategory of coherent $\\OX{X}$-modules\n\n\\item[-] $C(X)$ the abelian category of complexes of $\\OX{X}\n\n\\item[-] As usual, $C^{\\ast }(X)$, respectively $D^{\\ast }(X)$, where $\\ast\n=+,-,b$, denote the subcategories of complexes bounded below, bounded above,\nrespectively bounded\n\\end{itemize}\n\nLet $X$ be an $n$-dimensional complex manifold. We denote by $\\mathcal{E}%\n_{X}^{p,q}$ the sheaf of $(p.q)$-differential forms with $C^{\\infty }$\ncoefficients on $X$. It is a soft sheaf and, according to \\cite{M}, it is an \n$\\OX{X}$-flat module. The complex of $\\OX{X}$-modules: \n\\begin{equation}\n0\\longrightarrow \\mathcal{E}_{X}^{0,0}\\overset{\\overline{\\partial }}{%\n\\longrightarrow }...\\overset{\\overline{\\partial }}{\\longrightarrow }\\mathcal{%\nE}_{X}^{0,n}\\longrightarrow 0  \\label{Rezol_Dolb}\n\\end{equation}%\nis the Dolbeault-Grothendieck resolution of $\\OX{X}$.\n\nFor $f:X\\rightarrow Y$ a holomorphic mapping between two complex manifolds\nwe denote \\ $f^{\\ast }:\\mathcal{E}_{Y}^{p,q}\\rightarrow f_{\\ast }\\mathcal{E}%\n_{X}^{p,q}$ the $\\OX{Y}$-linear morphism given by the pullback of forms.\n\nIt is well known that if $X\\overset{f}{\\rightarrow }Y\\overset{g}{\\rightarrow \n}Z$ are holomorphic mappings between complex manifolds and $h=g\\circ f$, then\none has the commutative diagram:\\smallskip \n\\begin{equation}\n\\begin{tikzcd}[column sep=0.1cm] \nh_{\\ast}\\mathcal{E}_{X}^{p.q} \n&& g_{\\ast}\\mathcal{E}_{Y}^{p.q}\n\\arrow{ll}[swap]{g_{\\ast}(f^{\\ast})} \\\\ \n& \\mathcal{E}_{Z}^{p.q} \n\\arrow{ul}{h^{\\ast}} \\arrow{ur}[swap]{g^{\\ast}}\n\\end{tikzcd}\n\\label{Diagr_3_E}\n\\end{equation}\n\n\\section{Semi-simplicial Objects \\label{Sect_ss}}\n\n\\refstepcounter{subsection}\\textbf{\\arabic{section}.%\n\\arabic{subsection}} \\label{paragr_ss_an_sp}\\textbf{Semi-simplicial analytic\nspaces. }Let $(I,\\mathcal{S})$ be a simplicial complex, i.e. $I$ is a set\nand $\\mathcal{S}$ is a family of non-empty finite parts of $I$, called\nsimplexes, such that:\n\n\\begin{enumerate}\n\\item $\\{i\\}\\in \\mathcal{S}$ for all $i\\in I$\n\n\\item if $\\alpha ^{\\prime }\\subset \\alpha \\in \\mathcal{S}$ then $\\alpha\n^{\\prime }\\in \\mathcal{S}$\n\\end{enumerate}\n\nIf $\\alpha \\in \\mathcal{S}$ \\ we denote by $|\\alpha |\\,=Card(\\alpha )-1$ the\nlength of the simplex $\\alpha $. Recall that $\\dim ((I,\\mathcal{S}))=\\sup\n\\{|\\alpha |\\ |\\ \\alpha \\in \\mathcal{S}\\}$.\n\nA morphism of simplicial complexes $f:(I,\\mathcal{S})\\rightarrow $ $(J,%\n\\mathcal{T})$ is simply a mapping $f:I\\rightarrow J$ such that $f(\\alpha\n)\\in \\mathcal{T}$ whenever $\\alpha \\in \\mathcal{S}$. If $K(pt)$ is the\nsimplicial complex over the set with one element $\\{pt\\},$ then we denote by\n$a_{\\mathcal{S}}:(I,\\mathcal{S})\\rightarrow K(pt)$ the morphism\ninduced by the unique mapping $I\\rightarrow \\{pt\\}$.\n\n\\begin{definition}\n\\begin{enumerate}\n\\item Let $\\mathcal{C}$ be a category. A semi-simplicial (s.s.) system of\nobjects in $\\mathcal{C}$ indexed by the simplicial complex $(I,\\mathcal{S})$\nconsists of:\n\n\\begin{itemize}\n\\item[-] a family $(X_{\\alpha })_{\\alpha \\in \\mathcal{S}}$ of objects in $%\n\\mathcal{C}$\n\n\\item[-] a family $(\\rho _{\\alpha \\beta })_{\\alpha \\subset \\beta }$ of\nconnecting morphisms, $\\rho _{\\alpha \\beta }:X_{\\beta }\\rightarrow X_{\\alpha\n}$, such that \\linebreak \n$\\rho _{\\alpha \\alpha }=id$ for $\\alpha \\in \\mathcal{S}$, and $%\n\\rho _{\\alpha \\beta }\\circ \\rho _{\\beta \\gamma }=\\rho _{\\alpha \\gamma }$\nwhenever $\\alpha \\subseteq \\beta \\subseteq \\gamma $.\n\\end{itemize}\n\n\\item Let $\\mathscr{X}=((X_{\\alpha })_{\\alpha \\in \\mathcal{S}}$, $(\\rho\n_{\\alpha \\beta })_{\\alpha \\subset \\beta })$, $\\mathscr{Y}=((Y_{\\alpha\n})_{\\alpha \\in \\mathcal{S}}$, $(\\rho _{\\alpha \\beta }^{\\prime })_{\\alpha\n\\subset \\beta })$ be s.s.systems of objects in $\\mathcal{C}$ indexed by $(I,%\n\\mathcal{S})$. A morphism $F:\\mathscr{X}\\rightarrow \\mathscr{Y}$ consists of\na family of morphisms in $\\mathcal{C}$, $(F_{\\alpha })_{\\alpha \\in \\mathcal{S%\n}}$, $F_{\\alpha }:X_{\\alpha }\\rightarrow Y_{\\alpha }$, such\nthat $F_{\\alpha }\\circ \\rho _{\\alpha \\beta }=\\rho _{\\alpha \\beta\n}^{\\prime }\\circ F_{\\beta }$.\n\\end{enumerate}\n\\end{definition}\n\nIf the simplicial complex is clear from the context we shall omit mentioning\nit.\n\nIf $\\mathcal{C}$ is the category of analytic spaces then we say for short\ns.s.analytic space instead of s.s.system of analytic spaces. Let $\\mathscr{X}%\n=((X_{\\alpha })_{\\alpha \\in \\mathcal{S}}$, $(\\rho _{\\alpha \\beta })_{\\alpha\n\\subset \\beta })$ be a s.s.analytic space. Here $X_{\\alpha }$ is short for $%\n(X_{\\alpha },\\OX{\\alpha })$, where $\\OX{\\alpha }$ denotes the sheaf of holomorphic \nsections of $X_{\\alpha }$, and $\\rho _{\\alpha \\beta}$ is short for \n$(\\rho _{\\alpha \\beta },\\rho _{\\beta \\alpha }^{1})$ where \n$\\rho _{\\alpha \\beta }:X_{\\beta }\\rightarrow X_{\\alpha }$ is the topological\npart and $\\rho _{\\beta \\alpha }^{1}:\\OX{\\alpha }\\rightarrow \\rho\n_{\\alpha \\beta \\ast }(\\OX{\\beta })$ is the sheaf level part. If $%\nX_{\\alpha }$ is a complex manifold for all $\\alpha \\in \\mathcal{S}$ then $%\n\\mathscr{X}$ will be called a s.s.complex manifold.\n\n\\begin{remark}\n\\label{anal_sp_Kpt}An analytic space can be regarded as a s.s.analytic space\nindexed by $K(pt)$, the simplicial complex constructed over the index set\nwith one element.\n\\end{remark}\n\n\\begin{example}\n\\label{ex_acop}Let $X$ be an analytic space and $\\mathcal{U}=(U_{i})_{i\\in\nI} $ an open covering of $X$. One associates to $\\mathcal{U}$\n\n\\begin{enumerate}\n\\item[-] the simplicial complex $(I,\\mathcal{N(U)})$, where $\\mathcal{N(U)}$\ndenotes the nerve of $\\mathcal{U}$\n\n\\item[-] the s.s.analytic space indexed by $(I,\\mathcal{N(U)})$, \n\\begin{equation*}\n\\mathfrak{U}=((U_{\\alpha })_{\\alpha \\in \\mathcal{N(U)}},(i_{\\alpha \\beta\n})_{\\alpha \\subset \\beta })\n\\end{equation*}%\nwhere $U_{\\alpha }$ denotes, as usual, the intersection $\\bigcap\\limits_{i%\n\\in \\alpha }U_{i}$, and $i_{\\alpha \\beta }:U_{\\beta }\\rightarrow U_{\\alpha }$\nis the natural inclusion.\n\\end{enumerate}\n\\end{example}\n\n\\begin{example}\n\\label{ex_prod}Let $(I,\\mathcal{S})$ be a simplicial complex and $%\n(X_{i})_{i\\in I}$ a family of analytic spaces. For $\\alpha \\in \\mathcal{S}$\nlet $X_{\\alpha }=\\prod\\limits_{i\\in \\alpha }X_{i}$. Then $\\mathscr{X}%\n=((X_{\\alpha })_{\\alpha \\in \\mathcal{S}}$, $(p_{\\alpha \\beta })_{\\alpha\n\\subset \\beta })$ is a s.s.analytic space, where $p_{\\alpha \\beta }:X_{\\beta\n}\\rightarrow X_{\\alpha }$ is the natural projection.\n\\end{example}\n\n\\begin{definition}\n\\label{Def_morph_over_f}Let $\\mathcal{C}$ be a category, $f:(I,\\mathcal{S}%\n)\\rightarrow $ $(J,\\mathcal{T})$ a morphism of simplicial complexes, $%\n\\mathscr{X}=((X_{\\alpha })_{\\alpha \\in \\mathcal{S}}$, $(\\rho _{\\alpha \\beta\n})_{\\alpha \\subset \\beta })$, $\\mathscr{Y}=((Y_{\\gamma })_{\\gamma \\in \n\\mathcal{T}}$, $(\\rho _{\\gamma \\delta }^{\\prime })_{\\gamma \\subset \\delta })$\ns.s.systems of objects in $\\mathcal{C}$ indexed by $(I,\\mathcal{S})$,\nrespectively $(J,\\mathcal{T})$. A morphism $F:\\mathscr{X}\\rightarrow %\n\\mathscr{Y}$\\ of s.s.systems of objects in $\\mathcal{C}$ over $f$ consists\nof a family of morphisms in $\\mathcal{C}$, $(F_{\\alpha })_{\\alpha \\in \n\\mathcal{S}}$, $F_{\\alpha }:X_{\\alpha }\\rightarrow Y_{f(\\alpha )}$%\n, such that\\ $F_{\\alpha }\\circ \\rho _{\\alpha \\beta }=\\rho _{f(\\alpha\n)f(\\beta )}^{\\prime }\\circ F_{\\beta }$.\n\\end{definition}\n\n\\begin{example}\n\\label{Ex_incluz_acop}Let $X$ be an analytic space, $\\mathcal{U}%\n=(U_{i})_{i\\in I}$ an open covering of $X,$ and $\\mathfrak{U}=((U_{\\alpha\n})_{\\alpha \\in \\mathcal{N(U)}},(i_{\\alpha \\beta })_{\\alpha \\subset \\beta })$\nthe s.s.analytic space associated to $\\mathcal{U}$ (see Example \\ref{ex_acop}%\n). Then the inclusion mappings $i_{\\alpha }:U_{\\alpha }\\rightarrow X$\ndetermine a morphism of s.s.analytic spaces \\linebreak \n$i:$ $\\mathfrak{U\\rightarrow }X$ over \n$a_{\\mathcal{N(U)}}:\\mathcal{N(U)}\\rightarrow K(pt)$.\n\\end{example}\n\n\\refstepcounter{subsection} \\textbf{\\arabic{section}.\\arabic{subsection}} \n\\textbf{Modules over s.s.analytic spaces. }Unless otherwise stated, in this\nsection $\\mathscr{X}=((X_{\\alpha },\\OX{\\alpha })_{\\alpha \\in \n\\mathcal{S}},(\\rho _{\\alpha \\beta },\\rho _{\\beta \\alpha }^{1})_{\\alpha\n\\subset \\beta })$ will denote a s.s.analytic space indexed by the simplicial\ncomplex $(I,\\mathcal{S})$.\n\n\\begin{definition}\n\\begin{enumerate}\n\\item An $\\mathscr{X}$-module consists of\n\n\\begin{itemize}\n\\item[-] a family $(\\mathcal{F}_{\\alpha })_{\\alpha \\in \\mathcal{S}}$ \\ where \n$\\mathcal{F}_{\\alpha }$ is an $\\OX{\\alpha }$-module for each $\\alpha \\in \n\\mathcal{S}$\n\n\\item[-] a family of connecting morphisms $(\\varphi _{\\beta \\alpha\n})_{\\alpha \\subset \\beta }$, where%\n\\begin{equation*}\n\\varphi _{\\beta \\alpha }:\\mathcal{F}_{\\alpha }\\rightarrow \\rho _{\\alpha\n\\beta \\ast }(\\mathcal{F}_{\\beta })\n\\end{equation*}%\nis a morphism of $\\OX{\\alpha }$-modules such that $\\varphi _{\\alpha\n\\alpha }=id$ for all $\\alpha \\in \\mathcal{S}$, and \\linebreak \n$\\rho _{\\beta \\gamma \\ast}(\\varphi _{\\gamma \\beta })\\circ \n\\varphi _{\\beta \\alpha }=\\varphi _{\\gamma \\alpha }$ whenever \n$\\alpha \\subseteq \\beta \\subseteq \\gamma $.\n\\end{itemize}\n\n\\item If $\\mathcal{F}=((\\mathcal{F}_{\\alpha })_{\\alpha \\in \\mathcal{S}}\\\n,(\\varphi _{\\beta \\alpha })_{\\alpha \\subset \\beta })$, $\\mathcal{G}=((%\n\\mathcal{G}_{\\alpha })_{\\alpha \\in \\mathcal{S}}\\ ,(\\psi _{\\beta \\alpha\n})_{\\alpha \\subset \\beta })$ are $\\mathscr{X}$-modules, then a morphism of\\\n\\ $\\mathscr{X}$-modules \\ $u:$\\ $\\mathcal{F}\\rightarrow \\mathcal{G}$\nconsists of a family $(u_{\\alpha })_{\\alpha \\in \\mathcal{S}}$, where $%\nu_{\\alpha }:\\mathcal{F}_{\\alpha }\\rightarrow \\mathcal{G}_{\\alpha }$ is a\nmorphism of $\\OX{\\alpha }$-modules, such that for $\\alpha \\subseteq\n\\beta $ $\\rho _{\\alpha \\beta \\ast }(u_{\\beta })\\circ \\varphi _{\\beta \\alpha\n}=\\psi _{\\beta \\alpha }\\circ u_{\\alpha }$.\n\\end{enumerate}\n\\end{definition}\n\nWe denote by $Mod(\\mathscr{X})$ the abelian category of $\\mathscr{X}$%\n-modules and by $C(\\mathscr{X})$ the category of complexes with terms in $%\nMod(\\mathscr{X})$.\n\n\\begin{example}\n$((\\OX{\\alpha })_{\\alpha \\in \\mathcal{S}}\\ ,(\\rho _{\\beta \\alpha\n}^{1})_{\\alpha \\subset \\beta })$ is obviously an $\\mathscr{X}$-module that\nwe denote by $\\OXss{X}$.\n\\end{example}\n\n\\begin{example}\n\\label{Ex_F|U}In the context of Example \\ref{ex_acop} let $\\mathcal{F}\\in\nMod(\\OX{X})$. Then $(\\mathcal{F}|U_{\\alpha })_{\\alpha \\in \\mathcal{S%\n}}$ with the obvious connecting morphisms is an $\\mathfrak{U}$-module that\nwe denote by $\\mathcal{F}|\\mathfrak{U}$.\n\\end{example}\n\nThe tensor product induces a bifunctor: \n\\begin{equation*}\n\\otimes :Mod(\\mathscr{X})\\times Mod(\\mathscr{X})\\rightarrow Mod(\\mathscr{X})\n\\end{equation*}%\nnamely if $\\mathcal{F}=((\\mathcal{F}_{\\alpha })_{\\alpha \\in \\mathcal{S}}\\\n,(\\varphi _{\\beta \\alpha })_{\\alpha \\subset \\beta })$, $\\mathcal{G}=((%\n\\mathcal{G}_{\\alpha })_{\\alpha \\in \\mathcal{S}}\\ ,(\\psi _{\\beta \\alpha\n})_{\\alpha \\subset \\beta })$ $\\in Mod(\\mathscr{X})$ then \\linebreak \n$((\\mathcal{F}_{\\alpha }\\otimes \\mathcal{G}_{\\alpha })_{\\alpha \\in \\mathcal{S}},\n(\\varphi _{\\beta \\alpha }\\otimes \\psi _{\\beta \\alpha })_{\\alpha \\subset \\beta })$ \nis an $\\mathscr{X}$-module.\n\\medskip\n\n\\refstepcounter{subsection} \\textbf{\\arabic{section}.\\arabic{subsection}} \n\\textbf{Alternate $\\mathscr{X}$-modules. }In order to define the $F_{\\sharp\n} $ functor (see paragraph \\ref{paragr_im_dir}) we need to construct an\nalternate version for the notion of $\\mathscr{X}$-module. For this, let $(I,%\n\\mathcal{S})$ be a simplicial complex and fix a total order on $I$. We use\nthe following notations:\n\n\\begin{itemize}\n\\item[-] if $\\alpha \\in \\mathcal{S}$ and $j\\in \\left[ 0,|\\alpha |\\right] $,\nthen%\n\\begin{eqnarray*}\nv(\\alpha ;j) &=&\\text{the j-th vertex of }\\alpha \\text{ with respect to the\norder on }I\\text{,\\ } \\\\\n&&\\text{the counting starting from 0} \\\\\n\\sigma (\\alpha ;j) &=&\\alpha \\setminus \\{v(\\alpha ;j)\\}\n\\end{eqnarray*}\n\n\\item[-] if $\\alpha \\in \\mathcal{S}$, $|\\alpha |\\geq 1$ and $j,k\\in \\left[\n0,|\\alpha |\\right] $, $j\\neq k$, then%\n\\begin{equation*}\n\\sigma (\\alpha ;\\\/j,k)=\\alpha \\setminus \\{v(\\alpha ;j),v(\\alpha ;k)\\}\n\\end{equation*}\n\\end{itemize}\n\nThus, if for instance $j<k$ then $\\sigma (\\alpha ;j,k)=\\sigma (\\sigma\n(\\alpha ;k);j)=\\sigma (\\sigma (\\alpha ;j);k-1).$\n\nIf $\\alpha =\\{i_{0},...,i_{n}\\}$ and $i_{0}<i_{1}<...<i_{n}$, one checks\nimmediately that \n\\begin{eqnarray*}\n|\\alpha | &=&n \\\\\nv(\\alpha ;j) &=&v_{j} \\\\\n\\sigma (\\alpha ;j) &=&\\{i_{0},...,\\hat{\\imath}_{j},...,i_{n}\\} \\\\\n\\sigma (\\alpha ;j,k) &=&\\{i_{0},...,\\hat{\\imath}_{j},...,\\hat{\\imath}%\n_{k},...,i_{n}\\}\n\\end{eqnarray*}\n\nLet $\\mathscr{X}=((X_{\\alpha })_{\\alpha \\in \\mathcal{S}},(\\rho _{\\alpha\n\\beta })_{\\alpha \\subset \\beta })$ be a s.s.analytic space. We use the\nsubscript $(\\alpha ;j)$ to refer to the mappings along the edge $[\\alpha\n,\\sigma (\\alpha ;j)]$ of the simplicial complex $(I,\\mathcal{S})$. Thus we\nshall write $\\rho _{(\\alpha ;j)}$ instead of $\\rho _{\\sigma (\\alpha\n;j)\\alpha }:X_{\\alpha }\\rightarrow X_{\\sigma (\\alpha ;j)}$. Similarly, if $%\n\\mathcal{F}=((\\mathcal{F}_{\\alpha })_{\\alpha \\in \\mathcal{S}}\\ ,(\\varphi\n_{\\beta \\alpha })_{\\alpha \\subset \\beta })$ is an $\\mathscr{X}$-module we\nwrite $\\varphi _{(\\alpha ;j)}$ instead of $\\varphi _{\\alpha \\sigma (\\alpha\n;j)}:\\mathcal{F}_{\\sigma (\\alpha ;j)}\\rightarrow \\rho _{(\\alpha ;j)\\ast }(%\n\\mathcal{F}_{\\alpha })$.\n\n\\begin{remark}\nThe family of commuting morphisms $(\\rho _{\\alpha \\beta })_{\\alpha \\subset\n\\beta }$ can be \"reconstructed\" (by finite compositions) from the subfamily $%\n(\\rho _{(\\alpha ;j)})_{(\\alpha ;j)}$. More precisely, If $\\alpha \\in \n\\mathcal{S}$, $|\\alpha |\\geq 1$, $j,k\\in \\left[ 0,|\\alpha |\\right] $, and,\nfor instance, $j<k$, then the following rectangular diagram commutes:%\n\\begin{equation}\n\\begin{CD} X_{\\alpha} @>\\rho_{(\\alpha;j)}>> X_{\\sigma(\\alpha;j)}\\\\\n@VV\\rho_{(\\alpha;k)}V @VV\\rho_{(\\sigma(\\alpha;j),k-1)}V\\\\\nX_{\\sigma(\\alpha;k)} @>\\rho_{(\\sigma(\\alpha;k),j)}>> X_{\\sigma(\\alpha;j,k)}\n\\end{CD}  \\tag{$D(\\alpha ;j,k)$}  \\label{diagr_DX}\n\\end{equation}\n\nConversely, any family of morphisms $(\\rho _{(\\alpha ;j)})_{(\\alpha ;j)}$\nsuch that the diagrams $D(\\alpha ;j,k)$ commute, generates a family of\nconnecting morphisms for the family of analytic spaces $(X_{\\alpha\n})_{\\alpha \\in \\mathcal{S}}$.\n\nSimilarly, the connecting morphisms of the $\\mathscr{X}$-module $\\mathcal{F}$%\n\\ are uniquely determined by the subfamily $(\\varphi _{(\\alpha\n;j)})_{(\\alpha ;j)}$ and the obvious rectangular diagrams commute:%\n\\begin{equation}\n\\begin{CD} \\rho_{(\\alpha;j,k)\\ast}\\mathcal{F}_{\\alpha}\n@<\\rho_{(\\sigma(\\alpha;j),k-1)\\ast}(\\varphi_{(\\alpha;j)})<<\n\\rho_{(\\sigma(\\alpha;j),k-1)\\ast}(\\mathcal{F}_{\\sigma(\\alpha;j)})\\\\\n@AA\\rho_{(\\sigma(\\alpha;k),j)\\ast}({\\varphi_{(\\alpha;k)}})A\n@AA{\\varphi_{(\\sigma(\\alpha;j),k-1)}}A\\\\\n\\rho_{(\\sigma(\\alpha;k),j)\\ast}(\\mathcal{F}_{\\sigma(\\alpha;k)})\n@<\\varphi_{(\\sigma(\\alpha;k),j)}<< \\mathcal{F}_{\\sigma(\\alpha;j,k)} \\end{CD}\n\\tag{$D(\\mathcal{F};\\alpha ;j,k)$}\n\\end{equation}\n\\end{remark}\n\n\\begin{definition}\n\n\\begin{enumerate}\n\\item An alternate $\\mathscr{X}$-module consists of a family $(\\mathcal{F}%\n_{\\alpha })_{\\alpha \\in \\mathcal{S}}$, where each $\\mathcal{F}_{\\alpha }$ is\nan $\\OX{\\alpha }$-module, together with the family of connecting\nmorphisms $(\\varphi _{(\\alpha ;j)})_{(\\alpha ;j)}$,\\ $\\varphi _{(\\alpha ;j)}:%\n\\mathcal{F}_{(\\alpha ;j)}\\rightarrow \\rho _{(\\alpha ;j)\\ast }(\\mathcal{F}%\n_{\\alpha })$ such that the diagrams $D(\\mathcal{F};\\alpha ;j,k)$\nanti-commute.\n\n\\item \\label{pct_morf_alt_mod}Let $\\mathcal{F}=((\\mathcal{F}_{\\alpha\n})_{\\alpha \\in \\mathcal{S}}\\ ,(\\varphi _{(\\alpha ;j)})_{(\\alpha ;j)})$, $%\n\\mathcal{G}=((\\mathcal{G}_{\\alpha })_{\\alpha \\in \\mathcal{S}}\\ ,(\\psi\n_{(\\alpha ;j)})_{(\\alpha ;j)})$ be alternate $\\mathscr{X}$-modules. A\nmorphism of\\ alternate\\ $\\mathscr{X}$-modules \\ $u:$\\ $\\mathcal{F}%\n\\rightarrow \\mathcal{G}$ consists of a family $(u_{\\alpha })_{\\alpha \\in \n\\mathcal{S}}$, $u_{\\alpha }:\\mathcal{F}_{\\alpha }\\rightarrow \\mathcal{G}%\n_{\\alpha }$ morphism of $\\OX{\\alpha }$-modules, such that for each\npair $(\\alpha ;j)$ the diagram commutes: \n\\begin{equation}\n\\begin{CD} \\mathcal{F}_{(\\alpha ;j)} @>u_{(\\alpha;j)}>>\n\\mathcal{G}_{(\\alpha;j)}\\\\ @VV\\varphi _{(\\alpha ;j)}V @VV\\psi _{(\\alpha\n;j)}V\\\\ \\rho_{(\\alpha;j)\\ast }\\mathcal{F}_{\\alpha } @>u_{\\alpha }>>\n\\rho_{(\\alpha ;j)\\ast}\\mathcal{G}_{\\alpha }\\\\ \\end{CD} \n\\tag{$D(\\mathcal{F},\\mathcal{G};\\alpha ;j)$}\n\\end{equation}%\nOne denotes by $aMod(\\mathscr{X})$\\ the category of alternate $\\mathscr{X}$%\n-modules\n\n\\item With the notations at point \\ref{pct_morf_alt_mod}, an anti-morphism\nof alternate $\\mathscr{X}$-modules is a family of morphisms $u=(u_{\\alpha\n})_{\\alpha \\in \\mathcal{S}}$ such that the diagrams $D(\\mathcal{F},%\n\\mathcal{G};\\alpha ;j)$ anti-commute. A complex of alternate $\\mathscr{X}$%\n-modules with anti-morphism differentials will be called an alternate\ncomplex of alternate $\\mathscr{X}$-modules. One denotes by $aC(\\mathscr{X})$\nthe category of alternate complexes of alternate $\\mathscr{X}$-modules\n\\end{enumerate}\n\\end{definition}\n\nTo the edge $(\\alpha ;j)$ of the simplicial complex $(I,\\mathcal{S})$ we\nassociate the alternating coeficient $\\varepsilon (\\alpha ;j)=(-1)^{j}$.\nNote that if $\\mathcal{F}=((\\mathcal{F}_{\\alpha })_{\\alpha \\in \\mathcal{S}%\n},(\\varphi _{(\\alpha ;j)})_{(\\alpha ;j)})$ is an $\\mathscr{X}$-module then $%\nalt(\\mathcal{F})=((\\mathcal{F}_{\\alpha })_{\\alpha \\in \\mathcal{S}%\n},(\\varepsilon (\\alpha ;j)\\varphi _{(\\alpha ;j)})_{(\\alpha ;j)})$ is an\nalternate $\\mathscr{X}$-module. One checks easily that $\\ alt:Mod(\\mathscr{X}%\n)\\rightarrow aMod(\\mathscr{X})$ is an isomorphism of categories with an\nobvious inverse that we denote by $alt^{-1}$. The functor $alt$ extends to\nan isomorphism of categories $C(\\mathscr{X})\\rightarrow aC(\\mathscr{X})$.\nIndeed, if $\\mathcal{F}^{\\bullet }\\in C(\\mathscr{X}),$ $\\mathcal{F}^{\\bullet\n}=((\\mathcal{F}_{\\alpha }^{\\bullet })_{\\alpha \\in \\mathcal{S}},(\\varphi\n_{(\\alpha ;j)}^{\\bullet })_{(\\alpha ;j)})$ then the terms of $alt(\\mathcal{F}%\n^{\\bullet })$ are obtained from the terms of $\\mathcal{F}^{\\bullet }$ via\nthe functor $alt$, while the differentials of each complex $\\mathcal{F}%\n_{\\alpha }^{\\bullet }$ are multiplied by $(-1)^{|\\alpha |}$.\n\n\\begin{remark}\nThe notions of alternate $\\mathscr{X}$-module and alternate complex of $%\n\\mathscr{X}$-modules do not depend on the total order on $I$. The $alt$\nfunctors do. However for two total orders on $I$ there is a (non-unique)\nfunctorial isomorphism between the two corresponding $alt$ functors.\n\\end{remark}\n\n\\refstepcounter{subsection}\\textbf{\\arabic{section}.\\arabic{subsection}} %\n\\label{paragr_im_inv} \\textbf{Inverse images. }Consider the following\nsetting:\n\n\\begin{itemize}\n\\item[-] $f:(I,\\mathcal{S})\\rightarrow $ $(J,\\mathcal{T})$ a morphism of\nsimplicial complexes\n\n\\item[-] fixed total orders on $I$ and $J$.such that $f:I\\rightarrow $ $J$\nis increasing\n\n\\item[-] $\\mathscr{X}=((X_{\\alpha })_{\\alpha \\in \\mathcal{S}}$, $(\\rho\n_{\\alpha \\beta })_{\\alpha \\subset \\beta })$, $\\mathscr{Y}=((Y_{\\gamma\n})_{\\gamma \\in \\mathcal{T}}$, $(\\rho _{\\gamma \\delta }^{\\prime })_{\\gamma\n\\subset \\delta })$ s.s.analytic spaces indexed by $(I,\\mathcal{S})$,\nrespectively $(J,\\mathcal{T})$\n\n\\item[-] $F:\\mathscr{X}\\rightarrow \\mathscr{Y}$ a morphism of s.s.analytic\nspaces over $f$ (see Definition \\ref{Def_morph_over_f}), that is $%\nF=(F_{\\alpha },F_{\\alpha }^{\\ast })_{\\alpha \\in \\mathcal{S}}$ with $%\nF_{\\alpha }:X_{\\alpha }\\rightarrow Y_{f(\\alpha )}$ morphism of analytic\nspaces such that for $\\alpha \\subseteq \\beta $ the following diagram\ncommutes: \n\\begin{equation}\n\\begin{CD} X_{\\beta} @>F_{\\beta}>> Y_{f(\\beta )}\\\\ @VV\\rho_{\\alpha \\beta}V\n@VV\\rho_{f(\\alpha) f(\\beta)}^{\\prime }V\\\\ X_{\\alpha} @>F_{\\alpha}>>\nY_{f(\\alpha )} \\end{CD}  \\label{Diagr_XY_morph}\n\\end{equation}\n\\end{itemize}\n\nNote that $\\mathcal{S}$ is the disjoint union of the sets $(\\mathcal{S}%\n_{\\gamma })_{\\gamma \\in \\mathcal{T}}$, with \n\\begin{equation}\n\\mathcal{S}_{\\gamma }=\\{\\alpha \\in \\mathcal{S}|f(\\alpha )=\\gamma \\}\n\\end{equation}%\nMoreover, each $\\mathcal{S}_{\\gamma }$ is the union of the sets $(I(\\gamma\n,i))_{i\\geq 0}$ where \n\\begin{equation}\nI(\\gamma ,i)=\\{\\alpha \\in \\mathcal{S}|f(\\alpha )=\\gamma ,|\\alpha |=|\\gamma\n|+i\\}\n\\end{equation}%\nWe also set: \n\\begin{equation}\nI(i)=\\bigcup\\limits_{\\gamma \\in \\mathcal{T}}I(\\gamma ,i)\n\\end{equation}%\n\\ In particular $I(\\gamma ,0)$ consists of all the simplexes of $\\mathcal{S}$\nwhich are in a one-to-one correspondence with $\\gamma $ via $f$.\n\nLet $\\mathcal{G}\\in Mod(\\mathscr{Y})$ with $\\mathcal{G}=((\\mathcal{G}%\n_{\\gamma })_{\\gamma \\in \\mathcal{T}},(\\psi _{\\delta \\gamma })_{\\gamma\n\\subset \\delta })$. The inverse image $F^{\\ast }(\\mathcal{G})$ of $\\mathcal{G%\n}$ is, by definition, the $\\mathscr{X}$-module with the components: \n\\begin{equation}\nF^{\\ast }(\\mathcal{G})_{\\alpha }=F_{\\alpha }^{\\ast }(\\mathcal{G}_{f(\\alpha\n)})\\text{ for }\\alpha \\in \\mathcal{S}\n\\end{equation}%\nand connecting morphisms for all $\\alpha \\subset \\beta $ \n\\begin{equation}\n\\mu _{\\beta \\alpha }=F_{\\beta }^{\\ast }(\\widetilde{\\psi }_{f(\\beta )f(\\alpha\n)})\\text{ }\n\\end{equation}%\nwhere%\n\\begin{equation}\n\\widetilde{\\psi }_{f(\\beta )f(\\alpha )}:\\rho _{f(\\beta )f(\\alpha\n)}^{^{\\prime }\\ast }(\\mathcal{G}_{f(\\alpha )})\\rightarrow \\mathcal{G}%\n_{f(\\beta )}\n\\end{equation}%\nis the morphism which corresponds via the usual adjunction isomorphism to\nthe connecting morphism \n\\begin{equation}\n\\psi _{f(\\beta )f(\\alpha )}:\\mathcal{G}_{f(\\alpha )}\\rightarrow \\rho\n_{f(\\beta )f(\\alpha )\\ast }^{^{\\prime }}(\\mathcal{G}_{f(\\beta )})\n\\end{equation}\n\nOne checks easily that the family of morphisms $(\\mu _{\\beta \\alpha\n})_{\\alpha \\subset \\beta }$\\ satisfies the required conditions.\n\n\\begin{remark}\n\\label{Rem_im_inv_incluz}Let $X$ be an analytic space, $\\mathcal{U}%\n=(U_{i})_{i\\in I}$ an open covering of $X$, $\\mathfrak{U}$ the s.s.analytic\nspace determined by $\\mathcal{U}$ (see Example \\ref{ex_acop}) and \n$i:$ $\\mathfrak{U\\rightarrow }X$ the morphism of s.s.analytic spaces given by the natural inclusions \n(see Example \\ref{Ex_incluz_acop}). If $\\mathcal{F}\\in Mod(\\OX{X})$ then the \n$\\mathfrak{U}$-module $\\mathcal{F}|\\mathfrak{U}$ (see Example \\ref{Ex_F|U}) coincides \nwith $i^{\\ast }(\\mathcal{F})$.\n\\end{remark}\n\nLet $\\mathcal{G}\\in Mod(\\mathscr{Y})$ as above, $\\mathcal{F}\\in Mod(%\n\\mathscr{X})$ with $((\\mathcal{F}_{\\alpha })_{\\alpha \\in \\mathcal{S}%\n},(\\varphi _{\\alpha \\beta })_{\\alpha \\subset \\beta })$, and $v:F^{\\ast }(%\n\\mathcal{G}\\rightarrow \\mathcal{F}$ a morphism of $\\mathscr{X}$-modules. One remarks\nthat $v$ is completely determined by the family of morphisms $(v_{\\alpha\n})_{\\alpha }$%\n\\begin{equation*}\nv_{\\alpha }:F_{\\alpha }^{\\ast }(\\mathcal{G}_{f(\\alpha )})\\rightarrow \n\\mathcal{F}_{\\alpha }\n\\end{equation*}%\nwhere $\\alpha \\in \\mathcal{S}$ is such that $f|\\alpha $\\ is injective. More\nprecisely one checks directly the following lemma:\n\n\\begin{lemma}\n\\label{Lemma_crit_morf_im_inv}Let $\\mathcal{F}\\in Mod(\\mathscr{X})$, $%\n\\mathcal{G}\\in Mod(\\mathscr{Y})$ as above. Then the morphisms $(v_{\\alpha\n})_{\\alpha \\in I(0)}$ determine a morphism of $\\mathscr{X}$-modules $%\nv:F^{\\ast }(\\mathcal{G})\\rightarrow \\mathcal{F}$ iff they verify the following\nconditions:\n\n\\begin{enumerate}\n\\item For $\\gamma \\in \\mathcal{T}$, $\\beta \\in I(\\gamma ,1)$ let $\\alpha\n_{1} $, $\\alpha _{2}\\in $ $I(\\gamma ,0)$ be the only two simplexes s.t. \\linebreak\n$\\alpha _{1}$, $\\alpha _{2}\\subset \\beta $. Then the following diagram\ncommutes: \n\\begin{equation}\n\\begin{CD} F_{\\beta }^{\\ast }(\\mathcal{G}_{\\gamma}) @ >\n\\rho_{\\alpha_{1}\\beta}^{\\ast }(v_{\\alpha_{1}})>> \\rho_{\\alpha_{1}\n\\beta}^{\\ast}(\\mathcal{F}_{\\alpha_{1}})\\\\ @VV\\rho_{\\alpha_{2}\n\\beta}^{\\ast}(v_{\\alpha_{2}})V @VV\\widetilde{\\varphi} _{\\beta \\alpha_{1}}V\\\\\n\\rho_{\\alpha_{2} \\beta}^{\\ast }(\\mathcal{F}_{\\alpha_{2}})\n@>\\widetilde{\\varphi} _{\\beta \\alpha_{2}}>> \\mathcal{F}_{\\beta}\\\\ \\end{CD}\n\\label{Im_inv_cond_1}\n\\end{equation}\n\n\\item For $\\gamma ,\\delta \\in \\mathcal{T}$, $\\gamma \\subset \\delta $, and $%\n\\alpha \\in I(\\gamma ,0)$, $\\beta \\in I(\\delta ,0)$ with $\\alpha \\subset\n\\beta $ the following diagram commutes:%\n\\begin{equation}\n\\begin{CD} \\rho_{\\alpha \\beta}^{\\ast }F_{\\alpha }^{\\ast\n}(\\mathcal{G}_{\\gamma}) @>\\rho_{\\alpha \\beta}^{\\ast }(v_{\\alpha})>>\n\\rho_{\\alpha \\beta}^{\\ast }(\\mathcal{F}_{\\alpha})\\\\ @VVF_{\\beta }^{\\ast\n}(\\widetilde{\\psi} _{\\delta \\gamma})V @VV\\widetilde{\\varphi}_{\\beta\n\\alpha}V\\\\ F_{\\beta }^{\\ast }(\\mathcal{G}_{\\delta }) @>v_{\\beta }>>\n\\mathcal{F}_{\\beta }\\\\ \\end{CD}  \\label{Im_inv_cond_2}\n\\end{equation}\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{remark}\nIn the same way as above (i.e. componentwise) one can construct an inverse\nimage functor $F^{-1}$ for s.s.sheaves of abelian groups.\n\\end{remark}\n\n\\refstepcounter{subsection} \\textbf{\\arabic{section}.\\arabic{subsection}} %\n\\label{paragr_im_dir} \\textbf{The }$F_{\\sharp }$ \\textbf{functor. }We use\nthe setting described at the beginning of paragraph \\ref{paragr_im_inv}.\n\nLet $\\mathcal{F}\\in aMod(\\mathscr{X})$, $\\mathcal{F}=((\\mathcal{F}_{\\alpha\n})_{\\alpha \\in \\mathcal{S}}\\ ,(\\varphi _{(\\alpha ;j)})_{(\\alpha ;j)})$. For $%\n\\gamma \\in \\mathcal{T}$ \\ \n\\begin{equation}\n((F_{\\alpha \\ast }(\\mathcal{F}_{\\alpha }))_{\\alpha },(F_{\\sigma (\\alpha\n;j)\\ast }(\\varphi _{(\\alpha ,j)}))_{(\\alpha ;j)})_{f(\\alpha )=f(\\sigma\n(\\alpha ;j))=\\gamma }\n\\end{equation}%\nis a multicomplex of $Y_{\\gamma }$-modules (recall that multicomplex means\nanti-commuting rectangles) and consider the following simple complex associated to \nthis multicomplex:\n\\begin{equation}\n...\\rightarrow \\prod\\limits_{\\alpha \\in I(\\gamma ,i)}F_{\\alpha \\ast }(%\n\\mathcal{F}_{\\alpha })\\rightarrow \\prod\\limits_{\\alpha \\in I(\\gamma\n,i+1)}F_{\\alpha \\ast }(\\mathcal{F}_{\\alpha })\\rightarrow ... \n\\tag{$C^{\\bullet }(\\gamma )$}\n\\end{equation}%\nwith the product indexed by $I(\\gamma ,0)$ in degree $0.$ The connecting\nmorphisms of $\\mathcal{F}$\\ induce anti-morphisms $C^{\\bullet }(\\sigma\n(\\gamma ;j))\\rightarrow \\rho _{(\\gamma ;j)\\ast }(C^{\\bullet }(\\gamma ))$ and\none checks that $(C^{\\bullet }(\\gamma ))_{\\gamma \\in \\mathcal{T}}$ is an\nalternated complex of alternated $\\mathscr{Y}$-modules.\n\nIf we start with an alternated complex of alternated $\\mathscr{X}$-modules\\ $%\n\\mathcal{F}$\\ instead of an alternated $\\mathscr{X}$-module, then $%\nC^{\\bullet }(\\gamma )$ is a double complex where the product for $\\mathcal{F}%\n^{0}$ indexed by $I(\\gamma ,0)$ is considered in bidegree $(0,0)$.\n\n\\begin{definition}\n\\begin{enumerate}\n\\item If $\\mathcal{F}\\in aMod(\\mathscr{X})$ then $F_{\\sharp }(\\mathcal{F})$\nis the alternated complex of alternated $\\mathscr{Y}$-modules with \n\\begin{equation}\nF_{\\sharp }(\\mathcal{F})_{\\gamma }=C^{\\bullet }(\\gamma )\n\\end{equation}\nand with connecting morphisms induced by those of $\\mathcal{F}$\n\n\\item If $\\mathcal{F}\\in aC(\\mathscr{X})$ then $F_{\\sharp }(\\mathcal{F})$ is\nthe alternated complex of alternated $\\mathscr{Y}$-modules where $F_{\\sharp\n}(\\mathcal{F})_{\\gamma }$ is the simple complex associated to the double\ncomplex $C^{\\bullet }(\\gamma )$ and the connecting morphisms are induced by\nthose of $\\mathcal{F}$\n\n\\item If $\\mathcal{F}\\in Mod(\\mathscr{X})$ (respectively $\\mathcal{F}\\in C(%\n\\mathscr{X})$) then $F_{\\sharp }(\\mathcal{F})=alt^{-1}(F_{\\sharp }(alt(%\n\\mathcal{F})))$\n\\end{enumerate}\n\\end{definition}\n\nOne checks easily that the definition of $F_{\\sharp }$ is compatible with\nthe natural inclusion functors $aMod(\\mathscr{X})\\rightarrow aC(\\mathscr{X})$\nand $Mod(\\mathscr{X})\\rightarrow C(\\mathscr{X}).$\n\n\\begin{example}\nThe components $(F_{\\alpha }^{\\ast })_{\\alpha \\in \\mathcal{S}}$ of the\nmorphism $F:\\mathscr{X}\\rightarrow \\mathscr{Y}$ determine a morphism of $%\n\\mathscr{Y}$-modules \\ $F^{\\sharp}:\\OXss{Y}\\rightarrow F_{\\sharp }(\\OXss{X})$\n\\end{example}\n\n\\begin{example}\nIf \\ $f:(I,\\mathcal{S})\\rightarrow $ $(J,\\mathcal{T})$ is bijective (in\nparticular if $f$ is the identity of $(I,\\mathcal{S})$) then $F_{\\sharp }(%\n\\mathcal{F})_{\\gamma }=F_{\\alpha \\ast }(\\mathcal{F}_{\\alpha })$ where $%\nf(\\alpha )=\\gamma $. Remark that if $F:X\\rightarrow Y$ is a morphism of\nanalytic spaces and $\\mathcal{F}\\in Mod(\\OX{X})$ then the usual\ndirect image $F_{\\ast }(\\mathcal{F})$ coincides with $F_{\\sharp }\\mathcal{F}$\nas module over $X$\\ seen as s.s.analytic space indexed by $K(pt)$ (see\nExample \\ref{anal_sp_Kpt})\n\\end{example}\n\n\\begin{example}\n\\label{Ex_Cech_complex}Let $X$ be an analytic space, $\\mathcal{U}%\n=(U_{i})_{i\\in I}$ an open covering of $X$, and \\linebreak \n$\\mathcal{F}\\in Mod(\\OX{X})$. If $i:\\mathfrak{U}\\rightarrow X$ is the \nmorphism of s.s.analytic spaces over \\linebreak \n$a_{\\mathcal{N(U)}}:\\mathcal{N(U)}\\rightarrow K(pt)$ given by\nthe inclusions (see Example \\ref{Ex_incluz_acop}) then $i_{\\sharp }(\\mathcal{%\nF}|\\mathfrak{U})$ is the \\v{C}ech complex of $\\mathcal{F}$ with respect to\nthe covering $\\mathcal{U}$. Note that the natural morphism \\linebreak \n$\\mathcal{F}\\rightarrow i_{\\sharp }(\\mathcal{F}|\\mathfrak{U})$ is a\nquasi-isomorphism (see e.g. \\cite{FAC}, \\textit{chap 1, \\S 4, Lemme 1}).\n\\end{example}\n\nSince the Cartesian product is associative, the form of the terms of the\ncomplex ( $C^{\\bullet }(\\gamma )$) implies that the functor $F_{\\sharp }$\\\nof s.s.modules commutes with the composition of morphisms of s.s.analytic\nspaces. Thus the following lemma holds:\n\n\\begin{lemma}\n\\label{Lemma_Comp_im_dir}Let $f:(I_{1},\\mathcal{S}_{1})\\rightarrow $ $(I_{2},%\n\\mathcal{S}_{2})$, $g:(I_{2},\\mathcal{S}_{2})\\rightarrow (I_{3},\\mathcal{S}%\n_{3})$ be morphisms of simplicial complexes and assume that we have fixed\ntotal orders on $I_{1},$ $I_{2},$ $I_{3}$. Let $F:\\mathscr{X}\\rightarrow %\n\\mathscr{Y}$,\\linebreak \n$G:\\mathscr{Y}\\rightarrow \\mathscr{Z}$ be morphisms of s.s.analytic spaces over $f$, \nrespectively $g$, where $\\mathscr{X}$, respectively $\\mathscr{Y}$, repectively \n$\\mathscr{Z}$ is a s.s.analytic space relative to $(I_{1},\\mathcal{S}_{1})$, \nRespectively to $(I_{2},\\mathcal{S}_{2})$, and $(I_{3},\\mathcal{S}_{3})$ \nThen if $\\mathcal{F}\\in aMod(\\mathscr{X})$ or $\\mathcal{F}\\in aC(\\mathscr{X})$ \nor $\\mathcal{F} \\in Mod(\\mathscr{X})$ or $\\mathcal{F}\\in C(\\mathscr{X})$ or \n$\\mathcal{F}\\in aC(\\mathscr{X})$ one has \n\\begin{equation}\n(G\\circ F)_{\\sharp }(\\mathcal{F})=G_{\\sharp }F_{\\sharp }(\\mathcal{F})\n\\end{equation}%\n\\end{lemma}\n\\begin{flushright}\n$\\square$\n\\end{flushright}\n \nFor $\\mathcal{F}\\in Mod(\\mathscr{X})$ set \n\\begin{equation}\nF_{\\ast }(\\mathcal{F})=Z^{0}(F_{\\sharp }(\\mathcal{F}))\n\\end{equation}%\nwhere $Z^{0}$ denotes the sheaf of $0$ degree cocycles. One checks that the\ndefinition of $F_{\\ast }(\\mathcal{F})$ agrees with the one given in \\cite%\n{Flenner} \\S 2.A.\n\nThere is an obvious natural inclusion morphism:%\n\\begin{equation}\nF_{\\ast }(\\mathcal{F})\\hookrightarrow F_{\\sharp }(\\mathcal{F})\n\\label{Morph_F*F_sharp}\n\\end{equation}\n\n\\begin{remark}\nOne checks that the morphism (\\ref{Morph_F*F_sharp}) induces an isomorphism\nbetween the respective derived functors.\n\\end{remark}\n\n\\begin{remark}\nLemma \\ref{Lemma_Comp_im_dir} immediately implies that ($G\\circ F)_{\\ast }(%\n\\mathcal{F})=G_{\\ast }F_{\\ast }(\\mathcal{F})$\n\\end{remark}\n\nLet $\\mathcal{F}\\in Mod(\\mathscr{X})$ be as above, $\\mathcal{G}\\in Mod(%\n\\mathscr{Y})$ with $\\mathcal{G}=((\\mathcal{G}_{\\gamma })_{\\gamma \\in \n\\mathcal{T}},(\\psi _{\\delta \\gamma })_{\\gamma \\subset \\delta })$ and let $u:%\n\\mathcal{G}\\rightarrow F_{\\sharp }(\\mathcal{F})$ be a morphism of $%\n\\mathscr{Y}$-modules. Note that $u$ factors through $F_{\\ast }(\\mathcal{F})$%\n. Hence $u$ is completely determined by the family of morphisms $(u_{\\gamma\n\\alpha })_{\\gamma \\alpha }$ where $\\gamma \\in \\mathcal{T}$, $\\alpha \\in\nI(\\gamma ,0)$, $\\ $with%\n\\begin{equation}\nu_{\\gamma \\alpha }:\\mathcal{G}_{\\gamma }\\rightarrow F_{\\alpha \\ast }(%\n\\mathcal{F}_{\\alpha })\n\\end{equation}%\nConversely, one checks directly the following lemma that describes the\nfamilies\\ $(u_{\\gamma \\alpha })_{\\gamma \\alpha }$ as above that give a\nmorphism of $\\mathscr{Y}$-modules $u:\\mathcal{G}\\rightarrow F_{\\sharp }(%\n\\mathcal{F})$:\n\n\\begin{lemma}\n\\label{Lemma_crit_morf_im_dir}The morphisms $(u_{\\gamma \\alpha })_{\\gamma\n\\alpha }$ are the components of a morphism of $\\mathscr{Y}$-modules $u:%\n\\mathcal{G}\\rightarrow F_{\\sharp }(\\mathcal{F})$ iff they verify the\nfollowing conditions:\n\n\\begin{enumerate}\n\\item For $\\gamma \\in \\mathcal{T}$, $\\beta \\in I(\\gamma ,1)$ let $\\alpha\n_{1} $, $\\alpha _{2}\\in $ $I(\\gamma ,0)$ be the only two simplexes s.t. $%\n\\alpha _{1}$, $\\alpha _{2}\\subset \\beta $. Then the following diagram\ncommutes: \n\\begin{equation}\n\\begin{CD} \\mathcal{G}_{\\gamma} @>u_{\\gamma \\alpha_{1}}>> F_{\\alpha_{1}\n\\ast}(\\mathcal{F}_{\\alpha_{1}})\\\\ @VVu_{\\gamma \\alpha_{2}}V @VVF_{\\alpha_{1}\n\\ast}(\\varphi _{\\beta \\alpha_{1}})V\\\\ F_{\\alpha_{2}\n\\ast}(\\mathcal{F}_{\\alpha_{2}}) @>F_{\\alpha_{2} \\ast}(\\varphi _{\\beta\n\\alpha_{2}})>> F_{\\beta \\ast}(\\mathcal{F}_{\\beta})\\\\ \\end{CD}\n\\label{Im_dir_cond_1}\n\\end{equation}\n\n\\item For $\\gamma ,\\delta \\in \\mathcal{T}$, $\\gamma \\subset \\delta $, and $%\n\\alpha \\in I(\\gamma ,0)$, $\\beta \\in I(\\delta ,0)$ with $\\alpha \\subset\n\\beta $ the following diagram commutes:%\n\\begin{equation}\n\\begin{CD} \\mathcal{G}_{\\gamma} @>u_{\\gamma \\alpha}>> F_{\\alpha\n\\ast}(\\mathcal{F}_{\\alpha})\\\\ @VV \\psi _{\\delta \\gamma}V @VVF_{\\alpha\n\\ast}(\\varphi_{\\beta \\alpha})V\\\\ \\rho_{\\gamma \\delta \\ast}^{\\prime}\n(\\mathcal{G}_{\\delta }) @>\\rho_{\\gamma \\delta \\ast}^{\\prime}(u_{\\delta \\beta\n})>> F_{\\alpha \\ast} \\rho_{\\alpha \\beta \\ast}(\\mathcal{F}_{\\beta })\\\\\n\\end{CD}  \\label{Im_dir_cond_2}\n\\end{equation}%\n\\end{enumerate}\n\\end{lemma}\n\\begin{flushright}\n$\\square$\n\\end{flushright}\n\n\\begin{remark}\nConditions in Lemma \\ref{Lemma_crit_morf_im_dir} and those in Lemma \\ref%\n{Lemma_crit_morf_im_inv} can be obtained from one another by adjunction.\nUsing the two lemmas one checks that the functor $F_{\\ast }$ on $Mod(%\n\\mathscr{X})$ is indeed a right adjoint for $F^{\\ast }$.\n\\end{remark}\n\n\\section{Embedding Atlases \\label{Sect_Atlases}}\n\n\\refstepcounter{subsection}\\textbf{\\ \\arabic{section}.%\n\\arabic{subsection}} \\label{paragr_emb_triples}\\textbf{Embedding triples.}\nLet $i:$ $X\\hookrightarrow D$ be a closed embedding of the analytic space $X$\nin the complex manifold $D$. $(X,i,D)$ will be called an embedding triple. A\nmorphism of embedding triples $(f,\\tilde{f}):(X_{1},i_{1},D_{1})\\rightarrow\n(X_{2},i_{2},D_{2})$ is a pair of morphisms of analytic spaces such that the\nfollowing diagram commutes:\n\\begin{equation}\n\\begin{CD} X_{1} @>{f}>> X_{2} \\\\ @VV{i_{1}}V @VV{i_{2}}V\\\\ D_{1}\n@>{\\tilde{f}}>> D_{2} \\end{CD}  \\label{morph_embed}\n\\end{equation}%\nIf $\\tilde{f}$ is clear from the context we sometimes write $f$ instead of $%\n(f,\\tilde{f})$.\n\nA complex manifold $D$ will be identified with the embedding triple \n$(D,id,D) $.\n\nIf $(X,i,D)$, $(Y,i^{\\prime },D^{\\prime })$ are embedding triples and $%\nf:X\\rightarrow Y$ is a morphism of analytic spaces, then, in general, there\ndoes not exist $\\tilde{f}:D\\rightarrow D^{\\prime }$ such that $(f,\\tilde{f})$\nis a morphism of embedding triples. However the following result can be\nchecked easily:\n\n\\begin{lemma}\n\\label{Lemma_complet_morph} \\textbf{1.} Let $(X,i,D)$, $(Y,i^{\\prime\n},D^{\\prime })$ be embedding triples and \\linebreak $f:X\\rightarrow Y$ a\nmorphism of analytic spaces. Then $\\ j=(i_{,}i^{\\prime }\\circ\nf):X\\rightarrow D\\times D^{\\prime }$ is a closed embedding and we have\nnatural morphisms: \n\\begin{equation}\n(X,i,D)\\overset{(id,p_{1})}{\\longleftarrow }(X,j,D\\times D^{\\prime })\\overset%\n{(f,p_{2})}{\\rightarrow }(Y,i^{\\prime },D^{\\prime })\n\\end{equation}%\nwhere $p_{1},p_{2}$ are the projections. Moreover, assume we have another\nembedding triple $(X,i_{1},D_{1})$ and morphisms \n\\begin{equation}\n(X,i,D)\\overset{(id,q_{1})}{\\longleftarrow }(X,i_{1},D_{1})\\overset{(f,q_{2})%\n}{\\rightarrow }(Y,i^{\\prime },D^{\\prime })\n\\end{equation}%\nThen $\\alpha =(id,(q_{1},q_{2})):(X,i_{1},D_{1})\\rightarrow $ $(X,j,D\\times\nD^{\\prime })$ is the unique morphism s.t. the following diagram\ncommutes:\n\\begin{equation}\n\\begin{tikzcd}[column sep=1.5cm] \n(X,i,D) \n& (X,j,D \\times D^{\\prime}) \n\\arrow{l}[swap]{(id, p_{1})} \\arrow{r}{(f, p_{2})} \n& (Y,i^{\\prime},D^{\\prime}) \\\\ \n& (X,i_{1},D_{1})\n\\arrow{ul}{(id, q_{1})} \n\\arrow{u}{\\alpha} \\arrow{ur}[swap]{(f, q_{2})} \n\\end{tikzcd}  \n\\end{equation}\n\n\\textbf{2. }For $s=1$, $2$ let $(Y_{s},i_{s}^{\\prime },D_{s}^{\\prime })$ be\nembedding triples and $f_{s}:X\\rightarrow Y_{s}$ morphisms of analytic\nspaces Then $\\ j_{12}=(i_{,}i_{1}^{\\prime }\\circ f_{1},i_{2}^{\\prime }\\circ\nf_{2}):X\\rightarrow D\\times D_{1}^{\\prime }\\times D_{2}^{\\prime }$ is a\nclosed embedding and there exists unique morphisms $\\alpha _{s}$\\ s.t. the\nfollowing diagrams commute:\n\\begin{equation}\n\\begin{tikzcd}[column sep=1.5cm] \n(X,i,D) \n& (X,j,D \\times D^{\\prime}_{1} \\times D^{\\prime}_{2}) \n\\arrow{l}[swap]{(id, p_{1})} \\arrow{d}{\\alpha_{s}}\n\\arrow{r}{(f_{s}, p^{\\prime}_{s})} \n& (Y,i^{\\prime}_{s},D^{\\prime}_{s}) \\\\ \n& (X,j_{s},D \\times D^{\\prime}_{s})\n\\arrow{ul}{(id, p_{1})} \n\\arrow{ur}[swap]{(f_{s}, p_{2})} \n\\end{tikzcd}  \n\\end{equation}\nwhere $j_{s}=(i_{,}i_{s}^{\\prime }\\circ f_{s})$ and $p_{1}, p_{2}, p_{s}^{%\n\\prime }$ are the obvious projections.\n\\end{lemma}\n\n\\begin{remark}\nUnder the hypothesis of Lemma \\ref{Lemma_complet_morph}, if $D_{1}\\subset \n\\mathbb{C}^{n}$, $D_{2}\\subset \\mathbb{C}^{m}$ are Stein open sets\nthen one checks that there exists a Stein open subset $D_{1}^{^{\\prime\n}}\\subset D_{1}$ and $\\tilde{f}:D_{1}^{^{\\prime }}\\rightarrow D_{2}$ such\nthat the following diagram commutes: \n\\begin{equation}\n\\begin{CD} D_{1} @<{i}<< D_{1}^{\\prime}@>{\\tilde{f}}>> D_{2}\\\\ @AA{i_{1}}A\n@AA{i_{1}}A@AA{i_{2}}A\\\\ X @<{id}<< X @>{f}>> Y \\end{CD}\n\\end{equation}%\ni.e we have the diagram of embedding triples:%\n\\begin{equation}\n(X,i_{1},D_{1})\\overset{(id,i)}{\\hookleftarrow }(X,i_{1},D_{1}^{^{\\prime }})%\n\\overset{(f,\\tilde{f})}{\\rightarrow }(Y,i_{2},D_{2})\n\\end{equation}\n\\end{remark}\n\nFor brevity we shall say s.s.embedding triple instead of s.s.system of\nembedding triples\n\n\\begin{remark}\n\\label{Rem_ss_embed_triple}A s.s.embedding triple $(X_{\\alpha },k_{\\alpha\n},D_{\\alpha })_{\\alpha \\in S}$ can be seen as a triple $(\\mathscr{X},k,%\n\\mathscr{D})$ where $\\mathscr{X}=(X_{\\alpha })_{\\alpha \\in \\mathcal{S}}$ is\ns.s.analytic space, $\\mathscr{D}=(D_{\\alpha })_{\\alpha \\in \\mathcal{S}}$ is\na s.s.complex manifold, and $k:\\mathscr{X}\\rightarrow \\mathscr{D}$\\ is a\nmorphism of s.s.analytic spaces such that each $k_{\\alpha }:X_{\\alpha\n}\\rightarrow {D}_{\\alpha }$ is a closed embedding.\n\\end{remark}\n\n\\pagebreak\n\\refstepcounter{subsection} \\textbf{\\arabic{section}.\\arabic{subsection}} \n\\textbf{Embedding atlases.\\label{paragr_Atlas}}\n\n\\begin{definition}\n\\label{def_atlas}\n\n\\begin{enumerate}\n\\item Let $X$ be an analytic space. An embedding atlas of $X$ consists of a\nfamily of embedding triples $\\mathcal{A}=(U_{i},k_{i},D_{i})_{i\\in I}$ such\nthat the family \\linebreak $cov(\\mathcal{A})=(U_{i})_{i\\in I}$ is an open covering of $%\nX$. An embedding triple $(U_{i},k_{i},D_{i})$\\ of $\\mathcal{A}$ will be\ncalled a chart. The pair $(X,\\mathcal{A})$ will be called a locally embedded\nanalytic space or, sometimes, a local embedding of $X$.\n\n\\item Let $\\mathcal{A}=(U_{i},k_{i},D_{i})_{i\\in I}$ \\ and $\\mathcal{B}%\n=(V_{j},k_{j}^{\\prime },D_{j}^{\\prime })_{j\\in J}$ \\ be embedding atlases of\nthe analytic space $X$, respectively $Y$. A morphism of locally embedded\nanalytic spaces $F:(X,\\mathcal{A})\\rightarrow (Y,\\mathcal{B})$ consists of a\ntriple $(f,\\tau ,(\\tilde{f}_{i})_{i\\in I})$ where:\n\n\\begin{enumerate}\n\\item[-] $f:X\\rightarrow Y$ is a morphism of analytic spaces\n\n\\item[-] $\\tau :I\\rightarrow J$ is a refinement mapping such that $%\nf(U_{i})\\subset V_{\\tau (i)}$ for all $i\\in I$\n\n\\item[-] $\\tilde{f}_{i}:$ $D_{i}\\rightarrow D_{\\tau (i)}^{\\prime }$ is a\nmorphism of complex manifolds such that $(f|U_{i},\\tilde{f}%\n_{i}):(U_{i},k_{i},D_{i})\\rightarrow (V_{\\tau (i)},k_{\\tau (i)}^{\\prime\n},D_{\\tau (i)}^{\\prime })$\\ is a morphism of embedding triples.\n\\end{enumerate}\n\\end{enumerate}\n\\end{definition}\n\nWe say that $\\mathcal{A}$ is locally finite if the open covering $cov(%\n\\mathcal{A})$ is locally finite.\n\nIn particular, an embedding triple $(X,i,D)$ can be seen as an embedding\natlas of $X$\\ with one chart.\n\n\\begin{lemma}\n\\label{Lemma_assoc_ss_triples} \\textbf{1. }Let $(X,\\mathcal{A})$ be a\nlocally embedded analytic space with $\\mathcal{A}%\n=(U_{i},k_{i},D_{i})_{i\\in I}$. There exists a s.s.embedding\ntriple $(\\mathfrak{U},k,\\mathfrak{D})=(U_{\\alpha },k_{\\alpha\n},D_{\\alpha })_{\\alpha \\in \\mathcal{N(U)}}$ indexed by the simplicial\ncomplex $(I,\\mathcal{N}(cov(\\mathcal{A})))$\\ such that the embedding triples\ncorresponding to $0$-length simplexes coincide with the embedding triples of\nthe atlas $\\mathcal{A}$.\n\n\\textbf{2.} If $f:(X,\\mathcal{A})\\rightarrow (Y,\\mathcal{B})$ is a morphism\nof embedded analytic spaces then $f$ induces a morphism%\n\\begin{equation*}\nF:(\\mathfrak{U},k,\\mathfrak{D})\\rightarrow (\\mathfrak{V},k^{\\prime },%\n\\mathfrak{D}^{\\prime })\n\\end{equation*}%\nbetween the respective associated s.s.embedding triples. Moreover the\nfollowing diagram commutes:%\n\\begin{equation*}\n\\begin{CD} \\mathfrak{U} @>{F}>> \\mathfrak{V} \\\\ @VV{b}V @VV{b^{\\prime}}V\\\\ X\n@>{f}>> Y \\end{CD}\n\\end{equation*}\n\\end{lemma}\n\n\\begin{proof}\nTake $\\mathfrak{U}=((U_{\\alpha })_{\\alpha \\in \\mathcal{N}(cov(\\mathcal{A}%\n))},(i_{\\alpha \\beta })_{\\alpha \\subset \\beta })$ to be the s.s.analytic\nspace corresponding to the open covering $(U_{i})_{i\\in I}$ (see Example \\ref%\n{ex_acop}),\n$\\mathfrak{D}=((D_{\\alpha })_{\\alpha \\in \\mathcal{N}%\n(cov(\\mathcal{A}))},(p_{\\alpha \\beta })_{\\alpha \\subset \\beta })$ the\ns.s.complex manifold associated to the family $(D_{i})_{i\\in I}$\\ (see\nExample \\ref{ex_prod})\\ and $k:\\mathfrak{U\\rightarrow D}$\\ the morphism\ndeduced from the closed embeddings $k_{i}:U_{i}\\rightarrow D_{i}$(one checks\nthat each $k_{\\alpha }:U_{\\alpha }\\rightarrow D_{\\alpha }$ is also a closed\nembedding).\n\\end{proof}\n\nThe s.s.embedding triple from Lemma \\ref{Lemma_assoc_ss_triples}\\ will be\ncalled the s.s.embedding triple associated to $(X,\\mathcal{A})$.\n\n\\begin{remark}\nOne checks easily that the correspondence in Lemma \\ref%\n{Lemma_assoc_ss_triples} gives an equivalence between the category of\nlocally embedded analytic spaces and a subcategory of the category of\ns.s.embedding triples.\n\\end{remark}\n\n\\begin{lemma}\n\\label{Lemma_lifting_morph}\n\\begin{enumerate}\n\\item Let $f:(X,\\mathcal{A})\\rightarrow (Y,\\mathcal{B})$ be a morphism of\nlocally embedded analytic spaces, $\\mathcal{F}\\in Mod(\\OX{X})$, $%\n\\mathcal{G}\\in Mod(\\OX{Y})$ and $u:\\mathcal{G}\\rightarrow f_{\\ast }%\n\\mathcal{F}$ \\ a morphism of $\\OX{Y}$-modules. If $F:(\\mathfrak{U}%\n,k,\\mathfrak{D})\\rightarrow (\\mathfrak{V},k^{\\prime },\\mathfrak{D}^{\\prime\n}) $ is the morphism induced between the s.s.systems of embedding triples\nassociated to $(X,\\mathcal{A})$, $(Y,\\mathcal{B})$ then $u$\\ induces a\nnatural morphism: \n\\begin{equation}\nF^{\\ast }(u):\\mathcal{G}|\\mathfrak{V} \\rightarrow F_{\\ast }(%\n\\mathcal{F}|\\mathfrak{U})  \\label{morf_f*_sus}\n\\end{equation}\n\n\\item Let $(X,\\mathcal{A})\\overset{f}{\\rightarrow }(Y,\\mathcal{B})\\overset{g}%\n{\\rightarrow }(Z,\\mathcal{C})$ be morphisms of locally embedded analytic\nspaces and $h=g\\circ f$. Let moreover $\\mathcal{F} \\in Mod(\\OX{X})$, \n$\\mathcal{G}\\in Mod(\\OX{Y})$, $\\mathcal{H}\\in Mod(\\OX{Z})$\nand morphisms $u:\\mathcal{G}\\rightarrow f_{\\ast }\\mathcal{F}$ $\\OX{Y} $-linear, $v:\\mathcal{H}\\rightarrow g_{\\ast }\\mathcal{G}$, $w:\\mathcal{%\nH}\\rightarrow h_{\\ast }\\mathcal{F}$ $\\OX{Z}$-linear, such that $%\ng_{\\ast }(u)\\circ v=w$ then one has the commutative diagram\n\\begin{equation}\n\\begin{tikzcd}[column sep=0.2cm] \nH_{\\ast}(\\mathcal{F}|\\mathfrak{U}) \n&& G_{\\ast}(\\mathcal{G}|\\mathfrak{V})\n\\arrow{ll}[swap]{G_{\\ast}(F^{\\ast}(u))} \\\\ \n& \\mathcal{H}|\\mathfrak{W} \n\\arrow{ul}{H^{\\ast}(w)} \\arrow{ur}[swap]{G^{\\ast}(v)}\n\\end{tikzcd}\n\\label{diagr_3_sus}\n\\end{equation}\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\n\\textbf{1. }Let $cov(\\mathcal{A})\\mathcal{=}(U_{i})_{i\\in I}$, $cov(\\mathcal{%\nB})\\mathcal{=}(V_{j})_{j\\in J}$. For $\\alpha \\in \\mathcal{N} (cov(\\mathcal{A})), \n\\gamma \\in \\mathcal{N}(cov(\\mathcal{B}))$ such that $\\tau\n(\\alpha )\\subset \\gamma $ let%\n\\begin{equation*}\nf_{\\gamma \\alpha }:U_{\\alpha }\\rightarrow V_{\\gamma }\n\\end{equation*}%\nbe the restriction of $f$ and \n\\begin{equation*}\n\\ u_{\\gamma \\alpha }:\\mathcal{G}|V_{\\gamma }\\rightarrow f_{\\gamma \\alpha\n\\ast }(\\mathcal{F}|U_{\\alpha })\n\\end{equation*}%\nthe restriction of $u$. One verifies that the family of morphisms \n$(u_{\\gamma \\alpha })_{\\gamma \\alpha }$, where \n\\linebreak\n$\\gamma \\in \\mathcal{N}(cov(\\mathcal{B}))$, $\\alpha \\in I(\\gamma ,0)$, \nsatisfies the hypothesis of Lemma \\ref{Lemma_crit_morf_im_dir} and consequently \nthey determine a morphism $F^{\\ast }(u)$.\n\n\\textbf{2. }follows also from Lemma \\ref{Lemma_crit_morf_im_dir} since one\nverifies that travelling both ways along the edges of diagram (\\ref{diagr_3_sus})\nthe two morphisms are determined by the same family of morphisms.\n\\end{proof}\n\n\\begin{definition}\n\\label{Def_f_compliant}Let $f:X\\rightarrow Y$ be a morphism of analytic\nspaces and let \n\\linebreak \n$\\mathcal{A}=(U_{i},k_{i},D_{i})_{i\\in I}$ and \n$\\mathcal{B}=(V_{j},k_{j}^{\\prime },D_{j}^{\\prime })_{j\\in J}$ be \nembedding atlases of $X$, respectively $Y$. $\\mathcal{A}$\\ and $\\mathcal{B}$ \nare said to be $f$-compliant if $cov(\\mathcal{A})\\prec f^{-1}(cov(\\mathcal{B}))$, \ni.e.\\ there exists a refinement mapping $\\tau :I\\rightarrow J$ such that\n$f(U_{i})\\subset V_{\\tau (i)}$ for all $i\\in I.$ Obviously the refinement\nmapping $\\tau $ need not be unique.\n\\end{definition}\n\nNote that if $\\mathcal{A}_{1}$\\ and $\\mathcal{A}_{2}$ are embedding atlases\nof $X$, to say that $\\mathcal{A}_{1}$\\ and $\\mathcal{A}_{2}$ are $id_{X}$%\n-compliant simply means that $cov(\\mathcal{A}_{1})\\prec cov(\\mathcal{A}_{2})$%\n.\n\nIf $\\mathcal{A}$\\ and $\\mathcal{B}$ are $f$-compliant then, in general,\nthere does not exist a morphism of locally embedded analytic spaces $F:(X,%\n\\mathcal{A})\\rightarrow (Y,\\mathcal{B})$ over $f$. However Lemma \\ref%\n{Lemma_complet_morph} immediately implies:\n\n\\begin{lemma}\n\\label{Lemma_complet_morph_atlas}\n\n\\begin{enumerate}\n\\item \\label{Lemma_complet_morph_atlas_1} In the context of Definition \\ref%\n{Def_f_compliant} let \n\\begin{equation*}\n\\mathcal{A}\\times _{\\tau }\\mathcal{B}=(U_{i},k_{i\\tau (i)},D_{i}\\times\nD_{\\tau (i)}^{\\prime })_{i\\in I}\n\\end{equation*}%\nThen $\\mathcal{A}\\times _{\\tau }\\mathcal{B}$ is an embedding atlas on $X$,\nand the family of morphisms of embedding triples \n\\begin{equation*}\n(U_{i},k_{i},D_{i})\\overset{(id,p_{1i})}{\\longleftarrow }(U_{i},k_{i\\tau\n(i)},D_{i}\\times D_{\\tau (i)}^{\\prime })\\overset{(f,p_{2i})}{\\rightarrow }%\n(V_{\\tau (i)},k_{\\tau (i)}^{\\prime },D_{\\tau (i)}^{\\prime })\n\\end{equation*}%\nwhere $p_{1i}, p_{2i}$ are the projections, give the diagram of locally\nembedded analytic spaces: \n\\begin{equation*}\n(X,\\mathcal{A})\\overset{(id,id,p_{1})}{\\longleftarrow }(X,\\mathcal{A}\\times\n_{\\tau }\\mathcal{B})\\overset{(f,\\tau ,p_{2})}{\\rightarrow }(Y,\\mathcal{B})\n\\end{equation*}%\nMoreover, assume $\\mathcal{A}^{\\prime }=(U_{k},k_{k}^{\\prime },D_{k}^{\\prime\n})_{k\\in K}$ is another embedding atlas of $X$,\\ and there exist morphisms \n\\begin{equation*}\n(X,\\mathcal{A})\\overset{(id,\\upsilon ,q_{1})}{\\longleftarrow }(X,\\mathcal{A}%\n^{\\prime })\\overset{(f,\\tau \\circ \\upsilon ,q_{2})}{\\rightarrow }(Y,\\mathcal{%\nB})\n\\end{equation*}%\nThen there is a unique morphism $\\alpha =(X,\\mathcal{A}^{\\prime\n})\\rightarrow $ $(X,\\mathcal{A}\\times _{\\tau }\\mathcal{B})$\\ such that the\nfollowing diagram commutes:\n\\begin{equation}\n\\begin{tikzcd}[column sep=1.5cm] (X,\\mathcal{A}) & (X,\\mathcal{A}\n\\times_{\\tau} \\mathcal{B}) \\arrow{l}[swap]{(id,id,p_{1})} \\arrow{r}{(f,\n\\tau, p_{2})} & (Y,\\mathcal{B}) \\\\ & (X,\\mathcal{A}^{\\prime})\n\\arrow{ul}{(id,\\tau,q_{1})} \\arrow{u}{\\alpha} \\arrow{ur}[swap]{(f,\\tau \\circ\n\\upsilon, q_{2})} \\end{tikzcd}  \\label{Diagr_A_prim}\n\\end{equation}\n\n\\item \\label{Lemma_complet_morph_atlas_2} Let $\\tau _{1},$ $\\tau _{2}$:$%\nI\\rightarrow J$ be refinement mappings. Then \n\\begin{equation*}\n\\mathcal{A}\\times _{\\tau _{1}\\tau _{2}}\\mathcal{B}=(U_{i},k_{i\\tau\n_{1}(i)\\tau _{2}(i)},D_{i}\\times D_{\\tau _{1}(i)}^{\\prime }\\times D_{\\tau\n_{2}(i)}^{\\prime })_{i\\in I}\n\\end{equation*}%\nis an embedding atlas on $X$ and, for $s=1,2$, one has natural morphisms:%\n\\begin{equation}\n(X,\\mathcal{A})\\overset{(id,id,p_{1})}{\\longleftarrow }(X,\\mathcal{A}\\times\n_{\\tau _{1}\\tau _{2}}\\mathcal{B})\\overset{(f,\\tau _{s},p_{s})}{\\rightarrow }%\n(Y,\\mathcal{B})  \\label{Diagr_tau12}\n\\end{equation}%\nsuch that the following diagram commutes\n\\begin{equation}\n\\begin{tikzcd}[column sep=1.5cm] \n(X,\\mathcal{A}) & (X,\\mathcal{A} \\times_{\\tau_{1} \\tau_{2}} \\mathcal{B}) \n\\arrow{l}[swap]{(id,id,p_{1})} \\arrow{d}{\\alpha_{s}} \n\\arrow{r}{(f,\\tau_{s} ,p_{s})} \n& (Y,\\mathcal{B}) \\\\ \n& (X,\\mathcal{A} \\times_{\\tau_{s}} \\mathcal{B}) \n\\arrow{ul}{(id,id,p_{1})} \n\\arrow{ur}[swap]{(f,\\tau_{s} ,p_{2})} \n\\end{tikzcd}  \\label{Diagr_tau_1_2}\n\\end{equation}\nwhere $\\alpha _{s}=(id,id,p_{1s})$\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{corollary}\n\\label{Corol_Atlas_compunere}Let $X\\overset{f}{\\rightarrow }Y\\overset{g}{%\n\\rightarrow }Z$ be morphisms of analytic spaces, $h=g\\circ f$, and let $%\n\\mathcal{A}=(U_{i},k_{i},D_{i})_{i\\in I}$, $\\mathcal{B}=(V_{j},k_{j}^{\\prime\n},D_{j}^{\\prime })_{j\\in J}$, $\\mathcal{C}=(W_{k},k_{k}^{\\prime \\prime},D_{k}^{\\prime \\prime})_{k\\in\nK}$ be embedding atlases of $X$, respectively $Y$, and $Z$. Assume that $%\n\\mathcal{A}$, $\\mathcal{B}$ are $f$-compliant, $\\mathcal{B}$, $\\mathcal{C}$\nare $g$-compliant and let $\\tau :I\\rightarrow J$, $\\upsilon :J\\rightarrow K$\nbe refinement mappings. Set \n\\begin{equation}\n\\mathcal{A}\\times _{\\tau }\\mathcal{B}\\times _{\\upsilon }\\mathcal{C}%\n=(U_{i},k_{i\\tau (i)\\upsilon (i)},D_{i}\\times D_{\\tau (i)}^{\\prime }\\times\nD_{\\upsilon (i)}^{\\prime \\prime})_{i\\in I}\n\\end{equation}%\nThen $\\mathcal{A}\\times _{\\tau }\\mathcal{B}\\times _{\\upsilon }\\mathcal{C}$\nis an embedding atlas on $X$, and the following diagram commutes:%\n\\begin{equation}\n\\begin{tikzcd}[column sep=0cm, row sep=scriptsize] && (X,\\mathcal{A}\n\\times_{\\tau} \\mathcal{B} \\times_{\\upsilon} \\mathcal{C})\n\\arrow{dl}[swap]{(id,id, p_{12})} \\arrow{dr}{(f,\\tau , p_{23})} \\arrow[ddll,\nbend right, \"{(id,id,p_{1})}\"'] \\arrow[ddrr, bend left, \"{(g \\circ f,\n\\upsilon \\circ \\tau, p_{3})}\"] && \\\\ & (X,\\mathcal{A} \\times_{\\tau}\n\\mathcal{B}) \\arrow{dl}[swap]{(id,id, p_{1})} \\arrow{dr}{(f,\\tau , p_{2})}\n&& (Y,\\mathcal{B} \\times_{\\upsilon} \\mathcal{C})\n\\arrow{dl}[swap]{(id,id,p_{1})} \\arrow{dr}{(g,\\upsilon ,p_{2})} & \\\\\n(X,\\mathcal{A}) && (Y,\\mathcal{B}) && (Z,\\mathcal{C}) \\end{tikzcd}\n\\label{diagr_morph_compoz}\n\\end{equation}%\nwhere $p_{12},p_{23},p_{1},p_{2},p_{3}$ are the obvious projections.\n\\end{corollary}\n\n\\begin{remark}\n\\label{Rem_comut_morph_compoz}By Lemma \\ref{Lemma_complet_morph_atlas}. \\ref%\n{Lemma_complet_morph_atlas_1} there is a unique morphism \n\\begin{equation*}\n\\alpha :(X,\\mathcal{A}\\times _{\\tau }\\mathcal{B}\\times _{\\upsilon }\\mathcal{C%\n})\\rightarrow (X,\\mathcal{A}\\times _{\\upsilon }\\mathcal{C})\n\\end{equation*}%\n\\ such that the following diagram commutes:%\n\\begin{equation}\n\\begin{tikzcd}[column sep=1.5cm] (X,\\mathcal{A}) & (X,\\mathcal{A}\n\\times_{\\upsilon \\circ \\tau} \\mathcal{C}) \\arrow{l}[swap]{(id,id,p_{1})}\n\\arrow{r}{(g \\circ f,\\upsilon \\circ \\tau, p_{2})} & (Z,\\mathcal{C}) \\\\ &\n(X,\\mathcal{A} \\times_{\\tau} \\mathcal{B} \\times_{\\upsilon} \\mathcal{C})\n\\arrow{ul}{(id,id,p_{1})} \\arrow{u}{\\alpha} \\arrow{ur}[swap]{(g \\circ\nf,\\upsilon \\circ \\tau, p_{3})} \\end{tikzcd}  \\label{Diagr_comut_morph_compoz}\n\\end{equation}\n\\end{remark}\n\n\\begin{remark}\n\\label{Rem_morph_atlas_gen}Let $(X,\\mathcal{A})$, $(Y,\\mathcal{B})$ be\nlocally embedded analytic spaces,\\linebreak\\ $\\mathcal{A}%\n=(U_{i},k_{i},D_{i})_{i\\in I}$ and $\\mathcal{B}=(V_{j},k_{j}^{\\prime\n},D_{j}^{\\prime })_{j\\in J}$, and $f:X\\rightarrow Y$ a morphism of analytic\nspaces. If $\\mathcal{A}_{1}$ is an embedding atlas of $X$ over the open\ncovering $(U_{i}\\cap f^{-1}(V_{j}))_{i,j}$ then $\\mathcal{A}_{1}$, $\\mathcal{%\nA}$ are $id_{X}$-compliant and $\\mathcal{A}_{1}$, $\\mathcal{B}$ are $f$%\n-compliant. Moreover, if $\\mathcal{A}_{2}$ is another embedding atlas of $X$\ns.t. $\\mathcal{A}_{2}$, $\\mathcal{A}$ are $id_{X}$-compliant and $\\mathcal{A}%\n_{2}$, $\\mathcal{B}$ are $f$-compliant then $\\mathcal{A}_{2}$, $\\mathcal{A}%\n_{1}$ are $id_{X}$-compliant.\n\\end{remark}\n\n                      \n\\section{Construction of the Dolbeault resolution\\label{Sect_Resol}}\n\nWe shall extend successively the definition of the\nDolbeault-Grothendieck resolution from the classical case of complex\nmanifolds to that of embedding triples, then to s.s.of embedding triples\nand, finally, to the case of a general analytic space with a fixed embedding\natlas. It is essential that at each extension the definition be compatible\nwith $\\sharp$-direct images (i.e. there exists a commutative diagram similar to\ndiagram (\\ref{diagr_3_A})). For reference purposes property \\textbf{1.a}, for\ninstance, will be called \\textbf{1.a-mfld} in the smooth case, \\textbf{%\n1.a-emb} in the embedded case, and \\textbf{1.a-ss} in the semi-simplicial\ncase.\n\n\\refstepcounter{subsection}\\textbf{\\ \\arabic{section}.\\arabic{subsection}} \n\\textbf{The smooth case. }Let $X$ be an $n$-dimensional complex manifold \\\nWe regard $X$ as a locally embedded analytic space with one chart given by\nthe identity map. For $\\mathcal{F}\\in Mod(\\OX{X})$\\ denote by $\\Dolbm{X}{F}$ the complex:%\n\\begin{equation}\n0\\longrightarrow \\mathcal{E}_{X}^{0,0}\\otimes _{\\OX{X}}\\mathcal{F}%\n\\longrightarrow ...\\longrightarrow \\mathcal{E}_{X}^{0,n}\\otimes _{\\OX{X}} \\mathcal{F}\\longrightarrow 0  \\label{Compl_Dolb_F}\n\\end{equation}%\nobtained by applying the functor $\\bullet \\otimes _{\\OX{X}}\\mathcal{F}$ to the resolution (\\ref{Rezol_Dolb}) with the term containing $\\mathcal{E%\n}_{X}^{0,0}$ considered in degree $0$. This complex appeared first in the\nproof of the duality theorems of Serre-Malgrange.\n\nNote that $\\Dolb{X}{\\bullet }$ is a functor $Mod(\\OX{X})\\rightarrow C^{b}(X)$. We check that this functor satisfies the properties of the Theorem.\n\n\\textbf{1.d-mfld }is proved in Malgrange \\cite{M}. \\textbf{1.a-mfld} and \n\\textbf{1.b-mfld} follow immediately from \\textbf{1.d-mfld}, and \\textbf{%\n1.c-mfld} is well known. In \\textbf{1.e-mfld }the morphism \n\\begin{equation}\n\\Dolb{X}{\\OX{X}}\\otimes _{\\OX{X}}\\mathcal{F}\\rightarrow \\Dolbm{X}{F}  \\label{ident_Dolb_X}\n\\end{equation}%\nis obviously an isomorphism for any $\\mathcal{F}$.\n\nFor \\textbf{2-mfld }let $f:X\\rightarrow Y$ be a morphism of complex\nmanifolds. Let%\n\\begin{equation}\nf^{\\ast }:\\Dolb{Y}{\\OX{Y}}\\rightarrow f_{\\ast }\\Dolb{X}{\\OX{X}}   \\label{Dolb_X_f}\n\\end{equation}%\nbe the morphism given by the pullback of differential forms - it is a\nmorphism over the mapping $f^{\\ast }:\\OX{Y} \\rightarrow f_{\\ast }\\OX{X}$. For $\\mathcal{F}\\in Mod(\\OX{X})$, combining (\\ref{ident_Dolb_X}) and (\\ref{Dolb_X_f}), one gets a functorial morphism \n\\begin{equation}\n\\Dolb{Y}{f_{\\ast}\\mathcal{F}}\\rightarrow f_{\\ast}\\Dolbm{X}{F}\n\\label{Dolb_f_F}\n\\end{equation}%\nand moreover the following diagram commutes:%\n\\begin{equation}\n\\begin{tikzcd}[column sep=-0.2cm] \n\\Dolb{Y}{f_{\\ast}\\mathcal{F}}\n\\arrow{rr} && f_{\\ast}\\Dolbm{X}{F} \\\\ & f_{\\ast\n}\\mathcal{F} \\arrow{ul} \\arrow{ur} \\end{tikzcd}  \\label{Diagr_Dolb_f_F}\n\\end{equation}\n\nLet moreover $\\mathcal{G}\\in Mod(\\OX{Y})$ and $u:\\mathcal{%\nG}\\rightarrow f_{\\ast}\\mathcal{F}$ a morphism of $\\OX{Y}$%\n-modules Then the morphism (\\ref{Dolb_f_F}) together with $u$ induces the\nmorphism of complexes: \n\\begin{equation}\nf^{\\ast}(u):\\Dolbm{Y}{G}\\rightarrow f_{\\ast}\\Dolbm{X}{F}\n\\end{equation}%\nand combining (\\ref{Diagr_Dolb_f_F}) with \\textbf{1.b-mfld} one checks that\nthe diagram (\\ref{diagr_f*_u}) commutes in the smooth case.\n\n\\textbf{3-mfld }One starts from the obvious commutative diagram\n\\begin{equation}\n\\begin{tikzcd}[column sep=-0.2cm] h_{\\ast}\\Dolb{X}{\\OX{X}}\n&& g_{\\ast}\\Dolb{Y}{\\OX{Y}}\n\\arrow{ll}[swap]{g_{\\ast}(f^{\\ast})} \\\\ & \\Dolb{Z}{\\OX{Z}}\n\\arrow{ul}{h^{\\ast}} \\arrow{ur}[swap]{g^{\\ast}} \\end{tikzcd}\n\\label{Diagr_3_Dolb}\n\\end{equation}\nUsing isomorphism (\\ref{ident_Dolb_X}) and the above diagram one checks the\ncase $\\mathcal{G}=f_{\\ast }\\mathcal{F}$, $\\mathcal{H}=h_{\\ast }\\mathcal{F}\n$, and each of $u,v,w$ \\ equals the respective identity. For the general\ncase one uses the functorial morphism (\\ref{Dolb_f_F}).\n\nFinally we consider the case of open and closed embeddings of manifolds.\n\n\\begin{remark}\n\\label{Ex_open_emb} Let $i:X\\hookrightarrow Y$ be an open embedding of\ncomplex manifolds, \n\\linebreak\n$\\mathcal{F}\\in Mod(\\OX{Y})$, and denote by $u:%\n\\mathcal{F}\\rightarrow i_{\\ast }i^{-1}(\\mathcal{F})$ the canonical adjunction \nmorphism. Then obviously $\\Dolb{X}{i^{-1}\\mathcal{F}}\\simeq i^{-1}\\Dolbm{Y}{F}$ \nand, hence, $i^{\\ast }(u)$ coincides with the adjunction morphism \n$\\Dolbm{Y}{F}\\rightarrow i_{\\ast }i^{-1}\\Dolbm{Y}{F}$.\n\\end{remark}\n\n\\begin{proposition}\n\\label{Prop_qiso_manif} Let $i:X\\hookrightarrow Y$ be a closed embedding of\nmanifolds and $\\mathcal{F}\\in Mod(\\OX{X})$ Then the natural morphism \n\\begin{equation}\ni^{\\ast }(id):\\Dolb{Y}{i_{\\ast }\\mathcal{F}}\\rightarrow i_{\\ast } \\Dolbm{X}{F}  \\label{i*_closed_emb}\n\\end{equation}%\nis a quasiisomorphism.\n\\end{proposition}\n\n\\begin{proof}\n$i^{\\ast }(id)$ is a morphism between two soft resolutions of $i_{\\ast }%\n\\mathcal{F}$.\\medskip\n\\end{proof}\n\n\\refstepcounter{subsection} \\textbf{\\arabic{section}.\\arabic{subsection}} %\n\\label{paragr_emb_case}\\textbf{The embedded case. }Let $(X,i,D)$ be an\nembedding triple. We regard $(X,i,D)$ as a locally embedded analytic space\nwith one chart. Let $\\mathcal{F}\\in Mod(\\OX{X})$. We use the\nnotation $\\Dolbm{i\\,}{F}$ for the restriction to $X$ of the\ncomplex $\\Dolb{D}{i_{\\ast }\\mathcal{F}}$: \n\\begin{equation}\n0\\longrightarrow \\mathcal{E}_{D}^{0,0}\\otimes _{\\OX{D}}i_{\\ast }%\n\\mathcal{F}\\longrightarrow ...\\longrightarrow \\mathcal{E}_{D}^{0,n}\\otimes _{\\OX{D}}i_{\\ast }\\mathcal{F}\\longrightarrow 0  \\label{Compl_Dolb_i}\n\\end{equation}%\nRemark that each term of $\\Dolb{D}{i_{\\ast }\\mathcal{F}}$, is null\noutside $X$ and, furthermore, has a structure of $i_{\\ast }\\OX{X}$%\n-module deduced via the natural morphism $\\OX{D}\\rightarrow i_{\\ast}\n\\OX{X}$. If $\\mathcal{I}_{X}\\subset \\OX{D}$ is the ideal that defines $X$ \nas an analytic subspace of $D$ (i.e. $i_{\\ast }\\OX{X}\\thickapprox \\OX{D}\/\\mathcal{I}_{X}$) then it is immediate to see that $\\Dolb{i\\,}{\\OX{X}}$ is isomorphic with the restriction to $X$ of the complex \n\\begin{equation}\n0\\longrightarrow \\mathcal{E}_{D}^{0,0}\/\\mathcal{I}_{X}\\mathcal{E}%\n_{D}^{0,0}\\longrightarrow ...\\longrightarrow \\mathcal{E}_{D}^{0,n}\/\\mathcal{I%\n}_{X}\\mathcal{E}_{D}^{0,n}\\longrightarrow 0  \\label{Compl_Dolb_i1}\n\\end{equation}%\nand, moreover, $\\Dolbm{i\\,}{F}$ is isomorphic with the restriction to $X$ of the complex: \n\\begin{equation}\n0\\longrightarrow \\mathcal{E}_{D}^{0,0}\/\\mathcal{I}_{X}\\mathcal{E}%\n_{D}^{0,0}\\otimes _{\\OX{D}}i_{\\ast }\\mathcal{F}\\longrightarrow\n...\\longrightarrow \\mathcal{E}_{D}^{0,n}\/\\mathcal{I}_{X}\\mathcal{E}%\n_{D}^{0,n}\\otimes _{\\OX{D}}i_{\\ast }\\mathcal{F}\\longrightarrow 0\n\\end{equation}\n\nThus $\\Dolb{i\\,}{\\bullet}$ is a functor $Mod(\\OX{X})\\rightarrow C^{b}(X)$ and \n\\begin{equation}\n\\Dolbm{i\\,}{F}=i^{-1}\\Dolb{D}{i_{\\ast }\\mathcal{F}} \n\\label{Def_Dolb_i}\n\\end{equation}%\nMoreover, applying $i_{\\ast }$\\ one obtains an isomorphism on $D$: \n\\begin{equation}\n\\Dolb{D}{i_{\\ast }\\mathcal{F}}\\approx i_{\\ast }\\Dolbm{i\\,}{F}  \\label{i*_Def_Dolb_i}\n\\end{equation}\n\nThe functor $\\Dolb{i\\,}{\\bullet}$ coincides with $\\Dolb{X}{\\bullet}$ \nif $X$ is a complex manifold (recall that we identify $X$ with the embedding triple $(X,id,X)$).\n\nThe properties of $\\Dolb{i\\,}{\\bullet}$ are obtained from those of $\\Dolb{D}{\\bullet}$ via $i^{-1}$ and $i_{\\ast}$\n(see (\\ref{Def_Dolb_i}) and (\\ref{i*_Def_Dolb_i})). Indeed, since $i^{-1}$ \nis an exact functor the statements \\textbf{1.a-emb}, through \\textbf{1.d-emb}\nfor $\\Dolb{i\\,}{\\bullet}$ follow immediately from the respective statements in the smooth case. Moreover, for any $\\mathcal{F} \\in Mod(\\OX{X})$ the isomorphism (\\ref{ident_Dolb_X}) implies that one has an isomorphism \n\\begin{equation}\n\\Dolb{i\\,}{\\OX{X}}\\otimes _{\\OX{X}} \\mathcal{F}\\rightarrow \n\\Dolbm{i\\,}{F}  \\label{ident_Dolb_i}\n\\end{equation}\n\n\\qquad \\textbf{2-emb} Let \\ $f=(f,\\tilde{f}):(X,i_{1},D_{1})\\rightarrow\n(Y,i_{2},D_{2})$ be a morphism of embedding triples. On $D_{2}$ one has the\nsequence of morphisms:%\n\\begin{equation}\n\\Dolb{D_{2}}{i_{2\\ast}\\OX{Y}}\\rightarrow \\Dolb{D_{2}}{\\tilde{f}_{\\ast }i_{1\\ast }\\OX{X}} \\rightarrow  \\tilde{f}_{\\ast }\\Dolb{D_{1}}{i_{1\\ast}\\OX{X}}\n\\end{equation}\n\nBy definition $f^{\\ast }$ is the morphism obtained by restricting to $X$ the\ncomposition of morphisms above (which comes down to applying $i_{2}^{-1}$):%\n\\begin{equation}\nf^{\\ast }:\\Dolb{i_{2}}{\\OX{Y}} \\rightarrow f_{\\ast}\\Dolb{i_{1}}{\\OX{X}}  \\label{Dolb_f_i}\n\\end{equation}\n\nCombining (\\ref{ident_Dolb_i}) and (\\ref{Dolb_f_i}), one gets for $\\mathcal{F%\n}\\in Mod(\\OX{X})$ a functorial morphism \n\\begin{equation}\n\\Dolb{i_{2}}{f_{\\ast }\\mathcal{F}}\\rightarrow f_{\\ast }\\Dolbm{i_{1}\\,}{F}  \\label{Dolb_f*_i}\n\\end{equation}\n\nLet moreover $\\mathcal{G}\\in Mod(\\OX{Y})$ and $u:\\mathcal{%\nG}\\rightarrow f_{\\ast }\\mathcal{F}$ a morphism of $\\OX{Y}$%\n-modules Then $f^{\\ast }(u)$ is by definition the composition of the above\nmorphism with the morphism induced by $u$: \n\\begin{equation}\nf^{\\ast}(u):\\Dolbm{i_{2}\\,}{G} \\rightarrow f_{\\ast} \\Dolbm{i_{1}\\,}{F}\n\\end{equation}%\nFinally, diagram (\\ref{diagr_f*_u}) on $Y$ commutes iff its extension by $0$\nto $D_{2}$ commutes, and this is true by \\textbf{2-mfld}.\n\n\\begin{example}\nThe embedding morphism $i:X\\hookrightarrow D$ can be seen as the morphism of\nembedding triples $(i,id_{D})$ and $i^{\\ast }$ is the natural quotient\nmorphism \n\\begin{equation}\n\\Dolb{D}{\\OX{D}}\\rightarrow i_{\\ast }\\Dolb{i\\,}{\\OX{X}}\n\\end{equation}\n\\end{example}\n\nFor two embedings of the same analytic space one checks easily the following\nresult:\n\n\\begin{remark}\n\\label{Rem_qizo_id}Let $f=(id,\\tilde{f}):(X,i_{1},D_{1})\\rightarrow\n(X,i_{2},D_{2})$ be a morphism of embedding triples over the same analytic\nspace. If $\\mathcal{F}\\in Mod(\\OX{X})$ then one checks easily that\nthe natural morphism $f^{\\ast}(id):\\Dolbm{i_{2}\\,}{F}\\rightarrow \\Dolbm{i_{1}\\,}{F}$ is a quasi-isomorphism.\n\\end{remark}\n\n\\begin{remark}\nIn the above setting assume we only have a morphism of analytic spaces \n\\linebreak\n$f:X\\rightarrow Y$ instead of a morphism\\ of embedding triples \n$(f,\\tilde{f}):(X,i_{1},D_{1})\\rightarrow (Y,i_{2},D_{2})$. Then there \nmay not exist a \"direct\" morphism\n$f^{\\ast}(u):\\Dolbm{i_{2}\\,}{G}\\rightarrow f_{\\ast}\\Dolbm{i_{1}\\,}{F}$. \nHowever by Lemma \\ref{Lemma_complet_morph} we have the sequence of mappings:\n\\begin{equation}\n(X,i_{1},D_{1})\\overset{(id,p_{1})}{\\longleftarrow }(X,i_{12},D_{1}\\times\nD_{2})\\overset{(f,p_{2})}{\\longrightarrow }(Y,i_{2},D_{2})\n\\end{equation}%\nand consequently the sequence of morphisms on $Y$:%\n\\begin{equation}\nf_{\\ast}\\Dolbm{i_{1}\\,}{F}\\overset{f^{\\ast}(id)}{\\longrightarrow }\nf_{\\ast}\\Dolbm{i_{12}\\,}{F}\\overset{(f^{\\ast}(u))}{\\longleftarrow}\n\\Dolbm{i_{2}\\,}{G}\n\\label{morph_phi*_deriv}\n\\end{equation}%\nSince the components of $\\Dolb{\\bullet \\,}{\\mathcal{F}}$ are soft\nsheaves, Remark \\ref{Rem_qizo_id} implies that the first morphism above is a\nquasi-isomorphism and hence one has a morphism\n\\linebreak \n$\\Dolbm{i_{2}\\,}{G}\\rightarrow f_{\\ast}\\Dolbm{i_{1}\\,}{F}$ in the derived category \n$D^{b}(\\OX{Y})$, that we also denote by $f^{\\ast}(u)$. Moreover, if $(X,i_{3},D_{3})$ \nis another embedding triple with mappings:\n\\begin{equation}\n(X,i_{1},D_{1})\\overset{(id,q_{1})}{\\longleftarrow }(X,i_{3},D_{3})\\overset{%\n(f,q_{2})}{\\longrightarrow }(Y,i_{2},D_{2})\n\\end{equation}%\nthen Lemma \\ref{Lemma_complet_morph}\\ also implies that the morphism in the\nderived category $D^{b}(\\OX{Y})$: \n\\begin{equation}\nf_{\\ast}\\Dolbm{i_{1}\\,}{F}\\overset{f^{\\ast }(id)}{%\n\\longrightarrow }f_{\\ast}\\Dolbm{i_{3}\\,}{F}\\overset{(f^{\\ast\n}(u))}{\\longleftarrow }\\Dolbm{i_{2}\\,}{G}\n\\end{equation}%\ncoincides with the morphism (\\ref{morph_phi*_deriv}).\n\\end{remark}\n\nTo prove \\textbf{3-emb }in the embedded case extend the sheaves $\\mathcal{F}$%\n, $\\mathcal{G}$, $\\mathcal{H}$ with $0$ to $D_{1}$, $D_{2}$, respectively $%\nD_{3}$. It is easy to see that $i_{1\\ast }\\mathcal{F}$, $i_{2\\ast }\\mathcal{G%\n}$, $i_{3\\ast }\\mathcal{H}$ satisfy the hypothesis of \\textbf{3-mfld}. Thus\nwe get the commutative diagram on $D_{3}$:\n\n\\begin{equation}\n\\begin{tikzcd}[column sep=-0.2cm]\n\\tilde{h}_{\\ast}\\Dolb{D_{1}}{i_{1\\ast}\\mathcal{F}} &&\n\\tilde{g}_{\\ast}\\Dolb{D_{2}}{i_{2\\ast}\\mathcal{G}} \\arrow{ll} \\\\ &\n\\Dolb{D_{3}}{i_{3\\ast}\\mathcal{H}} \\arrow{ul} \\arrow{ur} \\end{tikzcd}\n\\label{diagr_3_i_sus}\n\\end{equation}\nRestricting the above diagram to $Z$ (i.e. applying $i_{3}^{-1}$) one gets\nthe result.\\smallskip \\smallskip\n\n\\refstepcounter{subsection} \\textbf{\\arabic{section}.\\arabic{subsection}} \n\\textbf{The semi-simplicial case. }We extend the functor $\\Dolbn$ to\nthe semi-simplicial context for technical reasons. While a s.s.analytic\nspace is not a particular case of analytic space with an embedding atlas,\nmost of the properties in Theorem \\ref{Theor_Dolb} apply.\n\nLet $\\mathscr{X}=((X_{\\alpha })_{\\alpha \\in \\mathcal{S}}$, $(\\rho _{\\alpha\n\\beta })_{\\alpha \\subset \\beta })$ be a s.s.complex manifold indexed by the\nsimplicial complex $(I, \\mathcal{S})$, and let $\\mathcal{F}\\in Mod(\\mathscr{X})$,  \n$\\mathcal{F}=((\\mathcal{F}_{\\alpha })_{\\alpha \\in \\mathcal{S%\n}}\\ ,(\\varphi _{\\beta \\alpha })_{\\alpha \\subset \\beta })$. We set: \n\\begin{equation}\n\\Dolbss{X}{F}=(\\Dolb{X_{\\alpha }}{\\mathcal{F}_{\\alpha }}_{\\alpha \\in \\mathcal{S}}\n\\ ,(\\rho _{\\alpha \\beta }^{\\ast }(\\varphi _{\\beta \\alpha }))_{\\alpha \\subset \\beta })\n\\end{equation}%\nBy property \\textbf{3-mfld} $\\Dolbss{X}{F}$ is a\ncomplex of $\\mathscr{X}$-modules; furthermore, it is null in degrees $<0$\n\nNote that if $\\mathscr{X}$ is a s.s.complex manifold relative to the\nsimplicial complex $K(pt)$ (i.e. a complex manifold - see Example \\ref%\n{anal_sp_Kpt})\\ the functor $\\Dolb{\\mathscr{X}\\,}{\\bullet}$\n coincides with the functor $\\Dolbn$ from the smooth case.\n\nLet now $(\\mathscr{X},k,\\mathscr{D})$ be a s.s.embedding triple indexed by\nthe simplicial complex $(I, \\mathcal{S})$ (see Remark \\ref{Rem_ss_embed_triple}), \nwhere $\\mathscr{X}=(X_{\\alpha })_{\\alpha \\in \\mathcal{S}}$ is a s.s.analytic space, \n$\\mathscr{D}=(D_{\\alpha })_{\\alpha \\in \\mathcal{S}}$ is a s.s.complex manifold, \nand each component $k_{\\alpha}:X_{\\alpha }\\rightarrow D_{\\alpha }$ of the morphism \n$k:$ $\\mathscr{X} \\rightarrow \\mathscr{D}$ is a closed embedding.\n\nIf $\\mathcal{F}\\in Mod(\\mathscr{X})$, $\\mathcal{F}=(\\mathcal{F}_{\\alpha\n})_{\\alpha \\in \\mathcal{S}}$ we set: \n\\begin{equation}\n\\Dolbm{k\\,}{F}=k^{\\ast }\\Dolb{\\mathscr{D}}{k_{\\sharp}\\mathcal{F}}\n\\end{equation}%\nNote that in this case $k_{\\sharp}=k_{\\ast}$ and for each $\\alpha $, $%\n(k_{\\sharp }\\mathcal{F})_{\\alpha }=k_{\\alpha \\ast }\\mathcal{F}_{\\alpha }$.\nHence, for $\\alpha \\in \\mathcal{S}$%\n\\begin{equation}\n\\Dolb{k\\,}{\\mathcal{F}}_{\\alpha }=\\Dolb{k_{\\alpha }\\,}{\\mathcal{F}_{\\alpha }}\n\\end{equation}%\nThus $\\Dolb{k\\,}{\\bullet }$ is a functor $Mod(\\mathscr{X})\\rightarrow C^{+}(\\mathscr{X})$. \nFurthermore, $\\Dolbm{k\\,}{F}$ is null in degrees $<0$.\n\nIn what follows we shall check the properties of the functor $\\Dolb{k\\,}{\\bullet }$. \nThe treatment for $\\Dolb{\\mathscr{X}\\,}{\\bullet }$, where $\\mathscr{X}$\\ is a s.s.complex manifold, is similar.\n\nProperties \\textbf{1.a-ss}, \\textbf{1.b-ss}, and \\textbf{1.e-ss} follow from\nthe corresponding properties in the embedded case applied for each component \n$\\alpha \\in \\mathcal{S}$. Properties \\textbf{1.c} and \\textbf{1.d} have no\nsense in this context. However, they can be replaced by the following\nstatements that follow immediately from the embedded case:\n\n\\textbf{1.c'-ss} \\textit{The terms of } $\\Dolbm{k\\,}{F}_{\\alpha }=\n\\Dolb{X_{\\alpha }}{\\mathcal{F}_{\\alpha }}$\\textit{\\ are\nsoft sheaves for all }$\\alpha \\in \\mathcal{S}$.\n\n\\textbf{1.d'-ss }\\textit{The terms of } $\\Dolb{k\\,}{\\OXss{X}}_{\\alpha }=\n\\Dolb{X_{\\alpha }}{\\OX{\\alpha }}\\ $%\n\\textit{are }$\\OX{X_{\\alpha }}$\\textit{--flat for all }$\\alpha \\in \\mathcal{S}$\n\n\\textbf{2-ss }Consider the following data:\n\n\\begin{itemize}\n\\item[-] $\\tau :(I,\\mathcal{S})\\rightarrow (J,\\mathcal{T})$ a mapping of\nsimplicial complexes\n\n\\item[-] $F:(\\mathscr{X},k,\\mathscr{D})\\rightarrow (\\mathscr{Y},k^{\\prime },%\n\\mathscr{D}^{\\prime })$ a morphism of s.s.embedding triples over $\\tau $\n\n\\item[-] $\\mathcal{F}\\in Mod(\\mathscr{X})$, $\\mathcal{F}=(\\mathcal{F}%\n_{\\alpha })_{\\alpha \\in \\mathcal{S}}$, $\\mathcal{G}\\in Mod(\\mathscr{Y})$, \n$\\mathcal{G}=(\\mathcal{G}_{\\gamma })_{\\gamma \\in \\mathcal{T}}$, and  \n$u: \\mathcal{G}\\rightarrow F_{\\ast }\\mathcal{F}$ a morphism of $\\mathscr{Y}$%\n-modules\n\\end{itemize}\n\nFor $\\gamma \\in \\mathcal{T}$ and $\\alpha \\in I(\\gamma ,0)$ consider the\nnatural morphism:%\n\\begin{equation}\nu_{\\gamma \\alpha }:\\Dolb{k_{\\gamma }^{\\prime }}{\\mathcal{G}_{\\gamma }}\n\\rightarrow F_{\\alpha \\ast }\\Dolb{k_{\\alpha }}{\\mathcal{F}_{\\alpha }}\n\\end{equation}%\nAccording to property \\textbf{3-emb} one sees that the family of morphisms $%\n(u_{\\gamma \\alpha })_{\\gamma \\alpha }$ satisfies the hypothesis of Lemma \\ref%\n{Lemma_crit_morf_im_dir}. Hence it induces a morphism%\n\\begin{equation}\nF^{\\sharp }(u):\\Dolb{k^{\\prime }\\,}{\\mathcal{G}}\\rightarrow F_{\\sharp }%\n\\Dolbm{k\\,}{F}  \\label{morph_pullback_sharp}\n\\end{equation}\n\n\\textbf{3-ss }One checks directly that on each component $\\gamma \\in \n\\mathcal{T}$ the diagram commutes, which comes down to property \\textbf{3-emb}.\n\\medskip\n\n\\refstepcounter{subsection} \\textbf{\\arabic{section}.\\arabic{subsection}%\n\\label{Paragr_loc_emb_sp}} \\textbf{The case of a locally embedded analytic\nspace. }Let $(X,\\mathcal{A})$ be a locally embedded analytic space, where\n$\\mathcal{A}=(U_{i},k_{i},D_{i})_{i\\in I}$. We fix the following notations\n(see paragraph \\ref{paragr_Atlas}):\n\n\\begin{itemize}\n\\item[-] $\\mathcal{U}=(U_{i})_{i\\in I}$ be the open covering of $X$\\\ncorresponding to the atlas $\\mathcal{A}$\n\n\\item[-] $(\\mathfrak{U},k,\\mathfrak{D})$ be the s.s.embedding triple\nassociated to $(X,\\mathcal{A})$ and $b:\\mathfrak{U}\\rightarrow X$ the\nnatural morphism given by the inclusions (see Lemma \\ref%\n{Lemma_assoc_ss_triples})\n\\end{itemize}\n\nFor $\\mathcal{F}\\in Mod(\\OX{X})$ set:%\n\\begin{equation}\n\\DolbA{A}{F}=b_{\\sharp }\\Dolb{k\\,}{\\mathcal{F}|\\mathfrak{U}}  \\label{Def_Dolb_A}\n\\end{equation}\nThus $\\DolbA{A}{F}$ is the simple complex associated to the double complex:\n\n\\begin{equation}\n0\\rightarrow \\prod\\limits_{|\\alpha |=0}b_{\\alpha \\ast }\\Dolb{k_{\\alpha }\\,}\n{\\mathcal{F}|U_{\\alpha }}\\rightarrow \\prod\\limits_{|\\alpha |=1} b_{\\alpha \n\\ast }\\Dolb{k_{\\alpha }\\,}{\\mathcal{F}|U_{\\alpha}}\\rightarrow ...  \\label{Complex_Dolb_A}\n\\end{equation}%\nwhere $\\alpha \\in \\mathcal{N(U)}$.\n\n\\textbf{1.f }If the atlas $\\mathcal{A}$ consists of only one chart $(X,i,D)$ \n(i.e. $X$ is embedded in the complex manifold $D$) then the functor $\\DolbA{A}{F}$ coincides with the functor $\\Dolbm{i\\,}{F}$ defined at paragraph \\ref{paragr_emb_case} (the embedded case). In particular, if $X$ is a complex manifold and $\\mathcal{A}$ consists of the chart $(X,id,X)$, then $\\Dolb{\\mathcal{A}\\,}{\\OX{X}}$ is the usual \nDolbeault-Grothendieck resolution on $X$.\n\n\\textbf{1.a }Let%\n\\begin{equation*}\n0\\rightarrow \\mathcal{F}_{1}\\rightarrow \\mathcal{F}_{2}\\rightarrow \\mathcal{F%\n}_{3}\\rightarrow 0\n\\end{equation*}%\nbe a short exact sequence of $\\OX{X}$-modules. By properties \n\\textbf{1.a-ss }(exactness of $\\Dolb{k\\,}{\\bullet }$ ) and \\textbf{1.c'-ss }it follows that \n\\begin{equation*}\n0\\rightarrow \\Dolb{k\\,}{\\mathcal{F}_{1}|\\mathfrak{U}} \\rightarrow \n\\Dolb{k\\,}{\\mathcal{F}_{2}|\\mathfrak{U}} \\rightarrow \\Dolb{k\\,}\n{\\mathcal{F}_{3}|\\mathfrak{U}} \\rightarrow 0\n\\end{equation*}%\nis an exact sequence of complexes of $\\mathfrak{U}$-modules such that the\nterms of every $\\Dolb{k\\,}{\\mathcal{F}_{i}}_{\\alpha }$ are soft sheaves. Hence the image by $i_{\\alpha \\ast }$ of the exact sequence on $U_{\\alpha }$ is an exact sequence on $X$, and the exactness of $\\Dolb{\\mathcal{A}\\,}{\\bullet}$ follows by using (\\ref{Complex_Dolb_A}).\n\n\\textbf{1.c }$b_{\\alpha \\ast }(\\Dolb{k\\,}{\\mathcal{F}}_{\\alpha})=\nb_{\\alpha \\ast }(\\Dolb{k_{\\alpha }}{\\mathcal{F}|U_{\\alpha }})$\nhas soft terms. Thus the terms of $\\DolbA{A}{F}$ are cartesian products of soft sheaves and consequently soft.\n\n\\textbf{1.b }The morphism is given by the composition:%\n\\begin{equation}\n\\mathcal{F}\\overset{u}{\\rightarrow }b_{\\sharp }(\\mathcal{F}|\\mathfrak{U})%\n\\overset{v}{\\rightarrow }b_{\\sharp} \\Dolb{k\\,}{\\mathcal{F}|\\mathfrak{U}}=\\DolbA{A}{F} \\label{morph_augment}\n\\end{equation}%\nAs remarked in Example \\ref{Ex_Cech_complex}, $b_{\\sharp }(\\mathcal{F}|%\n\\mathfrak{U})$ coincides with the \\v{C}ech complex of $\\mathcal{F}$ with\nrespect to the open covering $\\mathcal{U}$ and $u$ is a quasi-isomorphism.\n\nTo see that $v$ is also a quasi-isomorphism we restrict ourselves to an open\nset $U_{j}$ of the covering $\\mathcal{U}$. $\\DolbA{A}{F}|U_{j}$ is the simple complex associated to the double complex (see formula (\\ref{Complex_Dolb_A})):%\n\\begin{equation}\nK^{p,q}=\\prod\\limits_{|\\alpha |=p}b_{j\\alpha \\ast } \\Dolbn^{q}(k_{\\alpha }\\, ,\\mathcal{F}|U_{\\alpha })|U_{\\alpha }\\cap U_{j}\n\\end{equation}%\nwhere $b_{j\\alpha }:U_{\\alpha }\\cap U_{j}\\hookrightarrow U_{j}$ is the\nnatural inclusion.\n\nThe terms of the second drawer of the spectral sequence associated to \n$K^{\\cdot \\cdot }$ are:%\n\\begin{equation}\nE_{2}^{p,q}=\\left\\{ \n\\begin{array}{cc}\n\\mathcal{F}|U_{j} & \\text{if }p=q=0 \\\\ \n0 & \\text{otherwise}%\n\\end{array}%\n\\right.  \\label{K_spect_seq}\n\\end{equation}%\nIndeed, taking the cohomology of $K^{\\cdot \\cdot }$ in the $q$-direction,\none obtains for $q\\geq 0$, the following \\v{C}ech-type complex relative to\nthe covering $\\mathcal{U}\\cap U_{j}:$%\n\\begin{equation}\n\\cdots \\rightarrow \\prod\\limits_{|\\alpha |=p}R^{q}b_{j\\alpha \\ast }(\\mathcal{%\nF}|U_{\\alpha }\\cap U_{j})\\rightarrow \\prod\\limits_{|\\alpha\n|=p+1}R^{q}b_{j\\alpha \\ast }(\\mathcal{F}|U_{\\alpha }\\cap U_{j})\n\\rightarrow \\cdots   \\tag{C(q)}  \\label{Complex_R(q)}\n\\end{equation}%\nNote that $U_{j}$ is among the open sets of the covering $\\mathcal{U}\\cap %\nU_{j}$ and that $b_{jj}=id_{U_{j}}$. Using a homotopy argument similar to\nthat in \\cite{FAC}, chap 1, \\S 3 Proposition 3 and \\S 4, Lemma 1 one checks\nthat the cohomology of $C(q)$:%\n\\begin{equation*}\nH^{p}(C(q))=\\left\\{ \n\\begin{array}{cc}\nR^{q}b_{jj\\ast }(\\mathcal{F}|U_{j})=R^{q}id_{\\ast }(\\mathcal{F}|U_{j}) & \n\\text{if }p=0 \\\\ \n0 & \\text{otherwise}%\n\\end{array}%\n\\right.\n\\end{equation*}%\nwhich proves the claim. Moreover, one also checks that $v$ induces an\nisomorphism between the second drawers of the spectral sequences associated\nto $\\mathcal{C}^{\\bullet }(\\mathcal{U},\\mathcal{F})|U_{j}$ and $K^{\\cdot\n\\cdot }$ and, consequently, is a quasi-isomorphism.\n\n\\bigskip \\textbf{1.e} and \\textbf{1.d }The morphism (\\ref{ident_Dolb_A}) is\ninduced by the natural morphisms:%\n\\begin{equation}\n\\prod\\limits_{|\\alpha |=p}b_{\\alpha \\ast }\\Dolb{k_{\\alpha }\\,}\n{\\OX{X}|U_{\\alpha }}\\otimes _{\\OX{X}}\\mathcal{F}\\rightarrow\n\\prod\\limits_{|\\alpha |=p}b_{\\alpha \\ast }\\Dolb{k_{\\alpha }\\,}\n{\\mathcal{F}|U_{\\alpha }}  \\label{morph_comut_prod_cartez}\n\\end{equation}%\nIf $\\mathcal{F}$ is a coherent $\\OX{X}$-module then (\\ref%\n{morph_comut_prod_cartez}) is an isomorphism. Indeed if $\\mathcal{F}=\\OX{X}$\nor $\\mathcal{F}=\\OX{X}^{p}$ then the statement is clear. Hence, using local\npresentations of $\\mathcal{F}$%\n\\begin{equation*}\n\\OX{X}^{p}\\rightarrow \\OX{X}^{q}\\rightarrow \\mathcal{F} \\rightarrow 0\n\\end{equation*}%\nit follows that (\\ref{morph_comut_prod_cartez}) is a local isomorphism, and\nconsequently an isomorphism. Note that if $\\mathcal{F}$ is not coherent then %\n(\\ref{morph_comut_prod_cartez}) may not be an isomorphism.\n\nTo prove the flatness of $\\Dolb{\\mathcal{A}\\,}{\\OX{X}}$ it is enough to check \nthat the following sequence:%\n\\begin{equation}\n0\\rightarrow \\Dolb{\\mathcal{A}\\,}{\\OX{X}} \\otimes _{\\OX{X}} \\mathcal{I}\n\\rightarrow \\Dolb{\\mathcal{A}\\,}{\\OX{X}}\n\\end{equation}%\nis exact, where $\\mathcal{I}\\subset \\OX{X}|U$ is any coherent ideal on some \nopen set $U\\subset X$. This is implied by the following commutative diagram:%\n\\begin{equation}\n\\begin{CD} @. \\Dolb{\\mathcal{A}\\,}{\\OX{X}}\\otimes _{\\OX{X}} \\mathcal{I} @>>>\n\\Dolb{\\mathcal{A}\\,}{\\OX{X}}\\\\ @. @VV\\wr V @|\\\\ 0 @>>>\n\\Dolb{\\mathcal{A}\\,}{\\mathcal{I}} @>>>\n\\Dolb{\\mathcal{A}\\,}{\\OX{X}} \n\\end{CD}\n\\end{equation}%\nsince the lower row is exact (exactness of $\\Dolb{\\mathcal{A}\\,}{\\bullet }$ ) and \nthe left hand side vertical arrow is an isomorphism by property \\textbf{1.e} in the coherent case.\n\nFinally, to prove that the morphism (\\ref{ident_Dolb_A}) is a\nquasi-isomorphism consider the commutative diagram:\n\\begin{equation}\n\\begin{CD}\n\\Dolb{\\mathcal{A}\\,}{\\OX{X}\\otimes_{\\OX{X}}%\n\\mathcal{F}} @>>> \\DolbA{A}{F}\\\\ @AAA @AAA\\\\\n\\OX{X}\\otimes_{\\OX{X}}\\mathcal{F} @>>> \\mathcal{F} \\end{CD}\n\\medskip\n\\end{equation} \nBy the $\\OX{X}$-flatness of $\\Dolb{\\mathcal{A}\\,}{\\OX{X}})$ and \\textbf{1.c, }\nthe vertical arrows are quasi-isomorphisms, and the lower horizontal arrow is an isomorphism\nwhich yields the result.\n\n\\textbf{2 } Let $(\\mathfrak{U},k,\\mathfrak{D})$, $(\\mathfrak{V},k^{\\prime },%\n\\mathfrak{D}^{\\prime })$ be the s.s.embedding triples associated to $(X,%\n\\mathcal{A})$, $(Y,\\mathcal{B})$ and $b:\\mathfrak{U}\\rightarrow X$, \n$b^{\\prime }:\\mathfrak{V}\\rightarrow Y$ the morphisms given by the\ninclusions. Let%\n\\begin{equation}\nF:(\\mathfrak{U},k,\\mathfrak{D})\\rightarrow (\\mathfrak{V},k^{\\prime },%\n\\mathfrak{D}^{\\prime })\n\\end{equation}%\nbe the morphism induced by $f:(X,\\mathcal{A})\\rightarrow (Y,\\mathcal{B})$\n(see Remark \\ref{Lemma_lifting_morph} \\textbf{2}) and \n\\begin{equation}\nF^{\\ast }(u):\\mathcal{G}|\\mathfrak{V} \\rightarrow F_{\\ast }\n(\\mathcal{F}|\\mathfrak{U})\n\\end{equation}%\nbe the morphism induced by $u:\\mathcal{G}\\rightarrow f_{\\ast }\\mathcal{F}$\n(see Lemma \\ref{Lemma_lifting_morph}. \\textbf{1}). By property \\textbf{2-ss}\nthere exists a natural morphism:%\n\\begin{equation}\nF^{\\sharp }(F^{\\ast }(u)):\\Dolb{k^{\\prime}\\,}{\\mathcal{G}|{\\mathfrak{V}}} \\rightarrow F_{\\sharp }\\Dolb{k\\,}{\\mathcal{F}|\\mathfrak{U}}\n\\end{equation}%\nNow apply $b_{\\sharp }^{\\prime }$ and use Lemma \\ref{Lemma_Comp_im_dir} to\nobtain the morphism:%\n\\begin{equation}\nf^{\\ast }(u):\\DolbA{B}{G} \\rightarrow f_{\\ast }\\DolbA{A}{F}\n\\end{equation}\n\n\\begin{remark}\n\\label{Rem_q_izo_over_id}Assume $Y=X$, $f=(id,\\tau ,(\\tilde{f}_{i})_{i\\in\nI}):(X,\\mathcal{A})\\rightarrow (X,\\mathcal{B})$, $\\mathcal{G}=\\mathcal{F}$, \nand $u=id$. Then the morphism defined above%\n\\begin{equation}\nf^{\\ast }(id):\\DolbA{B}{F} \\rightarrow \\DolbA{A}{F}\n\\end{equation}%\nis a quasi-isomorphism, since it is a morphism between two resolutions of \n$\\mathcal{F}$.\n\\end{remark}\n\n\\textbf{3 } Let $(\\mathfrak{U},k,\\mathfrak{D}),$ $(\\mathfrak{V},k^{\\prime },%\n\\mathfrak{D}^{\\prime }),$ $(\\mathfrak{W},k^{^{\\prime \\prime }},\\mathfrak{D}%\n^{^{\\prime \\prime }})$ be the s.s.embedding triples associated to $(X,%\n\\mathcal{A})$, $(Y,\\mathcal{B})$, and $(Z,\\mathcal{C})$ and $b:\\mathfrak{U}%\n\\rightarrow X$, $b^{\\prime }:\\mathfrak{V}\\rightarrow Y$, $b^{\\prime \\prime }:%\n\\mathfrak{W}\\rightarrow Z$ the morphisms given by the inclusions. By Lemma %\n\\ref{Lemma_lifting_morph} \\textbf{2 \\ }$\\mathcal{F}|\\mathfrak{U}$, $\\mathcal{%\nG}|\\mathfrak{V}$, $\\mathcal{H}|\\mathfrak{W}$ satisfy the hypothesis of\nproperty \\textbf{3-ss }and hence the following diagram commutes\n\\begin{equation}\n\\begin{tikzcd}[column sep=-0.2cm]\nH_{\\sharp}\\Dolb{k\\,}{\\mathcal{F}|\\mathfrak{U}} &&\nG_{\\sharp}\\Dolb{k^{\\prime}\\,}{\\mathcal{G}|\\mathfrak{V}} \\arrow{ll} \\\\\n& \\Dolb{k^{\\prime\\prime}\\,}{\\mathcal{H}|\\mathfrak{W}} \\arrow{ul}\n\\arrow{ur} \\end{tikzcd}\n\\end{equation}\nTo get the result apply $b_{\\sharp }^{\\prime \\prime }$ to the above diagram\nand use Lemma \\ref{Lemma_Comp_im_dir}.\n\n\\begin{remark}\nBy properties \\textbf{2} and \\textbf{3} of Theorem \\ref{Theor_Dolb} the\nfunctor $\\Dolbn$ extends to the category of s.s.locally embedded\nanalytic spaces, with the pullback morphisms defined using $\\sharp $-direct\nimages (similar to morphism (\\ref{morph_pullback_sharp})). In particular, with\nthe notations at the beginning of paragraph \\ref{Paragr_loc_emb_sp}, the\npullback morphism over the natural mapping $b:\\mathfrak{U} \\rightarrow X$,%\n\\begin{equation*}\nb^{\\sharp }:\\DolbA{A}{F} \\rightarrow b_{\\sharp }%\n\\Dolb{k\\,}{\\mathcal{F}|\\mathfrak{U}}\n\\end{equation*}%\nis the identity of $\\DolbA{A}{F}$.\n\\end{remark}\n                     \n\n\\section{Further Results and Applications}\n\n\\refstepcounter{subsection} \\textbf{\\arabic{section}.%\n\\arabic{subsection} }\\label{paragr_uniqness_deriv_categ}\\textbf{The functor }%\n$\\Dolbn$ \\textbf{in the derived category}. To simplify notation in\nwhat follows we shall omit to write the localization functors such as \n\\begin{equation*}\nQ:K(\\OX{X})\\rightarrow D(\\OX{X})\n\\end{equation*}%\nIt should be clear from the context to which category each complex belongs.\n\n\\begin{corollary}\n\\label{Corol_q_inv}Let $(X,\\mathcal{A})$ be a locally embedded analytic\nspace and let \\linebreak $\\mathcal{F}\\in Mod(\\OX{X})$.\n\n\\begin{enumerate}\n\\item The natural functor $\\mathcal{F}\\rightarrow \\DolbA{A}{F}$ gives a functorial isomorphism in the derived category $D^{+}(\\OX{X})$\n\n\\item Let $S(X)$ be the full subcategory of $D^{+}(\\OX{X})$ consisting of complexes with soft terms and \n\\begin{equation*}\nj:S(X)\\rightarrow D^{+}(\\OX{X})\n\\end{equation*}%\nthe inclusion functor. Then the extension of $\\Dolb{\\mathcal{A}\\,}\n{\\bullet }$ to $D^{+}(\\OX{X})$ is a quasi-inverse for $j$.\n\\end{enumerate}\n\\end{corollary}\n\n\\begin{proof}\n\\textbf{1}. follows from Theorem \\ref{Theor_Dolb} \\textbf{1.b.} while \n\\textbf{2. }folows from \\textbf{1. }and Theorem \\ref{Theor_Dolb} \\textbf{1.c.}\n\\end{proof}\n\n\\begin{remark}\nCorollary \\ref{Corol_q_inv} implies in particular that using $\\Dolb{\\mathcal{A}\\,}{\\bullet }$ one defines derived functors for any\nfunctor $F:Mod(\\OX{X})\\rightarrow Mod(\\OX{X})$ s.t. soft sheaves are $F$-acyclic.\n\\end{remark}\n\nOne checks immediately:\n\n\\begin{corollary}\nLet $(X,\\mathcal{A})$ be a locally embedded analytic space and \\linebreak $%\n\\mathcal{F}\\in Mod(\\OX{X})$. Then the complex $\\Gamma (X,\\DolbA{A}{F})$ is a representative of $R\\Gamma (X,\\mathcal{F})$, and hence it computes the cohomology groups $H^{\\bullet }(X,\\mathcal{F})$.\n\\end{corollary}\n\n\\begin{corollary}\n\\label{Corol_topol_FS}Let $(X,\\mathcal{A})$ be a locally embedded analytic\nspace and \\linebreak $\\mathcal{F}\\in Coh(\\OX{X})$. Assume that $X$\nis countable at infinity and that $\\mathcal{A}$\\ has at most countably many\ncharts. Then the terms of the complex $\\Gamma (X,\\DolbA{A}{F})$ have natural topologies of type FS and the differentials are continuous. Furthermore, the terms of $\\Gamma (X,\\DolbA{A}{F})$ induce the natural topology on the cohomology groups $H^{\\bullet }(X,\\mathcal{F})$.\n\\end{corollary}\n\n\\begin{proof}\nIt is well known that $\\Gamma (X,\\mathcal{E}_{X}^{p,q}\\otimes \\mathcal{F})$\nhas a natural topology of type FS (see e.g. \\cite{B-S} 7\\S 4.b). Thus the global \nsections of the terms in (\\ref{Complex_Dolb_A}) are countable products of FS spaces and hence are themselves FS. Note that if $\\mathcal{B}$ is another embedding atlas of $X$ then, by Remark \\ref{Rem_q_izo_over_id}, $\\Gamma (X,\\DolbA{A}{F})$ and $\\Gamma(X,\\DolbA{B}{F})$ induce the same topology on $H^{\\bullet }(X,\\mathcal{F})$\n(since $f^{\\ast }(id)$ determines a continuous quasi-isomorphism). Thus, to\ncheck that this topology coincides with the natural one, we can assume that $%\ncov(\\mathcal{A})$ is a Stein covering. The morphism $v$ in diagram (\\ref%\n{morph_augment}) determines a continuous quasi-isomorphism on the global\nsections:%\n\\begin{equation*}\nC^{\\bullet }(cov(\\mathcal{A}),\\mathcal{F})\\rightarrow \\Gamma (X,\\DolbA{A}{F})\n\\end{equation*}%\nwhich ends the proof, since the left-hand side complex (the \\v{C}ech complex\nwith respect to $cov(\\mathcal{A})$ ) defines the natural topology on \n$H^{\\bullet }(X,\\mathcal{F})$.\n\\end{proof}\n\n\\begin{remark}\nBy \\textit{\\cite{Colt}} and \\cite{Colt-Joi} if $X$ is a finite dimensional\nanalytic space countable at infinity then it can be covered by finitely many\nStein open sets; if moreover $X$ is connected then the Stein open sets can\nalso be chosen connected. Hence if $X$ has also finite embedding dimension\nthen it\\ has an embedding atlas $\\mathcal{A}$ with finitely many charts,\nrespectively finitely many connected charts, and for any $\\mathcal{F}\\in Mod(\\OX{X})$ the terms of the complex $\\DolbA{A}{F}$ consist of products with finitely many factors (see (\\ref {Complex_Dolb_A})).\n\\end{remark}\n\n\\begin{theorem}\n\\label{Theor_Dolb_cat_deriv}Let $f:X\\rightarrow Y$ be a morphism of analytic\nspaces, $\\mathcal{F}\\in Mod(\\OX{X})$,\n\\linebreak\n$\\mathcal{G}\\in Mod(\\OX{Y})$ and \n$u:\\mathcal{G}\\rightarrow f_{\\ast }\\mathcal{F}$ a morphism of $\\OX{Y}$-modules. \nLet moreover $\\mathcal{A}=(U_{i},k_{i},D_{i})_{i\\in I}$ and \n$\\mathcal{B}=(V_{j},k_{j}^{\\prime },D_{j}^{\\prime })_{j\\in J}$ be embedding \natlases of $X$, respectively $Y$. Then\n\n\\begin{enumerate}\n\\item $f_{\\ast }\\DolbA{A}{F}$ is a representative for $Rf_{\\ast }\\mathcal{F}$\n\n\\item There exists a unique morphism $f^{\\ast }(u)$ in $D^{+}(\\OX{Y})$ such that the following diagram commutes (in $D^{+}(\\OX{Y})$):\n\\begin{equation}\n\\begin{CD} \n\\DolbA{B}{G} @>f^{\\ast }(u)>> f_{\\ast }\\DolbA{A}{F} \\\\ \n@AAb^{\\prime}A @AAf_{*}bA\\\\\n\\mathcal{G} @>{u}>> f_{\\ast }\\mathcal{F} \n\\end{CD}  \\label{diagr_f*_u_deriv}\n\\end{equation}\n\n\\item The morphism $f^{\\ast }(u)$ can be represented as a sequence of\npullback morphisms\n\n\\item Let $g:Y \\rightarrow Z$ be another morphism of analytic spaces and $%\nh=g\\circ f$. Let moreover $\\mathcal{H}\\in Mod(\\OX{Z})$ and $v:%\n\\mathcal{H} \\rightarrow g_{\\ast }\\mathcal{G}$, $w:\\mathcal{H} \\rightarrow %\nh_{\\ast }\\mathcal{F}$ morphisms of $\\OX{Z}$-modules, such that $%\ng_{\\ast }(u)\\circ v=w$. If $\\mathcal{C}$\\ is an embedding atlas of $Z$ then,\nin the derived category $D^{+}(\\OX{Z})$, one has the commutative\ndiagram: \n\\begin{equation}\n\\begin{tikzcd}[column sep=-0.2cm] h_{\\ast}\\DolbA{A}{F} && g_{\\ast}\\DolbA{B}{G}\n\\arrow{ll}[swap]{Rg_{\\ast}(f^{\\ast}(u)} \\\\ & \\DolbA{C}{H} \\arrow{ul}{h^{\\ast}(w)} \\arrow{ur}[swap]{g^{\\ast}(v)}\n\\end{tikzcd}\n\\end{equation}\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\n\\textbf{1.} and \\textbf{2.} are obvious.\n\n\\textbf{3. }Assume first that the embedding atlases $\\mathcal{A}$, $\\mathcal{%\nB}$ are $f$-compliant. If $\\tau $ is a refinement mapping, consider the\ndiagram of locally embedded analytic spaces (see Lemma \\ref%\n{Lemma_complet_morph_atlas} ): \n\\begin{equation}\n(X,\\mathcal{A})\\overset{P_{1}}{\\longleftarrow }(X,\\mathcal{A}\\times _{\\tau }%\n\\mathcal{B})\\overset{P_{2}}{\\longrightarrow }(Y,\\mathcal{B})\n\\label{Diagr_tau}\n\\end{equation}%\nand let \n\\begin{equation}\nf_{\\ast }\\DolbA{A}{F}\\overset{f_{\\ast }P_{1}^{\\ast }(id)}\n{\\longrightarrow }f_{\\ast }\\Dolb{\\mathcal{A}\\times _{\\tau }\\mathcal{B}\\,}{\\mathcal{F}}\\overset{P_{2}^{\\ast }(u)}{\\longleftarrow }\\DolbA{B}{G}\n\\label{Diagr_Def_f*_deriv}\n\\end{equation}%\nbe the corresponding $\\Dolbn$-diagram (i.e the diagram of pullback\nmorphisms over the arrows in diagram (\\ref{Diagr_tau})),\\ where \n\\begin{equation*}\nP_{1}=(id,id,p_{1})\\text{ and }P_{2}=(f,\\tau ,p_{2})\n\\end{equation*}%\nNote that since the components of the $\\Dolbn $-resolutions are soft\nsheaves, Remark \\ref{Rem_q_izo_over_id}\\ implies that $f_{\\ast }P_{1}^{\\ast\n}(id)$ is a quasi-isomorphism, and so diagram (\\ref{Diagr_Def_f*_deriv})\ngives a morphism in $D^{+}(\\OX{Y})$ which coincides with $f^{\\ast\n}(u)$ (to see this use diagrams similar to (\\ref{diagr_f*_u_deriv}) for the\nmorphisms in diagram (\\ref{Diagr_Def_f*_deriv}))\n\n\\begin{remark}\nThe morphism given by diagram (\\ref{Diagr_Def_f*_deriv}) does not depend on\nthe refinement mapping $\\tau $(use, for instance, the $\\Dolbn $-diagram over diagram \n(\\ref{Diagr_tau12}) in Lemma \\ref{Lemma_complet_morph_atlas}). Moreover let \n$\\mathcal{A}^{\\prime }$ be an embedding atlas on $X$ such that one has the diagram of \nlocally embedded analytic spaces:%\n\\begin{equation*}\n(X,\\mathcal{A})\\overset{(id,\\upsilon ,q_{1})}{\\longleftarrow }(X,\\mathcal{A}%\n^{\\prime })\\overset{(f,\\tau \\circ \\upsilon ,q_{2})}{\\longrightarrow }(Y,%\n\\mathcal{B})\n\\end{equation*}%\nThen the corresponding $\\Dolbn $-diagram is also a representative for \n$f^{\\ast }(u)$ (use, for instance, the $\\Dolbn $-diagram over\ndiagram (\\ref{Diagr_A_prim}) in Lemma \\ref{Lemma_complet_morph_atlas})\\ \n\\end{remark}\n\n\\begin{remark}\n\\label{Rem_izo_cat_deriv}Assume that $Y=X$, $f=id_{X}$, $\\mathcal{G}=%\n\\mathcal{F}$, and $u=id$. Remark\\nolinebreak\\ \\ref{Rem_q_izo_over_id}\nimplies $f^{\\ast }(u)$ is an isomorphism in the derived category.\n\\end{remark}\n\nNow drop the suplimentary assumption. Let $\\mathcal{A}^{\\prime }$ be an\nembedding atlas of $X$ s.t. $\\mathcal{A}^{\\prime }$, $\\mathcal{A}$ are $%\nid_{X}$-compliant and $\\mathcal{A}^{\\prime }$, $\\mathcal{B}$ are $f$%\n-compliant (choose for instance an embedding atlas over the open covering \n$cov(\\mathcal{A})\\cap f^{-1}cov(\\mathcal{B}))$. Then the $\\Dolbn $%\n-diagram in $D^{+}(\\OX{Y})$\n\\begin{equation*}\nf_{\\ast }\\DolbA{A}{F} \\overset{f_{\\ast }id^{\\ast }(id)}\n{\\longrightarrow }f_{\\ast }\\Dolb{\\mathcal{A}^{\\prime }}\n{\\mathcal{F}}\\overset{f^{\\ast }(u)}{\\longleftarrow }\\DolbA{B}{G}\n\\end{equation*}%\ndetermines a morphism:%\n\\begin{equation*}\n\\DolbA{B}{G} \\rightarrow f_{\\ast }\\DolbA{A}{F}\n\\end{equation*}%\nsince, by Remark \\ref{Rem_izo_cat_deriv}, the left-hand arrow is an\nisomorphism. Moreover one checks, as in the f-compliant case, that this\nmorphism coincides with $f^{\\ast }(u)$.\n\n\\textbf{4. }follows from the equality $g_{\\ast }(u)\\circ v=w$ by considering\nthe diagrams similar to diagram (\\ref{diagr_f*_u_deriv}) for each morphism.\nAlternatively, in the compliant case (i.e. if $\\mathcal{A}$, $\\mathcal{B}$\nare $f$-compliant and $\\mathcal{B}$, $\\mathcal{C}$ are $g$-compliant) the\nclaim follows from the $\\mathcal{D}olb$-diagram over diagram (\\ref%\n{Diagr_comut_morph_compoz})\\ in Remark \\ref{Rem_comut_morph_compoz}. The\ngeneral case reduces to the compliant one via isomorphisms.\n\\end{proof}\n\n\\refstepcounter{subsection}\\textbf{\\ \\arabic{section}.\\arabic{subsection} }%\n\\label{paragr_Reduced_space} \\textbf{Dolbeault resolutions on reduced analytic \nspaces.} Let $X$ be a reduced analytic space. Using an\nembedding atlas of a particular form one can construct a\nDolbeault-Grothendieck type resolution which coincides with the usual one on\nthe regular part of $X$. For this let $\\mathcal{F}\\in Mod(\\OX{X})$,\nlet $(W_{j})_{j}$ be a neighbourhood basis of $Sing(X)$, and $%\n(U_{i},k_{i},D_{i})_{i\\in I}$ a family of embedding triples s.t. $%\n(U_{i})_{i\\in I}$ are open sets of $X$ which cover $Sing(X)$. Denote \n\\begin{equation*}\nS=Sing(X)\\text{ and }U_{0}=Reg(X)\n\\end{equation*}%\nThe following Lemma is obvious:\n\n\\begin{lemma}\n\\begin{enumerate}\n\\item The family of charts $(U_{i},k_{i},D_{i})_{i\\in I}$ together with the\nembedding triple $(U_{0},id,U_{0})$ give an embedding atlas $\\mathcal{A}$ of \n$X$.\n\n\\item For each $j\\in J$, the family of embedding triples $(U_{i}\\cap\nW_{j},k_{i}|W_{j},D_{i}^{\\prime })_{i\\in I}$, where $D_{i}^{\\prime }\\subset\nD_{i}$ is a suitable open subset, together with the embedding triple $%\n(U_{0},id,U_{0})$ give an embedding atlas $\\mathcal{A}^{(j)}$ of $X$.\n\n\\item $\\Dolb{\\mathcal{A}^{(j)}}{\\mathcal{F}}$ does not depend on the\nchoice of the open subsets $D_{i}^{\\prime }$.\n\n\\item If $W_{j_{1}}\\subset W_{j_{2}}$ then there is a natural pullback\nmorphism \\linebreak $i_{j_{1}j_{2}}^{\\ast }(id):\\Dolb{\\mathcal{A}%\n^{(j_{2})}}{\\mathcal{F}}\\rightarrow \\Dolb{\\mathcal{A}^{(j_{1})}}{\\mathcal{F}}$\nover the identity of $X$, and $(\\Dolb{\\mathcal{A}^{(j)}}{\\mathcal{F}}%\n)_{j}$ is an inductive system of complexes of $\\OX{X}$-modules.\n\\end{enumerate}\n\\end{lemma}\n\nWe set:%\n\\begin{equation}\nr\\DolbA{A}{F} = \\lim\\limits_{\\overrightarrow{j}}%\n\\Dolb{\\mathcal{A}^{(j)}}{\\mathcal{F}}  \\label{Def_Dolb_redus}\n\\end{equation}%\nIt is easy to see that the definition of $r\\DolbA{A}{F}$ is independent of the neighbourhood basis $(W_{j})_{j}$.\n\n\\begin{corollary}\n$r\\Dolb{\\mathcal{A}\\,}{\\bullet }$ is a functor $Mod(\\OX{X})\\rightarrow C^{+}(X)$. \nMoreover properties \n\\linebreak\n\\textbf{1.a} - \\textbf{1.e} in Theorem \\ref{Theor_Dolb} hold \nfor $\\Dolb{\\mathcal{A}\\,}{\\bullet }$ replaced by  $r\\Dolb{\\mathcal{A}\\,}{\\bullet }$.\n\\end{corollary}\n\n\\begin{proof}\n\\textbf{1.a}, \\textbf{1.b}, \\textbf{1.d}, \\textbf{1.e} follow immediately\nfrom the respective properties in Theorem \\ref{Theor_Dolb} because of the\ncompatibility with inductive limits. For \\textbf{1.c }note that the terms of \n$r\\DolbA{A}{F}$ consist of Godement restrictions to the closed set $S$ of soft sheaves, and consequently are also soft.\n\\end{proof}\n\n\\begin{corollary}\n$r\\Dolb{\\mathcal{A}\\,}{\\OX{X}}|U_{0}$ coincides with the Dolbeault-Grothendieck resolution on the manifold $U_{0}$.\n\\end{corollary}\n\n\\begin{remark}\nAssume $\\mathcal{F}\\in Coh(\\OX{X})$. Then the topologies on the\nglobal sections spaces $\\Gamma (X,r\\Dolb{\\mathcal{A}\\,}{\\OX{X}})$ are more complicated than the FS topologies of Corollary \\ref{Corol_topol_FS}. However, since the natural quasi-isomorphism \n\\begin{equation}\n\\Gamma (X,\\DolbA{A}{F})\\rightarrow \\Gamma (X,r\\DolbA{A}{F})\n\\end{equation}%\nis continuous, both complexes induce the same topology on the cohomology\ngroups $H^{\\bullet }(X,\\mathcal{F})$.\n\\end{remark}\n\n\\begin{remark}\nIf $f:(X,\\mathcal{A})\\rightarrow (Y,\\mathcal{B})$ is a morphism of locally embedded\nanalytic spaces, $\\mathcal{G}\\in Mod(\\OX{Y})$ and $u:\\mathcal{%\nG}\\rightarrow f_{\\ast }\\mathcal{F}$ a morphism of $\\OX{Y}$-modules\nthen one has a pullback morphism \n\\begin{equation}\nf^{\\ast }(u):\\DolbA{B}{G}\\rightarrow f_{\\ast }r\\DolbA{A}{F}\n\\end{equation}%\nbut, in general, not a morphism \n\\begin{equation}\nr\\DolbA{B}{G} \\rightarrow f_{\\ast }r\\DolbA{A}{F}\n\\end{equation}%\nwhen $Y$ is also a reduced analytic space.\n\\end{remark}\n\n\\refstepcounter{subsection} \\textbf{\\arabic{section}.\\arabic{subsection}} %\n\\label{paragr_deRham} \\textbf{The functor }$\\Dolbn $\\textbf{\\ and the\nde Rham complex on analytic spaces}. Let $X$ be an analytic space and%\n\\begin{equation}\n0\\longrightarrow \\Omega_{X}^{0}\\overset{{\\partial }_{X}^{0i}}{%\n\\longrightarrow }\\Omega_{X}^{1}\\overset{{\\partial }_{X}^{1}}{%\n\\longrightarrow }...\n\\end{equation}%\nbe the de Rham complex on $X$ (see e.g. H.Grauert, H.Kerner \\cite{Grauert}\\\nor A.Grothendieck \\cite{Groth}). Recall that if $k:X\\hookrightarrow D$ is a\nclosed embedding of $X$ in the complex manifold $D$, and $\\mathcal{I}%\n_{X}\\subset \\OX{D}$ is the coherent ideal sheaf which gives $X$ as\na subspace of $D$, then \n\\begin{equation}\n\\Omega _{X}^{i}=\\Omega _{D}^{i}\/\\mathcal{N}_{X}^{i}\n\\end{equation}%\nwhere $\\mathcal{N}_{X}^{i}$ is the $\\OX{D}$-submodule of $\\Omega\n_{D}^{i}$ generated by $\\mathcal{I}_{X}\\Omega _{X}^{i}$ and ${%\n\\partial }_{D}^{i-1}(\\mathcal{I}_{X}\\Omega _{X}^{i-1})$, and the\ndifferentials \n\\begin{equation}\n\\partial _{X}^{i}:\\Omega _{X}^{i}\\rightarrow \\Omega _{X}^{i+1}\n\\end{equation}%\nare induced by those of $\\Omega _{D}^{\\bullet }$. To simplify notation in\nwhat follows we shall write $\\partial $ instead of ${%\n\\partial }_{X}^{i}$ if $i$ and $X$ are clear from the context.\n\nIf $f:X\\rightarrow Y$ is a morphism of analytic spaces, one has a pullback\nmorphism: \n\\begin{equation}\nf^{\\ast }:\\Omega _{Y}^{\\bullet }\\rightarrow f_{\\ast }\\Omega _{X}^{\\bullet }\n\\label{Morph_pullback_an_sp}\n\\end{equation}%\nIn particular, if $f:(X,i_{1},D_{1})\\rightarrow (Y,i_{2},D_{2})$ is a\nmorphism of embedding triples then the morphism (\\ref{Morph_pullback_an_sp})\nis induced by the usual pullback morphism \n\\begin{equation}\n\\Omega _{D_{2}}^{\\bullet }\\rightarrow f_{\\ast }\\Omega _{D_{1}}^{\\bullet }\n\\end{equation}%\nsince one checks that $f^{\\ast }(\\mathcal{N}_{Y}^{i})\\subset \\mathcal{N}%\n_{X}^{i}$ for all $i$.\n\n\\begin{remark}\n\\label{Rem_ss_de_Rham}If $\\mathscr{X}=(X_{\\alpha })_{\\alpha }$ is a\ns.s.analytic space then the functoriality of the pullback morphisms (\\ref%\n{Morph_pullback_an_sp}) implies that $\\Omega _{\\mathscr{X}}^{\\bullet\n}=(\\Omega _{X_{\\alpha }}^{\\bullet },\\partial )_{\\alpha }$ is a complex of $%\n\\mathscr{X}$-modules with $\\mathbb{C}$-linear differentials.\n\\end{remark}\n\n\\begin{theorem}\n\\label{Theor_Dolb_deRham}Let $(X,\\mathcal{A})$ be a locally embedded\nanalytic space.\n\n\\begin{enumerate}\n\\item The differential ${\\partial }_{X}^{i}:\\Omega\n_{X}^{i}\\rightarrow \\Omega _{X}^{i+1}$ induces a $\\mathbb{C}$-linear\nmorphism of resolutions:%\n\\begin{equation}\n{\\partial }^{i}:\\Dolb{\\mathcal{A}\\,}{\\Omega_{X}^{i}}\\rightarrow \\Dolb{\\mathcal{A}\\,}{\\Omega _{X}^{i+1}}\n\\label{Morpf_Dolb_de_Rham_A}\n\\end{equation}%\nsuch that $\\Dolb{\\mathcal{A}\\,}{\\Omega _{X}^{\\bullet }}$ is a double\ncomplex\n\n\\item The simple complex associated to $\\Dolb{\\mathcal{A}\\,}{\\Omega\n_{X}^{\\bullet }}$ is a resolution of $\\Omega _{X}^{\\bullet }$ with soft, \n$\\OX{X}$-flat sheaves\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\n\\textbf{1. }The morphism of resolutions is obtained by following the\nconstruction of the functor $\\Dolb{\\mathcal{A}\\,}{\\bullet }$ in\nSection \\ref{Sect_Resol}.\n\n\\textbf{a. (Smooth case)} Let $D$ be a complex manifold and consider the\nmorphism of resolutions: \n\\begin{equation}\n\\begin{tikzcd}[column sep=1.5cm] 0 \\arrow{r} & \\Omega^{i+1}_{D} \\arrow{r} &\n\\mathcal{E}_{D}^{i+1,0} \\arrow{r} & \\mathcal{E}_{D}^{i+1,1} \\arrow{r} &\n\\ldots \\\\ 0 \\arrow{r} & \\Omega^{i}_{D} \\arrow{r} \\arrow{u}{\\partial } &\n\\mathcal{E}_{D}^{i,0} \\arrow{r} \\arrow{u}{\\partial } & \\mathcal{E}_{D}^{i,1}\n\\arrow{r} \\arrow{u}{\\partial } & \\ldots \\end{tikzcd}\n\\end{equation}%\nUsing the natural isomorphisms:%\n\\begin{equation}\n\\mathcal{E}_{D}^{0,j}\\otimes _{\\OX{D}}\\Omega _{D}^{i}\\widetilde{%\n\\rightarrow }\\mathcal{E}_{D}^{i,j}\n\\end{equation}%\none gets a morphism of resolutions: $\\Dolb{D\\,}{\\Omega _{D}^{i}}\\rightarrow \\Dolb{D\\,}{\\Omega _{D}^{i+1}}$. Obviously\n\\linebreak\n${\\partial }^{i+1}\\circ {\\partial }^{i}=0$ and $\\Dolb{D\\,}{\\Omega _{D}^{\\bullet }}$ \nis a double complex.\n\nNote that if $(z_{1},\\ldots ,z_{n})$ are local coordinates on $D$ then the\nmorphism \n\\begin{equation}\n\\partial ^{i}:\\mathcal{E}_{D}^{0,j}\\otimes _{\\OX{D}}\\Omega\n_{D}^{i}\\rightarrow \\mathcal{E}_{D}^{0,j}\\otimes _{\\OX{D}}\\Omega\n_{D}^{i+1}\n\\end{equation}%\nis given by \n\\begin{equation}\n\\partial ^{i}(\\alpha \\otimes \\omega )=\\sum_{k=1}^{n}\\frac{\\partial }{%\n\\partial z_{k}}(\\alpha )\\otimes dz_{k}\\wedge \\omega +(-1)^{j}\\alpha \\otimes\n\\partial \\omega\n\\end{equation}\n\n\\textbf{b. (Embedded case)} If $(X,k,D)$ is an embedding triple then one\nchecks that \n\\begin{equation}\n\\partial (\\mathcal{E}_{D}^{0,j}\\otimes _{\\OX{D}}\\mathcal{N}%\n_{X}^{i})\\subset \\mathcal{E}_{D}^{0,j}\\otimes _{\\OX{D}}\\mathcal{N}%\n_{X}^{i+1}\n\\end{equation}%\nand consequently one obtains a morphism of resolutions:%\n\\begin{equation}\n\\partial :\\Dolb{k\\,}{\\Omega _{X}^{i}} \\rightarrow \\Dolb{k\\,}\n{\\Omega _{X}^{i+1}}  \\label{Morph_Dolb_de_Rham}\n\\end{equation}%\nand $\\Dolb{k\\,}{\\Omega _{X}^{\\bullet }}$ becomes a double complex. \nMoreover the differentials (\\ref{Morph_Dolb_de_Rham}) are compatible with\nthe pullback morphisms.\n\n\\textbf{c. (General case)} Let $(\\mathfrak{U},k,\\mathfrak{D})$ be the\ns.s.embedding triple associated to $(X,\\mathcal{A})$ and $b:\\mathfrak{U}%\n\\rightarrow X$ the natural morphism given by the inclusions. The morphisms (%\n\\ref{Morph_Dolb_de_Rham}) give a morphism of resolutions \n\\begin{equation}\n\\Dolb{k\\,}{\\Omega _{X}^{i}|\\mathfrak{U}}\\rightarrow \\Dolb{k\\,}{\\Omega_{X}^{i+1}|\\mathfrak{U}}\n\\end{equation}\nand by applying $b_{\\sharp }$ the morphism (\\ref{Morpf_Dolb_de_Rham_A}). It\nis immediate to check that $\\Dolb{\\mathcal{A}\\,}{\\Omega _{X}^{\\bullet}}$ is a double complex.\n\n\\textbf{2. }is obvious.\n\\end{proof}\n\n\\refstepcounter{subsection} \\textbf{\\arabic{section}.\\arabic{subsection}}%\n\\label{Paragr_smooth_forms} \\textbf{The functor }$\\Dolbn$ \\textbf{%\nand the complex of smooth differential forms}\n\nLet $X$ be an analytic space and let%\n\\begin{equation}\n0\\longrightarrow \\mathcal{E}_{X}^{0,0}\\overset{\\overline{\\partial }_{X}^{0}}{%\n\\longrightarrow }\\mathcal{E}_{X}^{0,1}\\overset{\\overline{\\partial }_{X}^{1}}{%\n\\longrightarrow }...\n\\end{equation}%\nbe the complex of smooth differential forms with first degree $0$ on $X$.\nRecall that if $k:X\\hookrightarrow D$ is a closed embedding of $X$ in the\ncomplex manifold $D$, and $\\mathcal{I}_{X}\\subset \\OX{D}$ is the\ncoherent ideal sheaf which gives $X$ as a subspace of $D$, then \n\\begin{equation}\n\\mathcal{E}_{X}^{0,i}=\\mathcal{E}_{D}^{0,i}\/\\mathcal{M}_{X}^{i}|X\n\\end{equation}%\nwhere $\\mathcal{M}_{X}^{i}$ is the $\\OX{D}$-submodule of $\\mathcal{E%\n}_{D}^{0,i}$ generated by $\\mathcal{I}_{X}\\mathcal{E}_{D}^{0,i}$, $\\overline{%\n\\mathcal{I}}_{X}\\mathcal{E}_{D}^{0,i}$, and \\linebreak $\\overline{%\n\\partial }^{i-1}(\\overline{\\mathcal{I}}_{X}\\mathcal{E}_{D}^{0,i-1})$; the\ndifferentials \n\\begin{equation}\n\\overline{\\partial }_{X}^{i}:\\mathcal{E}_{X}^{0,i}\\rightarrow \n\\mathcal{E}_{X}^{0,i+1}\n\\end{equation}%\nare induced by those of $\\mathcal{E}_{D}^{0,\\bullet }$. If $X$ is a reduced\nanalytic space $X$ then $\\mathcal{M}_{X}^{i}$ consists of the forms in $%\n\\mathcal{E}_{D}^{0,i}$ which have null pullback to $Reg(X)$. To simplify\nnotation, in what follows we shall write $\\overline{\\partial }$\ninstead of $\\overline{\\partial }_{X}^{i}$ if $i$ and $X$ are clear\nfrom the context.\n\nNote that the natural surjective morphisms%\n\\begin{equation}\n\\mathcal{E}_{D}^{0,i}\/\\mathcal{I}_{X}\\mathcal{E}_{D}^{0,i}|X \n\\rightarrow \\mathcal{E}_{X}^{0,i}\n\\end{equation}%\ndetermine a natural morphism of complexes:%\n\\begin{equation}\n\\Dolb{k\\,}{\\OX{X}}\\rightarrow \\mathcal{E}_{X}^{0,\\bullet }\n\\end{equation}%\nIn general one proves:\n\n\\begin{theorem}\n\\label{Theor_Dolb_diff_forms}Let $(X,\\mathcal{A})$ be a locally embedded\nanalytic space. Then there is a surjective morphism of complexes of sheaves%\n\\begin{equation}\n\\Dolb{\\mathcal{A}\\,}{\\OX{X}}\\rightarrow \\mathcal{C}^{\\bullet\n}(\\mathcal{U},\\mathcal{E}_{X}^{0,\\bullet })  \\label{Morph_E_X}\n\\end{equation}%\nwhere $\\mathcal{U} = cov(\\mathcal{A})$ is the open covering of $X$ \ncorresponding to the atlas $\\mathcal{A}$ and $\\mathcal{C}^{\\bullet }(%\n\\mathcal{U},\\bullet )$ denotes the \\v{C}ech complex on $\\mathcal{U}$.\n\\end{theorem}\n\n\\begin{proof}\nLet $(\\mathfrak{U,}k,\\mathfrak{D})$ be the s.s.embedding triple associated\nto $(X,\\mathcal{A})$ and $b:\\mathfrak{U}\\rightarrow X$ the natural morphism\ngiven by the inclusions. For each $\\alpha \\in \\mathcal{N(U)}$ one has a\nmorphism \n\\begin{equation}\n\\Dolb{k_{\\alpha }\\,}{\\OX{U\\alpha }} \\rightarrow \\mathcal{E}_{U\\alpha }^{0,\\bullet }\n\\end{equation}%\nand these morphisms give a morphism%\n\\begin{equation}\n\\Dolb{k\\,}{\\mathcal{O}|\\mathfrak{U}})\\rightarrow \\mathcal{E}%\n_{X}^{0,\\bullet }|\\mathfrak{U}  \\label{Morph_E_U}\n\\end{equation}%\nThe morphism (\\ref{Morph_E_X}) is obtained by applying $b_{\\sharp }$ to (\\ref%\n{Morph_E_U}).\n\\end{proof}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}