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+{"text":"\\section{Introduction}\n\nAt a distance of 3.8~Mpc \\citep{sak04}, \\object{NGC~5253}, in the Centaurus A \/ M~83 group \\citep{kar07}, is one of the closest Blue Compact Dwarf (BCD) galaxies. This galaxy is well known for presenting several peculiarities, whose detailed study are closely connected to its proximity and high surface brightness. For example, it contains a deeply embedded very dense compact H\\,\\textsc{ii}\\ region at its nucleus (hereafter, \"the supernebula\"), detected in the radio at 1.3~cm and 2~cm \\citep{tur00} that host two very massive Super Star Clusters \\citep[SSCs,][]{alo04} and is embedded in a larger (i.e. $\\sim$100~pc$\\times$80~pc) Giant H\\,\\textsc{ii}\\ Region (hereafter, the central GH\\,\\textsc{ii}R). Recently, mid-infrared observations showed how its kinematics is compatible with a model for the supernebula in which gas is outflowing from the molecular cloud \\citep{bec12}. Indeed, the whole central region of the galaxy is dominated by an intense burst of star formation in the form of a large collection of compact young ($\\sim1-12$~Myr) star clusters \\citep[e.g.][]{har04}. In contrast to this, the main body of \\object{NGC~5253} resembles that of a dwarf elliptical galaxy and recently, three potentially massive ($\\gsim10^5$~M$_\\odot$) and old ($1-2$ Gyr) star clusters have been found in the outskirts of the galaxy \\citep{har12}.\nFinally, \\object{NGC~5253} is best-known  for being one of the few examples (and the closest) of a galaxy presenting  a confirmed local excess in nitrogen \\citep[see e.g.][]{wal89}.\n\n\\defcitealias{mon10}{Paper~I}\n\\defcitealias{mon12}{Paper~II}\n\\defcitealias{wes13}{Paper~III}\n\nWe are carrying out  a detailed study of this galaxy using Integral Field Spectroscopy (IFS). The results obtained so far have further highlighted its peculiar nature.\nIn \\citet[][hereafter Paper I]{mon10}, we found that the emission line profiles were complex and consistent with an scenario where the two SSCs produce an outflow \\citep[see also][]{bec12}. Also, we delimited very precisely the area polluted with extra nitrogen. Moreover, we detected nebular He\\,\\textsc{ii}$\\lambda$4686 in several locations, some associated with WN-type Wolf-Rayet (WR) stars (as traced by the blue bump at around 4680\\AA) and some not, but not necessarily coincident with the area exhibiting extra nitrogen. In \\citet[][hereafter Paper II]{mon12}, we studied the 2D distribution of  the main physical (electron temperature and density, degree of ionization) and chemical properties (metallicity and relative abundances of several light elements) of the ionized gas. A new area of enhanced nitrogen abundance at $\\sim$130~pc from the main area of enhancement and not reported so far was found. In \\citet[][hereafter Paper III]{wes13} several locations showing emission characteristic of WC-type WR stars (via the red bump at around 5810\\AA) were identified. The fact that WR stars are spread over $\\sim$350~pc gives an idea of the area over which the recent starburst has occurred.\nThe chemical analysis was extended with the finding that, with the exception of the aforementioned localised N excess, the $O\/H$ and $N\/H$ distributions are flat within the whole central 250~pc.\n\n\nAn issue not addressed in detail so far in NGC~5253 is the 2D determination and distribution of the He$^+$ abundance. In a cosmological context, this is particularly relevant since the joint determination of metallicity (as traced by the $O\/H$ abundance) and $^4He$ abundance ($Y$\\footnote{Here, we use $Y$ for  the helium mass fraction and $y$ for the number density of helium relative to hydrogen. Assuming $Z = 20(O\/H)$, they are related as $Y = \\frac{4y(1-20(O\/H))}{1+4y}$. }) for extragalactic H\\,\\textsc{ii}\\ regions and star-forming galaxies at low-metallicity was proposed as a means to estimate the primordial helium abundance, $Y_P$ \\citep{pei76,pag86}\n serving as a test-bench for the standard hot big band model of nucleosynthesis. However, the  density of baryonic matter depends weakly on $Y_P$. Therefore, to put useful constrains on $Y_P$, $^4He$ abundance of  individual objects  has to be determined with accuracies $\\lsim$1\\%. Nowadays, emission\nline flux data of this quality can be achieved and, indeed, the astronomical community is actively working on getting and improving the estimation of $Y_P$ \\citep[see e.g. ][for recent estimations by the different groups]{ave10,izo10,fuk06,pei07,izo07,oli01,pei02}.\nHowever, He abundance determinations are influenced by several effects and systematic errors which are not, in principle, straightforward to quantify and correct  \\citep[see for example][]{oli01}. Specifically, the intensity of He\\,\\textsc{i}\\ emission lines may intrinsically deviate from the recombination values due to collisional and radiative transfer effects. Moreover, the emitted spectrum also depends on the physical conditions of the ionized gas (e.g. temperature, density and ionization structure). Also, on top of these specific properties of the H\\,\\textsc{ii}\\ region, extinction by dust and a possible underlying stellar absorption component can also affect the observed spectrum. \nAll these effects contribute to the uncertainties associated with the estimation of \\emph{ionized} helium abundance ($y^+=He^+\/H^+$).\nA final extra source of uncertainty is associated with the estimation of the amount of existing neutral helium (i.e. the estimation of the ionization correction factor, icf(He)) and the calculation of the total helium abundance, $y=y_{tot}=\\rm{icf(He)}\\times y^+$.\n  \nTentative values for  $y^+$ were presented in \\citetalias{mon10} based on the He\\,\\textsc{i}$\\lambda$6678 line, which is almost insensitive to collisional and self-absorption effects. However, 2D distributions of all the relevant physical properties for the ionized gas were not available at that time. Moreover, other He\\,\\textsc{i}\\ lines, in particular  He\\,\\textsc{i}$\\lambda$7065, can indeed be affected by collisional and self-absorption effects, specially in conditions of relatively high electron temperature ($T_e$) and density ($n_e$), as it is the case in the main GH\\,\\textsc{ii}R\\ of NGC~5253. Supporting this expectation, long-slit measurements predict too high He$^+$ abundances from the  He\\,\\textsc{i}$\\lambda$7065 when only recombination effects are taken into account \\citep{sid10} and a non-negligible optical depth $\\tau$(3889) when the He$^+$ abundance is derived using several emission lines in a consistent manner \\citep{lop07}.\n\nAt present, a complete 2D characterization of the physical properties of the ionized gas in the main GH\\,\\textsc{ii}R\\ of \\object{NGC~5253} is available. Therefore, we are in an optimal situation for both mapping the collisional and radiative transfer effects in 2D, and re-visiting the derivation of the He$^+$ abundance map taking into account many He\\,\\textsc{i}\\ lines. This will be the main purpose of this work. Given \nthat the metallicity of the object  \\citepalias[$12+\\log(O\/H)=8.26$,][]{mon12} is only moderately low, our focus will not to be in achieving the $\\lsim1\\%$ accuracy required in the determination of the primordial He abundance, but to explore the effects of a parameter not taken into account so far: namely spatial resolution.\nIn addition, we will explore the 2D relationship between the properties of the ionized gas derived so far and the He\\,\\textsc{i}\\ collisional and self-absorption effects. To our knowledge this work constitutes the first attempt to study in 2D the collisional and self-absorption effects  in He\\,\\textsc{i} in any extra-galactic object.\nMoreover, irrespective of the spatial resolution,  due to the characteristics inherent in IFS data, there is the guarantee that the set of $\\sim100$ spectra utilized in this work have been processed in a homogeneous manner all the way from the observations (i.e. a given observable was taken with the same observing conditions for the whole set of data) to the final $y^+$ and $y_{tot}$ derivations. \n\nThe paper is structured as follows:  Sec. 2 describes the characteristics of the data utilized in our analysis; Sec. 3 contains an evaluation of the He\\,\\textsc{i}\\ collisional and self-absorption effects in 2D as well as the derivation of the $y^+$ and $y$ maps. Sec. 4 discusses the relation between radiative transfer effects and other quantities (e.g. kinematics of the gas, relative $N\/O$ abundance). Our main conclusions are summarized in Sec. 5.\n\n\n\\section{The data}\n\nWe will focus our study on the area associated with the main GH\\,\\textsc{ii}R\\ in \\object{NGC~5253} (see Fig. \\ref{apuntado}).\nThis is a portion of the full area studied in \\citetalias{mon10}\nand the location where: i) the gas presents relatively high electron temperature and density, and therefore, important collisional and radiative transfer effects are expected; and ii) several  helium lines can be detected over a relatively large area with sufficient quality, and therefore a 2D analysis based on the information in individual spaxels is feasible. The utilized data were collected during several observing runs using the FLAMES-Argus and the GMOS Integral Field Units (IFUs) and together cover the whole optical spectral range. In the following, we briefly describe the basic instrumental characteristics of each set of data and compile the information that was extracted. \n\n\n \n  \\begin{figure}[th!]\n   \\centering\n\\includegraphics[angle=0,width=0.48\\textwidth,clip=]{.\/fg1.eps}\n   \\caption[Area under study]{\n\\emph{Left:} False colour image in filters $F658N$ (H$\\alpha$, cyan channel),  $F550M$ ($V$, yellow channel), and $F814W$ ($I$, magenta channel) for the central part of \\object{NGC~5253} using the HST-ACS images (programme 10608, P.I.: Vacca). The area  studied here is marked with a black rectangle.\n\\emph{Right:}   \n   Ionized gas  distribution as traced by the extinction corrected H$\\beta$\\ map derived from a portion of the original FLAMES data.   The position of the two main peaks of continuum emission are marked with crosses.\nThe map is presented in logarithmic scale in order to emphasize  the relevant morphological features and cover  a range of 2.0~dex.\nFlux units are arbitrary. Note the existence of three dead spaxels at $\\sim[5\\farcs0,-1\\farcs0]$ as well as absence of signal in the spaxels at the two left corners of the field of view.  \\label{apuntado}}\n \\end{figure}\n\n\n  \\begin{figure}[th!]\n   \\centering\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg2a.ps}\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg2b.ps}\\\\\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg2c.ps}\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg2d.ps}\\\\\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg2e.ps}\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg2f.ps}\\\\\n   \\caption[Maps of the extinction corrected fluxes of the utilized lines]{Extinction corrected flux maps normalized to H$\\beta$\\ for the lines utilized in this work.  The position of the two main peaks of continuum emission are marked with crosses in this and all subsequent maps.\nWe marked in white the areas where a given line was not observed. Specifically, for the He\\,\\textsc{i}$\\lambda$6678,  He\\,\\textsc{i}$\\lambda$4922 and He\\,\\textsc{i}$\\lambda$7065, they correspond to dead fibers. For He\\,\\textsc{i}$\\lambda$5876, He\\,\\textsc{i}$\\lambda$4471 and He\\,\\textsc{i}$\\lambda$3889, these areas were not covered by the corresponding GMOS or FLAMES-Argus field of view.\n \\label{hetohb}}\n \\end{figure}\n\n\n\\subsection{FLAMES-Argus data}\n\nData were obtained with FLAMES \\citep{pas02} at VLT UT2 in Paranal. We used the Argus IFU with the sampling of 0.52$^{\\prime\\prime}$\/lens, covering a field of view (f.o.v.) of 11\\farcs5$\\times$7\\farcs3, and four low resolution gratings (LR1, LR2, LR3, and LR6).\nAll together, they  offer a spectral coverage of $3\\,610-5\\,070$~\\AA\\ plus  $6\\,440-7\\,180$~\\AA\\ at a dispersion of 0.2~\\AA~pix$^{-1}$.\n Details of the observations, data reduction, and cube processing,\nas well as maps for most of the physical and chemical properties utilized in this work, have already been presented in \\citetalias{mon10} and \\citetalias{mon12}. \nMaps for the helium lines were derived by independently fitting in each spaxel the  He\\,\\textsc{i}\\ line profiles with a single Gaussian function with MPFITEXPR \\citep{mar09}. \nFor the particular case of He\\,\\textsc{i}$\\lambda$3889,  which is blended with H8 at our spectral resolution, we created an extinction corrected map by subtracting from the H8+He\\,\\textsc{i}$\\lambda$3889 map, a map of 0.659$\\times$ the H7 map. This H8 line intensity is that predicted by \\citet{sto95}  for Case B, $T_e = 10^4$~K and $n_e = 100$~cm$^{-3}$.\nThe final set of FLAMES-Argus maps utilized here are:\n\\begin{enumerate}[i)]\n\\item an extinction map derived from the H$\\alpha$\/H$\\beta$\\ line ratio;\n\\item a map for the H$\\beta$\\ equivalent width  ($EW$(H$\\beta$));\n\\item a map of electron temperature $T_e$ as derived from the \\textsc{[O\\,iii]}$\\lambda\\lambda$4959,5007\/ \\textsc{[O\\,iii]}$\\lambda$4363 line ratio: $T_e$(\\textsc{[O\\,iii]}). In those spaxels where no determination of $T_e(\\textsc{[O\\,iii]})$ was available, we assumed $T_e(\\textsc{[O\\,iii]})=10\\,500$~K \\citepalias[see][for typical $T_e$(\\textsc{[O\\,iii]}) values outside the main GH\\,\\textsc{ii}R]{mon12};\n\\item a map for the electron density ($n_e$) as derived from the \\textsc{[S\\,ii]}$\\lambda$6717\/\\textsc{[S\\,ii]}$\\lambda$6731 line ratio; \n\\item maps for the $O^+\/H^+$, $O\/H$ and $S^+\/H^+$ abundances, as derived from collisional lines using the direct method, to estimate the icf(He);\n\\item maps for different tracers of the excitation degree (i.e. \\textsc{[S\\,ii]}$\\lambda\\lambda$6717,6731\/H$\\alpha$\\ and \\textsc{[O\\,iii]}$\\lambda$5007\/H$\\beta$\\ line ratio) to be used in the estimation of the icf(He) at those locations where no measure of the $O^+\/H^+$, $O\/H$ and $S^+\/H^+$ abundances is available and to explore the dependence of $y^+$ and $y_{tot}$ on the excitation;\n\\item a map of the relative abundance of nitrogen, $N\/O$, as derived from collisional lines using the direct method;\n\\item maps for the $\\lambda$3889, $\\lambda$4471, $\\lambda$4922, $\\lambda$6678, $\\lambda$7065 He\\,\\textsc{i}\\ equivalent widths and extinction corrected  line fluxes using our extinction map and  the extinction curve of \\citet{flu94}. The extinction corrected line flux maps normalized to H$\\beta$\\ are presented in Fig. \\ref{hetohb}. \nNote that in order to minimize uncertainties associated to aperture matching, absolute flux calibration and extinction, lines were measured relative to a bright Balmer line observed simultaneously with a given helium line. Then, we assumed the theoretical Balmer line intensities obtained from \\citet{sto95} for Case B, $T_e = 10^4$~K and $n_e = 100$~cm$^{-3}$. \nAdditional He\\,\\textsc{i}\\ lines were covered by the FLAMES set-up but not used here. Specifically, He\\,\\textsc{i}$\\lambda$5016 and He\\,\\textsc{i}$\\lambda$4713 emission lines were detected over most of the FLAMES f.o.v. He\\,\\textsc{i}$\\lambda$5016 is relatively close to the much brighter (i.e. $\\sim$150-400 times)  \\textsc{[O\\,iii]}$\\lambda$5007 line and, in the main GH\\,\\textsc{ii}R, the wings of the \\textsc{[O\\,iii]}\\ line profile prevented us from measuring a reliable line flux. Regarding the relatively weak $\\lambda$4713 line, at the spectral resolution of these data, this line is strongly blended with [Ar\\,\\textsc{iv}]$\\lambda$4711 and  the uncertainties associated with the deblending of these lines in the GH\\,\\textsc{ii}R\\ are relatively large due to the existence of several distinct kinematic components \\citepalias[see][]{mon10,wes13};\n\\item maps with the extinction corrected line fluxes in $\\lambda$6678, $\\lambda$7065 He\\,\\textsc{i}\\  and H$\\alpha$, for the different kinematic components presented in \\citetalias{mon10}.\n\\end{enumerate}\n\n\\subsection{GMOS data}\n\nThe Gemini-South Multi-Object Spectrograph (GMOS) data were taken using the one-slit mode of its IFU \\citep{all02}. In this mode, the IFU covers a f.o.v. of $5\\farcs0\\times3\\farcs5$ sampled by 500 contiguous hexagonal lenslets of 0\\farcs2 diameter.\nThe utilized grating (R381) gives a spectral coverage of $4\\,750-6\\,850$~\\AA\\ at a dispersion of 0.34~\\AA~pix$^{-1}$,  thus complementing the FLAMES-Argus data. We refer to \\citetalias{wes13}  for further details on the observations and data reduction. The product of the reduction is a datacube per pointing with a uniformly sampled grid of 0\\farcs1. In this work, we utilized the two pointings (out of four) that mapped the central GH\\,\\textsc{ii}R. As we did with the FLAMES's data, in each spaxel all the lines of interest were independently fit with a single Gaussian function with MPFITEXPR. \nThe final set of GMOS maps utilized here are:\n\n\\begin{enumerate}[i)]\n\\item Maps for H$\\alpha$\\ and H$\\beta$\\ fluxes. These images were utilized to check the consistency between the FLAMES and GMOS data, both in terms of observed structure and derived extinction map, to estimate the offset and rotation that was necessary to be applied to the GMOS data, and to correct for extinction the  He\\,\\textsc{i}$\\lambda$5876 map;\n\\item A map for He\\,\\textsc{i}$\\lambda$5876 flux. This is the strongest He\\,\\textsc{i}\\ line and therefore, one of the key observables for the present study;\n\\item  Equivalent widths and extinction corrected line flux maps of He\\,\\textsc{i}$\\lambda$5876 flux normalized to H$\\beta$. These were derived from the maps previously mentioned and were rotated and reformatted to match the FLAMES data using the \\texttt{drizzle} task of the Space Telescope Science Data Analysis System (STSDAS) package of IRAF\\footnote{The Image Reduction and Analysis Facility \\emph{IRAF} is distributed by the National Optical Astronomy Observatories which is operated by the association of Universities for Research in Astronomy, Inc. under cooperative agreement with the National Science Foundation.}. They were the only GMOS maps utilized jointly with the FLAMES's maps. The flux map is shown in Fig. \\ref{hetohb} together with the other He\\,\\textsc{i}\\ line flux maps. \n\\end{enumerate}\n\n\n\\section{Results}\n\n\\subsection{Collisional effects as traced by the theoretical C\/R ratio \\label{seccoli}}\n\nCollisional excitation in hydrogen can be important in regions of very low metallicities ($Z\\lsim1\/6$~Z$_\\odot$)\n due to their relatively high temperatures \\citep{lur09}. However, for the typical temperatures and densities found in H\\,\\textsc{ii} regions in general, and in the main Giant H\\,\\textsc{ii} Region (GH\\,\\textsc{ii}R) of \\object{NGC~5253} in particular, the collisional excitation in hydrogen is negligible in comparison with recombination \\citep[see e.g. Fig. 9 of ][]{ave10}.  This is not the case for  helium.  The He\\,\\textsc{i}\\ $2^3S$ level \\citep[see Fig. 1 in ][for a representation of the Grotian diagram for He\\,\\textsc{i}\\ singlet and triplet ladders]{ben99} is highly metastable, and collisional transition from it can be important \\citep{ost06}. Specifically, at relatively high densities this level can be depopulated via collisional transitions to the $2^3P^0$, $2^1P^0$ and $2^1S$ and, to a lesser extent, to higher singlets and triplets (mainly $3^3P^0$). The effect on the observed He\\,\\textsc{i}\\ emission lines is always an increase in the observed flux. The relative importance of the collisional effects on a given emission line is characterized by the $C\/R$ factor, i.e. the ratio of the collisional component to that arising from recombination which is given by:\n\n\\begin{equation}\n\\frac{C}{R} = \\frac{n_{2^3S}k_{eff}}{n^+_{He}\\alpha_{eff}}\n\\end{equation}\n\nwhere $n_{2^3S}$, and $n^+_{He}$ are the densities of the $2^3S$ state and He$^+$ respectively, $\\alpha_{eff}$ is the effective recombination coefficient for the line, and $k_{eff}$ is the effective collisional rate coefficient.\n\n\n  \\begin{figure}[th!]\n   \\centering\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg3a.ps}\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg3b.ps}\\\\\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg3c.ps}\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg3d.ps}\\\\\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg3e.ps}\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg3f.ps}\\\\\n   \\caption[C\/R ratios]{Collisional effects as traced by the $C\/R$ ratio for the He\\,\\textsc{i}\\ lines utilized in this work. Note that a common scale was used for all the lines in order to facilitate comparison of the relative effects between the different lines. Also, a logarithmic color stretch is used to emphasize the variations \\emph{within} the region for a given line.\n \\label{c2rratios}}\n \\end{figure}\n\n  \\begin{figure}[th!]\n   \\centering\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg4a.ps}\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg4b.ps}\\\\\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg4c.ps}\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg4d.ps}\\\\\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg4e.ps}\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg4f.ps}\\\\\n   \\caption[Differences on the $C\/R$ factors  for different electron temperature]{\n Maps for the ratio between the $C\/R$ factors derived for two assumptions of the $T_e(He\\,\\textsc{i})$: $Q(\\lambda)=\\frac{C\/R(\\lambda)_{Case 1}}{C\/R(\\lambda)_{Case 2}}$. Note that areas with constant values for which a $T_e$(\\textsc{[O\\,iii]})=10\\,500~K was assumed have not been included in the comparison.\n %\n A logarithmic color stretch is employed to emphasize the variations \\emph{within} the region for a given line.\n \\label{c2rratiosdif}}\n \\end{figure}\n\nHere, we will estimate the 2D contribution to the collisional component using the relations derived by  \\citet{por07}, where  theoretical $C\/R$ factors are calculated as functions of $n_e$ and $T_e$, and assuming that the density from \\textsc{[S\\,ii]}\\ traces well that from helium.\nPlasma temperatures as traced by helium lines can typically be $\\sim$50\\% of those traced by oxygen in planetary nebulae \\citep{zha05}. For H\\,\\textsc{ii}\\ the situation is not so clear. For the specific case of \\object{NGC~5253}, values between 82\\% and 96\\% have been reported in specific apertures \\citep{lop07}. We made two assumptions for the electron temperature:\n\\begin{itemize}\n\\item Case 1: The temperature from oxygen traces well that from helium: $T_e$(He\\,\\textsc{i}) = $T_e$(\\textsc{[O\\,iii]});\n\\item Case 2: The temperature from helium is proportional to the temperature from oxygen, with the constant of proportionality estimated as the mean of the ratios between both temperatures provided by \\citet{lop07}:  $T_e$(He\\,\\textsc{i}) = 0.87\\,$T_e$(\\textsc{[O\\,iii]}).\n\\end{itemize}\n\nMaps for the $C\/R$ factors using the first assumption are presented in Fig. \\ref{c2rratios} while the ratio between the two estimations is presented in Fig. \\ref{c2rratiosdif}.\nThis is the first time that collisional effects for a set of helium lines have been mapped in an\nextra-galactic source. Several results can be extracted from these maps.\n\nFirstly, all the $C\/R$ maps display the same structure. That is: higher ratios at the peak of emission for the ionized gas and towards the northwest half of the GH\\,\\textsc{ii}R\\ and a decrease of the collisional contribution outwards. This reproduces the observed density structure \\citepalias[see e. g. Fig. 6 in ][]{mon10}.\n\nSecondly, \nlines corresponding to transitions in the singlet cascade have a negligible contribution from collisional effects (e.g. $C\/R$ factor for $\\lambda$4922 varies between  $\\sim0.001-0.006$) while for those lines in the triplet cascade ($\\lambda$7065, $\\lambda$5876, $\\lambda$4471,  and $\\lambda$3889), the contributions from  collisional effects can be important.  In particular,  the  $C\/R$ factor  for $\\lambda$7065 ranges between $\\sim0.02$ and $\\sim0.22$, meaning it reaches  $\\sim$20\\% in the nucleus of the galaxy. \n\nThirdly, the assumed temperature has some influence in the estimation of the collisional effects. In our particular case the assumed temperature in Case 2 was only $\\sim15\\%$ smaller than that in Case 1. However, this implies a smaller contribution of the collisional effects by $\\sim$25-30~\\% for $\\lambda$7065  and $\\sim$30-35~\\% for $\\lambda$3889 (the two lines most affected by collisional effects) and up to $\\sim50$\\% for the other lines under study.  Interestingly, areas of lower temperature are more sensitive to the assumption on $T_e$. \nIt is important to note that the uncertainties associated to the errors due to the measurement of the line fluxes involved in the determination of $T_e$(\\textsc{[O\\,iii]}) were typically $\\lsim$1\\,000~K \\citepalias{mon12}. This is smaller than the difference  in the assumed temperature between the two reasonable assumptions, Case 1 and 2, which range between $\\sim$1\\,500~K at the peak of emission to $\\sim$1\\,300~K in the areas of lowest surface brightness. \nThis implies that, at this level of data quality, more than errors associated to line fluxes, it is systematic errors associated with assumptions in density and temperature that dominate the uncertainties in the evaluation of the collisional effects.\n\n\\subsection{Radiative transfer effects as traced by $\\tau(3889)$ and derivation of He$^+$ abundance \\label{secymas}}\n\nSingle ionized helium abundance, $y^+$,   can be calculated as follows:\n\n\\begin{equation}\ny^+(\\lambda) = \\frac{F(\\lambda)}{F(H\\beta)}\n\\frac{E(H\\beta)}{E(\\lambda)}\n\\frac{\\frac{EW(\\lambda) + a_{He\\,I}(\\lambda)}{EW(\\lambda)}}{\\frac{EW(H\\beta) + a_{H}(H\\beta)}{EW(\\lambda)}} \\frac{1}{f_\\tau(\\lambda)}\n\\label{eqymas}\n\\end{equation}\n\nwhere $F(\\lambda)\/F$(H$\\beta$) is the extinction corrected flux for the He\\,\\textsc{i}\\ line scaled to H$\\beta$, $E$(H$\\beta$) and $E$(He\\,\\textsc{i}) are the theoretical emissivities; $f_{\\tau}(\\lambda)$ is a factor that takes into account  radiative transfer effects. The remaining term in Eq. \\ref{eqymas}, with $a(\\lambda)$ is the equivalent width in absorption, which takes into account the effect of the underlying stellar population.\nIn the following, we will describe how each of these terms were evaluated.\n\n\\subsubsection{Calculation of the emissivities}\n\nFor H$\\beta$\\  emissivities, we utilized the function from \\cite{sto95} which is, in units of 10$^{25}$ erg cm$^{-3}$ s$^{-1}$:\n\n\\begin{equation}\nE(H\\,\\beta) =  4\\pi j_{H\\,\\beta}\/n_e n_{H^+} = 1.37 t_e^{-0.982} \\exp(-0.104\/t_e)\n\\end{equation}\n\nwith $t_e = T_e\/10^4 $.\nFor the He\\,\\textsc{i}\\ lines, we utilized those emissivities originally provided by \\citet{por12} and recently corrected by \\citet{por13}. These are the most recent He\\,\\textsc{i}\\ emissivities and have the collisional effects included. Therefore, there was no need to include an extra term in Eq. \\ref{eqymas} to take into account collisional effects. They are tabulated for discrete values of $n_e$ and $T_e$. However, the $n_e$ in the main GH\\,\\textsc{ii}R\\  varies between $\\lsim100$~cm$^{-3}$ and $\\sim660$~cm$^{-3}$ and $T_e$ ranges between $\\sim9\\,000$ and $\\sim12\\,000$~K. Therefore, to evaluate the emissivities at each individual spaxel, we fitted the values provided for $n_e= 100$ and 1\\,000~cm$^{-3}$  and $T_e$ ranging from 5\\,000~K to 25\\,000~K to functions with the same parametrization as $E(H\\,\\beta)$, $a  t_e^{b} \\exp(c\/t_e)$, and then interpolated on a logarithmic scale as follows:\n\n\\begin{equation}\nE(\\lambda,\\log n_e) = E(\\lambda,2) + (E(\\lambda,3) - E(\\lambda,2)) (\\log n_e -2)\n\\end{equation}\n\nThe coefficients and standard deviations of the fits are compiled in Table \\ref{coeficientes} while Fig. \\ref{compaemi} shows a \ncomparison between the fitted function and the discrete values provided by \\citet{por12} for the range of densities and temperatures covered in the GH\\,\\textsc{ii}R. A comparison between this figure and Fig. \\ref{c2rratios} shows the correspondence between the degree of dependence on the density and the contribution of the collisional effects.\n\n\n  \\begin{figure}[th!]\n   \\centering\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=]{fg5a.ps}\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=]{fg5b.ps}\\\\\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=]{fg5c.ps}\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=]{fg5d.ps}\\\\\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=]{fg5e.ps}\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=]{fg5f.ps}\\\\\n   \\caption[Fitted function to the Porter et al. 2012 emissivities]{Fitted functions to the \\citet{por12} emissivities. The black lines are the functions for $n_e=100$~cm$^{-3}$ (continuous line) and for $n_e=1000$~cm$^{-3}$ (long dashed line). The intemediate short dashed lines represent the interpolated emissivities for $\\log(n_e) = 2.2, 2.4, 2.6, 2.8$. \n \\label{compaemi}} \n \\end{figure}\n\n\n\\begin{table}\n\\small\n     \\centering\n     \\caption[]{Coefficients for the equations fitted to the \\citet{por12} Table 2. \\label{coeficientes}}\n            \\begin{tabular}{lcccccccccccc}\n            \\hline\n            \\noalign{\\smallskip}\n\n  Line                   & $a$  &  $b$  &  $c$  & Std Dev (\\%)  \\\\\n\\hline\n\\multicolumn{5}{c}{$n_e=100$~cm$^{-3}$}\\\\\n\\hline\n\\noalign{\\smallskip}  \n3889 & 1.453 & $-0.724$ & $-0.036$ & 0.010 \\\\\n4471 & 0.679 & $-1.078$ & $-0.105$ & 0.003\\\\\n4922 & 0.184 & $-1.091$ & $-0.105$ & 0.001\\\\\n5876 & 1.680 & $-1.061$ & $+0.004$ & 0.042\\\\\n6678 & 0.473 & $-1.065$ & $+0.013$ & 0.015\\\\\n7065 & 0.269 & $-0.367$ & $+0.100$ & 0.007\\\\ \n\\hline\n\\multicolumn{5}{c}{$n_e=1000$~cm$^{-3}$}\\\\\n\\hline\n3889 &  1.198 & $-0.336$ & $+0.207$ & 0.032\\\\\n4471 &  0.507 & $-0.694$ & $+0.202$ & 0.048\\\\\n4922 & 0.131 & $-0.642$ & $+0.252$ & 0.016\\\\\n5876 & 0.895 & $-0.195$ & $+0.671$ & 0.255\\\\\n6678 & 0.237 & $-0.138$ & $+0.738$ & 0.094\\\\\n7065 & 0.320 &  $+0.171$ & $+0.163$ & 0.096\\\\ \n\n\\noalign{\\smallskip}  \n\\hline\n\n\\end{tabular}\n\\end{table}\n\n\n\n\\subsubsection{Correction for underlying stellar population}\n\nTo take into account the effect of the underlying stellar population it is necessary to estimate the equivalent width both in emission and absorption for H$\\beta$\\ and the helium lines.\nThe equivalent width of H$\\beta$\\ in emission (not shown) ranges typically from $\\sim$240~\\AA\\ at the peak of emission for the ionized gas to $\\sim$65~\\AA\\ in the most outer regions of the area sampled here.\nWe assumed a component in absorption of 2~\\AA, which is adequate for very young starbursts, as the one at the nucleus of \\object{NGC~5253}  \\citep[e.g.][]{gon05,alo10}. \nThis implies a correction for H$\\beta$\\ in absorption from a negligible value (i.e. $\\lsim1\\%$) at the peak of emission to $\\sim$4\\% in the outer parts of the covered area. \n\n  \\begin{figure}[th!]\n   \\centering\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg6a.ps}\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg6b.ps}\\\\\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg6c.ps}\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg6d.ps}\\\\\n   \\caption[Maps for the estimated correction factors for an underlying stellar population]{\n Maps of the correction factor due to a component in absorption in the He\\,\\textsc{i}\\ lines.\n %\nDifferent scales were utilized in the different maps to emphasize the structure.\n The corresponding map for the H$\\beta$\\ correction factor (not shown) is similar to those for $\\lambda$6678 and $\\lambda$7065.\n %\n \\label{corrabs}}\n \\end{figure}\n\nThe correction due to the underlying stellar population is not straightforward to estimate for the helium lines. \nAs explained in \\citetalias{mon12}, for the \ndata obtained with FLAMES Giraffe and the LR1 or LR2 gratings, the contribution of the stellar population was separated from that of the gas using the STARLIGHT code \\citep{cid05,cid09} to match the stellar continuum. Therefore, \nwe can assume $a_{HeI}(\\lambda4471)=a_{HeI}(\\lambda3889)=0$.\nFor the other lines,  a common approach assumes identical stellar equivalent widths for all the helium lines. However,  He\\,\\textsc{i}\\ lines are produced by the bluest stars. Therefore, in the context of a galaxy  suffering a burst of star formation on top of an older population, redder lines should have smaller equivalent widths in absorption, since older stars contribute more to the stellar continuum.\nSpecifically, typical estimations of the equivalent width in absorption for redder lines would be $\\sim40-80$\\% that of $a_{He\\,\\textsc{i}}(\\lambda4471)$ \\citep{ave10}. This is one of the lines observed with the LR2+FLAMES configuration for which nebular and stellar information have been disentangled. Therefore $a_{He\\,\\textsc{i}}(\\lambda4471)$ could be measured from our emission line free cube. Typical values were extremely low (i.e. $\\sim0.11\\pm0.03$~\\AA) and without any obvious variation following the structure of the GH\\,\\textsc{ii}R\\ or the location of the Super Star Clusters.  Taking this as reference, and the relative values between the different $a_{He\\,\\textsc{i}}(\\lambda)$'s reported by \\citet{ave10}, which were in turn derived using the models presented by \\citet{gon05} and \\citet{mar05b}, we assumed $a_{He\\,\\textsc{i}}(\\lambda)=0.09, 0.08, 0.06$ and 0.05~\\AA\\ for $\\lambda$4922, $\\lambda$5876, $\\lambda$6678, $\\lambda$7065, respectively. \nEquivalent widths for the  $\\lambda$4922, $\\lambda$5876, $\\lambda$6678, $\\lambda$7065 He\\,\\textsc{i}\\ emission lines were $\\sim0.3-4$~\\AA, $\\sim20-60$~\\AA, $\\sim3-20$~\\AA, and $\\sim3-35$~\\AA, respectively. With these values, the largest correction for absorption was for He\\,\\textsc{i}$\\lambda$4922, with typical values between 3\\% and 6\\%. However, corrections in the most external spaxels could reach up to  $\\sim$25\\%. For the other lines the correction was more moderate, with values between $\\sim$1\\% and $\\lsim$8\\% for He\\,\\textsc{i}$\\lambda$5876 and comparable to the correction in H$\\beta$\\ for the  He\\,\\textsc{i}$\\lambda$6678 and  He\\,\\textsc{i}$\\lambda$7065 lines. This is illustrated in Fig. \\ref{corrabs}.\n\n\\subsubsection{Estimation of radiative transfer effects}\n\nRadiative transfer effects can be important in recombination radiation. Given the metastable character of the $2^3 S$ level, under certain conditions, the optical depths in lower $2^3S - n^3 P^0$ lines imply non-negligible effects on the emission line strengths \\citep{ost06}. Specifically $\\lambda$10\\,830 photons are only scattered, but absorbed photons, corresponding to transitions to higher levels, can be converted to several photons of lower energy. The most prominent example is\n$\\lambda$3889 photons that can be converted to $\\lambda4.3~\\mu$m $3^3 S-3^3 P^0$, plus $\\lambda 7065\\, 2^3 S-3^3 P^0$, plus $\\lambda10\\,830\\, 2^3 S-2^3 P^0$. The net effect  is that lines associated with transitions from the $2^3 P$ level upwards (e.g. $\\lambda3889$) are weakened by self-absorption, while lines associated with several transitions from higher levels (e.g. $\\lambda$7065) are strengthened by resonance fluorescence. Contrary to collisional effects, radiative transfer effects do not affect photons in the singlet cascade.\nThe relative importance of radiative transfer effects is quantified by a correction factor, $f_\\tau(\\lambda)$,  for each line which is a function of the optical depth at $\\lambda$3889, $\\tau(3889)$. Here, we  parametrized these factors by fitting the values provided by \\citet{rob68} to a non-expanding nebula to the functional form $f_\\tau(\\lambda) _{\\omega=0} = 1 + a \\tau^b$. The corresponding functions for the lines utilized in this work are:\n\n\\begin{equation}\nf_\\tau(7065)_{\\omega=0} = 1 + 0.741\\tau^{0.341} \\label{eqfac7065}\n\\end{equation}\n\n\\begin{equation}\nf_\\tau(6678)_{\\omega=0} = 1    \\label{eqfac6678}\n\\end{equation}\n\n\\begin{equation}\nf_\\tau(5876)_{\\omega=0} = 1 + 0.0126\\tau^{0.496}   \\label{eqfac5876}\n\\end{equation}\n\n\\begin{equation}\nf_\\tau(4922)_{\\omega=0} = 1    \\label{eqfac4922}\n\\end{equation}\n\n\\begin{equation}\nf_\\tau(4471)_{\\omega=0} = 1 + 0.0022\\tau^{0.728}   \\label{eqfac4471}\n\\end{equation}\n\n\\begin{equation}\nf_\\tau(3889)_{\\omega=0} = 1 - 0.261\\tau^{0.305}  \\label{eqfac3889}\n\\end{equation}\n\nNote that no values were provided for He\\,\\textsc{i}$\\lambda$4922 in the original work of \\citet{rob68}. However, as it happens for He\\,\\textsc{i}$\\lambda$6678, this is a singlet line, and therefore $f_\\tau(4922) = 1$ can be assumed. \n\nTypically, optical depth ($\\tau(3889)$) and He\\,\\textsc{i}\\ abundance ($y^+$)  (and other parameters) are determined simultaneously by minimizing $\\chi^2$, defined as the difference between each helium line's abundance (weighted according to a reasonable criterion like the line flux) and the average. In this methodology it is implicit that all the lines trace the same location in the nebula\/galaxy. However, the area under study in this work suffers from heavy extinction \\citepalias[see Fig. 3 in][]{mon10}.\nTherefore,  \\emph{a priori}, it is not possible to assume that all the lines equally penetrate the nebula interior and that bluer and redder lines trace zones with the same $y^+$. Because of that, we grouped the lines in two sets according to their wavelengths, called hereafter the \\emph{blue} ($\\lambda$3889, $\\lambda$4471, $\\lambda$4922) and \\emph{red} ($\\lambda$5876, $\\lambda$6678, $\\lambda$7065) \\emph{sets}, which will be analyzed independently. Each set is made out of: i) a line from the singlet cascade, and therefore not affected by radiative transfer effects; ii) a line from the triplet cascade, thus highly sensitive to radiative transfer effects; iii) a line from the triplet cascade mildly sensitive to radiative transfer effects. \n\n\n  \\begin{figure}[th!]\n   \\centering\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg7a.ps}\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg7b.ps}\\\\\n   \\caption[Abundance maps from singlets]{Maps for $y^+$ derived from lines of the singlet cascade for the blue (left) and red (right) sets.\nA common scale was used in both maps to facilitate comparison between them.\n  \\label{abunmapsinglet}}\n \\end{figure}\n\nA first estimation of the $y^+$ abundance structure as traced by the red and blue lines can be obtained from the singlet lines, since they are not affected by radiative transfer effects. This is shown in Fig. \\ref{abunmapsinglet}. The mean ($\\pm$ standard deviation) are \n80.7($\\pm$5.1) and 76.8($\\pm$1.8)\nfor the $\\lambda$4922 and $\\lambda$6678 lines averaged over the mapped area. These values indicate that even if the red and blue lines are not tracing exactly the same gas columns, at least they sample areas with a the same  $y^+$ within $\\sim$5\\%. \n\nA comparison of the initial $y^+$ maps  with those obtained from a line highly sensitive to radiative transfer effects in each set ($\\lambda$3889 and $\\lambda$7065), allowed us to determine the respective $f_\\tau(\\lambda)$ map for each set, which can, in turn, be converted into $\\tau(3889)$ maps. These are shown in Fig. \\ref{maptau}. The optical depth has the same structure in both maps, with the peak at the main super star cluster(s). The shape of the area presenting $\\tau(3889)>0$ is circular but resolved. With a FWHM=1\\farcs6$\\sim$30~pc, this is larger than the seeing ($\\sim$0\\farcs9). This is consistent with a picture where large optical depths are not restricted to the deepest core of the galaxy, associated with the supernebula, but extend over a larger region. Also, the fact that\nthe optical depths derived from the red set of lines are larger than those derived from the blue set is in harmony with a picture where deeper layers in the nebula, which we interpreted as denser and hotter \\citep[][see also Beck et al. 2012]{mon12}, suffer from larger radiative transfer effects and suggests a link between He\\,\\textsc{i}\\ optical depth and dust optical depth.\n\n  \\begin{figure}[th!]\n   \\centering\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg8a.ps}\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg8b.ps}\\\\\n   \\caption[Maps for $\\tau(3889)$]{Maps for $\\tau(3889)$ as derived from the $\\lambda$4922 and $\\lambda$3889  (left) and the $\\lambda$6678 and $\\lambda$7065 (right) emission lines.\n \\label{maptau}}\n \\end{figure}\n\n\\subsubsection{Final derivation of He$^+$ abundance}\n\nFinally for both sets, the information of the  mildly sensitive line was added and abundances from each line were recalculated using their corresponding $\\tau(3889)$. The final abundance maps for each set  were  made using a weighted average.  We used the mean fluxes of the He\\,\\textsc{i}\\ in the utilized area to determine the respective weights. These were 3:1:1 for the $\\lambda$5878:$\\lambda$6678:$\\lambda$7065 and 25:5:1 for $\\lambda$3889:$\\lambda$4471:$\\lambda$4922.  The mean ($\\pm$ standard deviation) for the red and blue sets for $y^+$ are\n79.3($\\pm$2.5) and 82.0($\\pm$3.8),\ni.e., results from the red and the blue sets agree to within $\\sim$3\\%. \nSince differences between the  maps derived from the blue and red sets are consistent with tracing the same $y^+$ per spaxel, we derived a final map using the information of all the lines by averaging these two maps with a \\emph{blue:red} weight of 0.8:1.0, the mean ratio between the brightest lines in the blue and red sets (i.e. $\\lambda$3889 and $\\lambda$5876).  This is shown in Fig. \\ref{abundancefinal}. The mean ($\\pm$ standard deviation) is\n 80.3($\\pm$2.7).\n  This value compares well with other values of $y^+$ reported in the literature for this area \\citep{kob97,lop07,sid10}.\n\n\n  \\begin{figure}[th!]\n   \\centering\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg9.ps}\n   \\caption[Final $y^+$ map]{Final map for $y^+$ derived from a weighted average of those derived for the lines in the blue and red sets.  Note we used a different scale than in Fig. \\ref{abunmapsinglet}  to increase the contrast and emphasize the structure following the excitation structure. Instead, the scale is common with that in the right column of Fig. \\ref{icfhe} to make easier the comparison between total and ionic helium abundances.\n  \\label{abundancefinal}}\n  \\end{figure}\n\nTo estimate the emissivities, the calculation of these abundances was made with assumed $n_e$(\\textsc{[S\\,ii]}) and $T_e$(He\\,\\textsc{i}) = $T_e$(\\textsc{H\\,i}) = $T_e$(\\textsc{[O\\,iii]}). As for the mapping of the collisional effects, in order to evaluated the influence of the selection of $T_e$,  the derivation presented here was repeated assuming   $T_e$(He\\,\\textsc{i}) = $T_e$(\\textsc{H\\,i}) = 0.87 $T_e$(\\textsc{[O\\,iii]}) and finding a mean  ($\\pm$ standard deviation) of\n79.4($\\pm$3.0).\nThis implies that the precise selection of $T_e$ has a small effect ($\\sim$1\\%) in the determination of $y^+$, in comparison to other factors like e.g.  the correction for absorption or radiative transfer effects, as long as one keeps this selection within reasonable values.\n\nFinally, it is worth mentioning that throughout all the derivation of $y^+$, we have assumed that He\\,\\textsc{i} singlets are formed under Case B conditions (i.e. lines are formed in the limit of infinite Lyman line  optical depth and there is no optical depth in transitions arising from excited states). This is the standard framework for the derivation of helium abundances.\nHowever, under certain conditions, helium Ly$\\alpha$ $\\lambda$584 (and higher Lyman transitions) may be attenuated due to two effects: absorption of the helium Ly$\\alpha$ line by dust and hydrogen absorption \\citep{fer80,shi93}.\nIf these effects were important, the Case B assumption would no longer be applicable. The fact that we have recovered similar helium\nabundances from the two sets of lines supports the Case B assumption.\n \n\\subsection{Determination of He abundance}\n\nThe favorite scenario to explain the extra nitrogen found within the area studied here is that this has been produced by Wolf-Rayet stars and presumedly those at the two Super Star Clusters. In this scenario, as it happens with the nitrogen, an overabundance of helium is also expected to be observed.\nSpecifically, typical stellar atmospheric $N\/He$ ratios  would be $\\sim3\\times10^{-3}$ and $\\sim5\\times10^{-4}$ for WN and WC stars, respectively \\citep{smi82}. This could be use as a reference for the expected $N\/He$ ratios of the material newly incorporated into the warm ISM. Are our data supporting an enrichment in helium abundance compatible with these ratios and, therefore, supporting the Wolf-Rayet stars hypothesis?\n\n\n  \\begin{figure}[ht!]\n   \\centering\n\\includegraphics[angle=0,width=0.39\\textwidth,clip=]{.\/fg10.ps}\n\\caption{\nHe$^+$ abundance vs. \\textsc{[O\\,iii]}$\\lambda$5007\/H$\\beta$. The first-degree polynomial fit is shown with a black line. The 1-$\\sigma$ and 3-$\\sigma$ levels are marked with thick and thin red lines respectively. Mean and standard deviation of each 0.05 dex bin in $\\log$(\\textsc{[O\\,iii]}$\\lambda$5007\/H$\\beta$) are shown with blue diamonds and error bars respectively. The green horizontal dashed line shows the mean value. Data points corresponding to spaxels with $\\log(N\/O)>-1.3$ have been marked with orange circles.\n}\n \\label{ionidegvsymas}\n \\end{figure}\n\nAs can be clearly seen in Fig. \\ref{abundancefinal}, the $y^+$ map presents some structure that follows the excitation, even if the standard deviation for $y^+$ is small ($\\sim$4\\% of the mean value). This is better seen in Fig. \\ref{ionidegvsymas} which is an updated version of Fig. 14 in \\citetalias{mon10} but restricted to the area studied here. \nThe figure presents $y^+$ for each individual spaxel versus  a tracer of the excitation, in this case \\textsc{[O\\,iii]}$\\lambda$5007\/H$\\beta$. \nFor each bin of 0.05~dex, we overplotted the mean and standard deviation of $y^+$ values with blue diamonds and error bars and fitted all our data points with a first-degree polynomial.\nConsidering that if we find $<y^+>$  in the bin of highest \\textsc{[O\\,iii]}$\\lambda$5007\/H$\\beta$\\ to be larger than $<y^+>+\\sigma(y^+)$ in the bin of lowest \\textsc{[O\\,iii]}$\\lambda$5007\/H$\\beta$\\ indicates the presence of a gradient, then this plot is consistent with a positive gradient in $y^+$.\nMoreover, spaxels with an extra amount of nitrogen have, on average higher $y^+$.\nIs this increase due \\emph{only} to variations in the ionization structure or are we witnessing an enrichment in the helium abundance?  \nTo answer this question, one needs to estimate and correct for the  unseen neutral helium, removing in this way, the dependence on the excitation.\nThis is not straight forward  as examplified by previous studies \\citep[e.g.][]{vie00,sau02,gru02}.\n\n  \\begin{figure}[ht!]\n   \\centering\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg11a.ps}\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg11b.ps}\n   \\caption[Ionic abundance maps]{Maps for the ionic abundances utilized in the derivation of the ionization correction factors. \\emph{Left:} $O^+\/H^+$. \\emph{Right:} $S^+\/H^+$ }\n \\label{ionicabundancemap}\n \\end{figure}\n\n\n  \\begin{figure}[th!]\n   \\centering\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg12a.ps}\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg12b.ps}\\\\\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg12c.ps}\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg12d.ps}\\\\\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg12e.ps}\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg12f.ps}\\\\\n   \\caption[icf(He)]{\\emph{Left:} Maps for the icf(He) as estimated from the expressions proposed by \\citet[][\\emph{top}]{kun83},  \\citet[][\\emph{center}]{pei77} and assuming no gradient in helium abundance \\emph{bottom}. \\emph{Right:} Corresponding helium abundance maps. }\n \\label{icfhe}\n \\end{figure}\n\n\n  \\begin{figure}[th!]\n   \\centering\n\\includegraphics[angle=0,width=0.39\\textwidth,clip=]{.\/fg13a.ps}\\\\\n\\includegraphics[angle=0,width=0.39\\textwidth,clip=]{.\/fg13b.ps}\\\\\n\\includegraphics[angle=0,width=0.39\\textwidth,clip=]{.\/fg13c.ps}\\\\\n\\caption{\nHelium abundance as derived using the icf(He) proposed by  \\citet[][\\emph{top}]{kun83},  \\citet[][\\emph{center}]{pei77} and assuming no gradient in helium abundance (\\emph{bottom}) vs. \\textsc{[O\\,iii]}$\\lambda$5007\/H$\\beta$. Symbol and color codes are as in Fig. \\ref{ionidegvsymas}.\n}\n \\label{ionidegvsy}\n \\end{figure}\n\nWithin the spirit of keeping the analysis simple, here we compare the results using three approaches. The first two make use the icf's proposed by  \\citet{kun83} and  \\citet{pei77}:\n\n\\begin{equation}\n\\rm icf(He)_{K\\&S83} = (1 - 0. 25\\,O^+ \/ O)^{-1}                             \\label{eqks83}\n\\end{equation}\n\\begin{equation}\n\\rm icf(He)_{P\\&TP77} = (1 - 0. 35\\,O^+ \/ O- 0.65\\,S^+ \/ S)^{-1}  \\label{eqptp77}\n\\end{equation}\n\nFor the third one, we assumed \\emph{a priori} a functional form similar to that of \\citet{kun83} and iteratively determined the required coefficients to obtain a relation consistent with no gradient of helium abundance. This would represent the case of largest reasonable correction for unseen helium:\n\n\\begin{equation}\n\\rm icf(He)_{M13} = (1 - 0. 46\\, O^+ \/ O)^{-1}                                  \\label{eqm13}\n\\end{equation}\n\nThese icf's depend on the ionic and total abundances of oxygen and sulfur. A map for the total abundance of oxygen was presented in \\citetalias{mon12}. We utilized a constant sulfur abundance determined as the mean of those presented in \\citetalias{wes13}. Regarding the ionic abundances of oxygen and sulfur, they were derived as part of the work presented in \\citetalias{mon12}. However, maps were not included there and are displayed in Fig. \\ref{ionicabundancemap} for completeness. An extra assumption is needed to calculate the icf's in those areas where no measurement for the ionic abundance is available. For that, we utilized the information in the other spaxels to fit  a  first-degree polynomial to the relation between  the \\textsc{[S\\,ii]}$\\lambda\\lambda$6717,6731\/H$\\alpha$\\ line ratio and icf(He):\n\n\n\\begin{equation}\n\\rm icf(He)_{K\\&S83} =  (0.354\\pm0.012) \\mathrm{[S\\,\\textsc{ii}]\/H}\\alpha + (1.033\\pm0.002)\n\\end{equation}\n\\begin{equation}\n\\rm icf(He)_{P\\&TP77} =  (0.447\\pm0.026) \\mathrm{[S\\,\\textsc{ii}]\/H}\\alpha + (1.054\\pm0.003)\n\\end{equation}\n\\begin{equation}\n\\rm icf(He)_{M13} =  (0.763\\pm0.025) \\mathrm{[S\\,\\textsc{ii}]\/H}\\alpha + (1.057\\pm0.004)\n\\end{equation}\n\nThe  maps for the three icf(He) estimations are presented in the left column of Fig. \\ref{icfhe}. In all three cases, the icf is smallest at the peak of emission for the ionized gas and increases outwards, following the ionization structure. \n\nThe icf(He) values are close to one another but variable within a range $\\sim1.04-1.09$,  $\\sim1.06-1.12$ and $\\sim1.09-1.20$ when using Eqs. \\ref{eqks83}, \\ref{eqptp77}, and  \\ref{eqm13}, respectively.\nThe corresponding maps with the total helium abundance are presented in the right column of this figure, while the dependence on the excitation is presented in Fig. \\ref{ionidegvsymas}.\nNote that, when using the \\citet{kun83} and \\citet{pei77} icf's, the slope of the relation between $y$ and \\textsc{[O\\,iii]}$\\lambda$5007\/H$\\beta$\\ is slightly positive within the errors of the fit. However, according to the criterion presented for the $y^+$ vs. \\textsc{[O\\,iii]}$\\lambda$5007\/H$\\beta$\\ relation, the derived helium abundance for these icf's would be effectively consistent with no positive gradient.\nThe mean total helium abundance ranges between 0.086 and 0.091, depending on the assumed icf.  This range is larger by a factor $\\sim$2 than  the uncertainties due to the measurements  (assumed to be traced by  the standard deviation).  This implies that the main source of uncertainty is still in the assumptions taken on the way to the derivation of the final abundance and highlight how,\neven with a level of data quality as high as those utilized here (i.e. data allowing to determine locally the physical conditions of the gas, the fluxes of several helium lines and estimations of the contribution of underlying stellar populations), achieving an uncertainty $\\lsim1$\\% is extremely difficult.\n\nThe main conclusion of Fig. \\ref{ionidegvsy} is that the relation between the total helium abundance, $y_{tot}$, and the excitation of the gas, as traced by \\textsc{[O\\,iii]}$\\lambda$5007\/H$\\beta$\\  is consistent with a lack of gradient in  helium abundance.\nHowever, if the extra helium were produced by the Wolf-Rayet stars, the required amount to be detected would be tiny in comparison to the pre-existing helium. This implies that  a positive slope in Fig. \\ref{ionidegvsy} consistent with this enrichment would be indistinguishable of the presented fits.\nExploiting these data to their limits, we can derive the average excess in $N\/He$, by comparing the mean abundances in the N-enriched and non N-enriched areas.\nA map for the $N\/H$ abundance can be derived using the $O\/H$ and $N\/O$ maps presented in \\citetalias{mon12}. Assuming a value of $\\log(N\/O=-1.3)$ as the limit above which the interstellar medium is enriched in nitrogen, we find a mean $N\/H$ of $8.4\\times10^{-6}$ and $17.4\\times10^{-6}$ in the spaxels without and with enrichment. This implies a flux weighted  excess in nitrogen of $N\/H_{exc}=8.9\\times10^{-6}$.\nProceeding in the same manner with the different estimations of the total helium abundance, we find mean $He\/H$ values ranging between $8.49\\times10^{-2}$ and $9.05\\times10^{-2}$ for the non-enriched spaxels and between $8.68\\times10^{-2}$ and $9.14\\times10^{-2}$, depending on the assumed icf(He). Differences in helium abundances between the enriched and non-enriched zones are $\\sim0.9-1.9\\times10^{-3}$. The lowest value was derived using our largest icf(He) (i.e. Eq. \\ref{eqm13}) which, by construction, was defined to minimize any helium abundance gradient. The largest value compares well with the stellar atmospheric $N\/He$ ratios for an N-type Wolf-Rayet star \\citep[e.g.][]{smi82}. However, it is also at the limit of the uncertainties ($\\sim2\\times10^{-3}$assumed to be traced by the standard deviation for the spaxels under consideration in each separate group). \nTherefore, these data appear to be marginally in accord with the hypothesis of a putative enrichment of helium due to the Wolf-Rayet population in the main H\\,\\textsc{ii}\\ region, but more importantly, they stress the difficulties in pushing this methodology further in order to  confirm this contamination without doubt.\n\n\\section{Discussion}\n\n\\subsection{Kinematics and radiative transfer effects}\n\nThe  standard approach in the literature to estimate radiative transfer effects assumes\na negligible influence of the movements of the gas in the H\\,\\textsc{ii}\\ region. i.e. a static nebula \\citep[$\\omega=v\/v_{th}=0$, in the formalism presented by ][]{rob68}. This is the approach utilized in Sec. \\ref{secymas}. \nThe general use of this assumption is motivated by the difficulty of obtaining data of sufficient  quality to allow, on top of measuring the flux of several helium lines with good signal to noise, tracing their profiles and clearly identify the different kinematic components in a consistent manner in all of them.\nHowever, it is not unusual for starburst galaxies or H\\,\\textsc{ii}\\ regions to present velocity gradients and\/or relatively high velocity dispersion that can be attributed to outflows (or expanding structures).\nIn particular, the kinematic study presented in \\citetalias{mon10} and  \\citetalias{wes13} showed that movements in this H\\,\\textsc{ii}\\ region are significant and can indeed be attributed to an outflow caused by the two embedded Super Star Clusters. Specifically, in  \\citetalias{mon10}, we detected a relatively static (i.e. small velocity gradient) narrow component on top of a much broader component ($\\sigma\\sim20-25$ km s$^{-1}$) with a velocity gradient of $\\Delta v\\sim70$~km~s$^{-1}$. An additional component was also detected with the higher resolution GMOS data  \\citepalias{wes13}. Therefore, these data provide a very good opportunity to explore what is the impact  of the kinematics in the derivation of the optical depths from an empirical point of view.  \n\n  \\begin{figure}[th!]\n   \\centering\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg14a.ps}\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=,bb = 45 30 405 390]{.\/fg14b.ps}\\\\\n   \\caption[$\\tau(3889)$ for the two kinematic components]{ $\\tau(3889)$ for the two main kinematic components presented in Fig. 19 of \\citetalias{mon10} derived from the fits to relations presented by \\citet{rob68} for $T_e$=10\\,000~K. \\emph{Left:} Narrow kinematic component and $\\omega=0$.  \\emph{Right:} Broad kinematic component and $\\omega=3$. The $\\tau$(3889) map for this component using the relation for $\\omega=0$ (not shown) displays the same structure but with values $\\sim4-6$ times smaller.}\n \\label{taucompos}\n \\end{figure}\n\nSince this is only an exploratory analysis and a consistent approach based on the identification and modeling of multiple kinematic components in \\emph{several} emission lines would be more complex (and\nnot supported by the signal-to-noise of all the He\\,\\textsc{i}\\ data) than the one presented in Sec. \\ref{secymas}, we opt for a simpler analysis based on the  He\\,\\textsc{i}$\\lambda$6678 and He\\,\\textsc{i}$\\lambda$7065 lines. These were the pair of lines utilized to estimate $\\tau(3889)$ in the \\emph{red set}. Also both were observed with the same instrumental set-up, which was the same as for H$\\alpha$\\ in the multicomponent analysis presented in  \\citetalias{mon10}. Therefore, we could link on a spaxel-by-spaxel basis the multicomponent analysis utilized in \\citetalias{mon10} to similar components in  He\\,\\textsc{i}$\\lambda$6678 and He\\,\\textsc{i}$\\lambda$7065.\nThe contribution to the total flux from each component varies from spaxel to spaxel as well as with the line in consideration. Specifically, the broad component presents a larger contribution  (i.e. $\\sim35-60$\\%) to He\\,\\textsc{i}$\\lambda$7065 than to He\\,\\textsc{i}$\\lambda$6678 ($\\sim30-50$\\%). This implies different correction factors, $f_\\tau(7065)$, for the narrow and broad component that can be converted into optical depths as in Sec. \\ref{secymas}.\nGiven the velocity gradient and the velocity dispersion measured for the broad component, values for $\\omega=v\/v_{th}=3$ are more appropriate for this component. Fitting the same functional form as in Eqs. \\ref{eqfac7065}-\\ref{eqfac3889} to the values reported in \\citet{rob68} for  $\\omega=v\/v_{th}=3$ and $T_e=10\\,000$~K, we derived the following relation:\n\n\\begin{equation}\nf_\\tau(7065)_{\\omega=3} = 1 + 0.279\\tau^{0.521}\n\\end{equation}\n\nThe structure of the derived optical depth for both the narrow and the broad components is shown in Fig. \\ref{taucompos} and several conclusions can be drawn from this figure. Firstly, a comparison between the left panel of this figure with those presented in Fig. \\ref{maptau} highlights the similarity between the optical depth maps derived for the narrow component and for the line integrated analysis. To our knowledge, this is the first empirical evidence supporting the traditional assumption that movements in extragalactic H\\,\\textsc{ii}\\ regions with  a known outflow have a negligible effect in the estimation of the global optical depth. \n\nSecondly, a comparison between the two maps in Fig. \\ref{taucompos}  shows how the \\emph{structure} of  the two $\\tau(3889)$ is different: while the narrow component $\\tau(3889)$ displays a clear peak associated with the peak of emission for the GH\\,\\textsc{ii}R\\ and decreases outwards, the broad component  remains relatively high over a large area of $\\sim4^{\\prime\\prime}\\times2^{\\prime\\prime}$ ($\\sim$74~pc$\\times$37~pc) centered on the double cluster.\n\nThirdly, the comparison between the specific values displayed in these two graphics shows that the broad component suffers from radiative transfer effects to a larger degree than the narrow one. This is a consequence of the adopted relation between $f_\\tau(7065)$ and $\\tau(3889)$, which is motivated by our kinematic results. If we had used Eq. \\ref{eqfac7065} (i.e. assuming $\\omega=0$ for the \\emph{broad} component), we would have obtained comparable values for the $\\tau(3889)$ in both components (although, still, with a different structure since this is determined by the $f_\\tau(7065)$ itself, which in turns depends on the relative flux between He\\,\\textsc{i}$\\lambda$6678 and He\\,\\textsc{i}$\\lambda$7065).\n\nThere is certainly room for improvement in this analysis. For example, although the multicomponent analysis treats in a consistent way the line profiles of He\\,\\textsc{i}$\\lambda$6678 and He\\,\\textsc{i}$\\lambda$7065, the same physical conditions (i.e. $T_e$, $n_e$, extinction) had to be assumed for both components. Moreover, we used here only two helium lines, in contrast with the more canonical methodology for determining helium abundances based in a larger set of helium lines. These improvements suggest a new approach to the study of the helium content in galaxies with IFS. Specifically,  the 2D analysis presented in the previous sections together with the separate analysis of different kinematic components along the line of sight (i.e. in a given spaxel) constitute a 3D view of the radiative transfer effects in the nebula.\n\n\\subsection{Relation between collisional and radiative transfer effects and other properties of the ionized gas} \n\nWe have evaluated  locally both collisional and radiative transfer effects in the helium lines, as well as several important physical conditions ($n_e$, $T_e$, excitation) and chemical properties (i.e. relative abundance $N\/O$) of the ionized gas. Therefore, the central part of \\object{NGC~5253} constitutes a good case study to explore whether there is a (strong) spatial relation of collisional and\/or radiative transfer effects with other gas properties.\n\nAs discussed in Sec. \\ref{seccoli}, theory predicts a strong dependence of the collisional effects on the electron density and, to a lesser extent, on the electron temperature. This implies a strong spatial correlation with the these properties. Indeed, when all the spaxels in the area under study are considered, the relations between $C\/R(\\lambda7065)$ vs.  $n_e$ and $T_e$ have Pearson's correlation coefficients of 0.99 and 0.71, respectively. Regarding the other two properties,  Fig. \\ref{c2rvscosas} presents the relation between the collisional effects (as traced by $C\/R$(7065)) and the excitation (as traced by \\textsc{[O\\,iii]}$\\lambda$5007\/H$\\beta$) and the relative abundance of nitrogen: $N\/O$.  The excitation presents a degree of correlation similar to  $n_e$. The correlation for $N\/O$ is not as strong as that for $n_e$ but still comparable to that for $T_e$.\n\n  \\begin{figure}[th!]\n   \\centering\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=]{.\/fg15a.ps}\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=]{.\/fg15b.ps}\\\\\n   \\caption[C\/R ratios]{Relation between collisional effects as traced by the C\/R ratio for the $\\lambda$7065 line and the excitation (\\emph{left}) and relative abundance of nitrogen, N\/O (\\emph{right}). The Pearson's correlation coefficients are indicated in the corners of the individual plots.\n \\label{c2rvscosas}}\n \\end{figure}\n\n  \\begin{figure}[th!]\n   \\centering\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=]{.\/fg16a.ps}\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=]{.\/fg16b.ps}\\\\\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=]{.\/fg16c.ps}\n\\includegraphics[angle=0,width=0.24\\textwidth,clip=]{.\/fg16d.ps}\\\\\n   \\caption[$\\tau(3889)$ vs. physical and chemical properties]{Relation between the self-absorption effects as traced by $\\tau(3889)$  and different physical and chemical properties in \\object{NGC~5253}.  From left to right and from top to bottom, these are electron density and temperature, excitation and relative abundance of nitrogen, N\/O.  The correlation coefficients  are indicated in the corners of the individual plots.\n \\label{tauvscosas}}\n \\end{figure}\n\nLikewise, Fig. \\ref{tauvscosas} shows the relation between radiative effects, as traced by $\\tau$(3889) and different properties of the gas. As expected, there is a good correlation with electron density. For electron temperatures lower than $\\sim$10\\,700~K, He\\,\\textsc{i}\\ emission lines do not suffer from radiative transfer effects in a significant way. However, above this threshold there is a clear correlation between the temperature and the relevance of the radiative transfer effects. \n\nMore interestingly, as happened with the collisional effects, there is a good correlation between the excitation and the relevance of radiative transfer effects (lower left panel in Fig. \\ref{tauvscosas}).  It is not clear why the excitation should correlate with the contribution of  collisional effects (mainly) or the radiative transfer effects (to a lesser extent). \n\nSomething similar occurs with the relative abundance, $N\/O$. At the typical $\\log(N\/O)$ values for this galaxy ($\\lsim-1.3$), He\\,\\textsc{i}\\ radiative transfer effects appear to be negligible. However, those locations displaying extra nitrogen appear to show significant radiative transfer effects. Rather than a discernible correlation between individual spaxels, there appears a correlation between the upper limit  to the contribution of the radiative transfer effects and the amount of extra nitrogen. To our knowledge, there is no reason to link these two properties from the theoretical point of view. The fact that both the relations with $T_e$ and $N\/O$ present bimodal behavior (i.e. with \"turning points\"), and that there is a similar structure for the electron temperature and relative abundance \\citepalias{mon12}, implying a good correlation (i.e. $r=0.91$) between $T_e$ and $N\/O$, points towards the local electron temperature as a common cause.\n\n\\section{Conclusions}\n\nThis is the fourth in a series of articles that make use of IFS-based data to study in detail the 2D physical and chemical properties of the gas in the main GH\\,\\textsc{ii}R\\ of the nearby BCD \\object{NGC~5253}. The main goal of this article was to estimate the contribution of the collisional and radiative transfer effects on the helium emission lines and to map, in turn, the helium abundance.\nThe major conclusions can be summarized as follows:\n\n   \\begin{enumerate}\n      \\item The collisional effects on the different helium transitions have been mapped for the first time in an extragalactic object. As expected, they reproduce the electron density structure. They are negligible (i.e. $\\sim$0.1-0.6\\%) for transitions in the singlet cascade while relatively important  for those transitions in the triplet cascade. In particular, they can contribute up to 20\\% of the flux in the He\\,\\textsc{i}$\\lambda$7065 line.\n      \n      \\item The contribution of the collisional effects is sensitive to the assumed $T_e$ for helium. Specifically, we found  differences $\\sim25-35$\\% for the $\\lambda$3889 and $\\lambda$7065 lines two reasonable assumptions for the $T_e$ sensitivity. Relative differences for the other lines were larger. However, this does not have important consequences  since the contribution of the collisional effect to the observed spectrum for these lines is negligible.\n\n      \\item We present a map for the optical depth at $\\lambda$3889 in the main GH\\,\\textsc{ii}R\\ of \\object{NGC~5253}. $\\tau(3889)$ is elevated over an extended and circular area of $\\sim$30~pc in diameter, centered at the Super Star Cluster(s), where it reaches its maximum. \n\n      \\item The singly ionized helium abundance, $y^+$, has been mapped using extinction corrected fluxes of six He\\,\\textsc{i}\\ lines, realistic assumptions for $T_e$, $n_e$, and the stellar absorption equivalent width as well as the most recent emissivities. We found a mean($\\pm$ standard deviation of $10^3 y^+ \\sim80.3(\\pm2.7)$ over the mapped area.\n      \n      \\item We derived total helium abundance maps using three possible icf(He)'s. \nThe relation between the excitation and the total helium abundances is consistent with no abundance gradient.\nDifferences between the derived total abundances according to the three methods are \\emph{larger} than statistical errors associated with the data themselves, emphasizing how uncertainties in the derivation of helium abundances are dominated by the adopted assumptions.\n      \n      \\item We illustrated the difficulty of detecting a putative helium enrichment due to the presence of Wolf-Rayet stars in the main GH\\,\\textsc{ii}R. This is due to the comparatively large amount of preexisting helium. The data are marginally consistent with an excess in the $N\/He$ ratio in the nitrogen enriched area of the order of the atmospheric $N\/He$ ratios in W-R stars. However, this excess is also of the same order of the uncertainty estimated for the $N\/He$ ratios in the nitrogen enriched and non-enriched areas.\n      \n      \\item We explored the influence of the kinematics in the evaluation of the He\\,\\textsc{i}\\ radiative transfer effects. Our data empirically support  the use of the traditional assumption that motions in an extragalactic H\\,\\textsc{ii}\\ region have a negligible effect in the estimation of the global optical depths. However, individually, the broad kinematic component (associated with an outflow) is affected by radiative transfer effects in a much more significant way than the narrow one.\n      \n      \\item The local relationships between the contribution of collisional and radiative transfer effects to the helium lines and different physical and chemical properties of the gas have been explored. Interestingly, we found a relation between the amount of extra nitrogen and the upper limit of the contribution from radiative transfer effects that requires further investigation. We suggest the electron temperature as perhaps a common agent causing this relation.\n      \n \\end{enumerate}\n\n\\begin{acknowledgements}\n\nWe are very grateful to the referee for the careful and diligent reading of the manuscript as  well as for the useful comments that helped us to clarify and improve the first submitted version of this paper.\nAlso, we thank R. L. Porter for advice on use of his tabulated He\\,\\textsc{i}\\ emissivities and for so promptly informing us of the Corrigendum.\n\nBased on observations carried out at the European Southern\nObservatory, Paranal (Chile), programmes 078.B-0043(A) and\n383.B-0043(A). This paper uses \nthe plotting package \\texttt{jmaplot}, developed by Jes\\'us\nMa\\'{\\i}z-Apell\\'aniz,\n\\texttt{http:\/\/dae45.iaa.csic.es:8080\/$\\sim$jmaiz\/software}. This \nresearch made use of the NASA\/IPAC Extragalactic \nDatabase (NED), which is operated by the Jet Propulsion Laboratory, California\nInstitute of Technology, under contract with the National Aeronautics and Space\nAdministration.\n\n\nA.~M.-I. is supported by the Spanish Research Council within the program JAE-Doc, Junta para la Ampliaci\\'on de Estudios, co-funded by the FSE.\nA.M-I is also grateful to ESO - Garching, where part of this work was carried out, for their hospitality and funding via their visitor program. \nThis work has been partially funded by the Spanish PNAYA, project AYA2010-21887 of the Spanish MINECO. \nThe research leading to these results has received funding from the European Community's Seventh Framework Programme (\/FP7\/2007-2013\/) under grant agreement No 229517.\n\\end{acknowledgements}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION}\n\n\nTime series forecasting is playing a vital role in many application scenarios of a variety of domains, such as solar power generation forecasting in renewable energy \\citep{10,11,mtgnn}, traffic forecasting in transportation system \\citep{traffic1,traffic2},\nelectricity consumption forecasting in social life \\citep{10,11,mtgnn}, market trend prediction in financial investment~\\citep{finance1,finance2}, etc.\nIn many of them, there exist multiple forecasting entities, e.g. power stations in the solar system, stations in the traffic system, stocks in financial market, and commodities in retailing business. \n\n\nA straightforward forecasting solution is to mine temporal dependency for each individual entity by 1d-convolution neural network (1D-CNN) \\citep{tcn-2018}, recurrent neural network (RNN) \\citep{lstm,gru1}, transformer \\citep{attention-is-all-you-need}, and etc., while it will overlook important relations between these entities, such as explicit relation defined by human knowledge (e.g. competitive, cooperative, causal, and geospatial relation) and other implicit relation behind the data.\nIn reality, such relations could provide valuable signals towards accurate forecasting for each individual entity.\nFor example, in traffic system, the station can affect its geospatial adjacent stations;\nthe stock price of upstream companies in the supply-chain can substantially indicate that of downstream ones in the financial market scenario; the similarity relation between entities behind the data can also increase the robustness of model. \nSpatial-temporal graph neural networks \\cite{dcrnn,stmetanet,graphwavenet,mtgnn}, in which the entity relations are fully utilized, are good examples with better forecasting performance.\n\n\nIn many real-world scenarios, however, beside some explicit relations, there usually exist crucial yet implicit relations between entities.\nRecently, a growing number of research works pay attention to graph neural network (GNN) \\citep{gcn,graphsage,gat,protein1,physical-system1} to leverage the implicit relations between entities~\\citep{ddgf,graphwavenet,Gumbel-graph-neural-network-2019,mtgnn}. \nOne common idea among these works is to assume all entities composing a complete graph and let GNNs automatically learn the pairwise correlations between any two entities~\\citep{ddgf,graphwavenet}. \nTo avoid the computation complexity and over-smoothing issues of above-mentioned studies, the operation analogous to sparse encoding~\\citep{mtgnn} has been further proposed.\nNevertheless, an imprudent employment of sparse constraints at the earlier learning phase may limit the model capacity of discovering crucial relation, while the uni-directional relation is not suitable for every real scenario.\n\n\nMoreover, there also exist critical challenges regarding the utilization of multiple relations, one of which is the information aggregation through different types of relations, especially the co-existence of explicit relation as well as implicit relation.\nA straightforward method is to employ different graph convolution blocks for different relation types and then conduct a direct fusion \\citep{graphwavenet}. Some others conduct fusion based on correlation between nodes by leveraging the meta information of both nodes and edges \\citep{stmetanet}. In addition, Graph multi-attention network (GMAN) \\citep{gman} employs a gated fusion mechanism to fuse the spatial and temporal representations. \nHowever, all of these efforts rely on static fusion of multiple relations reflecting either pre-existed or latent connections. Indeed, given the distinct characteristics of multiple relations, accurate forecasting usually relies on dynamic reliance on them under various circumstances.\n\n\n\n\\begin{figure}[!htbp]\n\\centerline{\\includegraphics[width=1\\linewidth]{photo\/demo_show_case_of_a2gnn.png}}\n\\caption{Diagrammatic representation of the relationships between entities.}\n\\label{fig:Diagrammatic representation}\n\\end{figure}\n\nThus, the problem lies in two crucial aspects, i.e., implicit relation {\\em discover} and relations' effective {\\em utilization}.\nWe take the $6$ entities in Fig. \\ref{fig:Diagrammatic representation} to illustrate the importance of them.\nIn Fig. \\ref{fig:Diagrammatic representation}, entity-(a), entity-(b), and entity-(c) have a similar curve, thus, the similarity can be employed to enhance the robustness of the model.\nAlthough there does not exist a similar curve for entity-(e), it is not hard to see that when entity-(c) or entity-(d) is in lowest peak, entity-(e) is also in lowest peak. Thus, entity-(e) can use this relation for better forecasting.\nThe periodicity of entity-(f) is strong so that only employing its own information can also get an accurate forecasting result (i.e. the explicit and implicit relation may be not necessary for entity-(f)). \n\n\nIn this paper, in order to mine the implicit relation between entities as much as possible and effectively utilize the relations to improve the forecasting performance, we propose an attentional multi-graph neural network with automatic graph learning (A2GNN).\nIn particular, we propose a novel auto graph learner based on Gumbel-Softmax~\\citep{gumbelsoftmax} to sample all feasible entity pairs so that the relation between any entity pair has the chance to be reserved in the learned graph, while we leverage the sparse matrix to ensure the computing efficiency. Moreover, we propose an attentional relation learner so that every entity can dynamically pay attention to useful relations, resulting in more flexible utilization of multiple relations and consequently better forecasting performance.\n\n\n\n\nThe main contributions of this paper include:\n\\begin{itemize}\n\\item We propose a new graph neural network framework, A2GNN, with automatic discovery and dynamic utilization of relations for time series forecasting.\n\\item Within A2GNN, a novel auto graph learner based on Gumbel-Softmax~\\citep{gumbelsoftmax} can effectively discover the implicit relation between entities while significantly reducing complexity.\n\\item Within A2GNN, an attentional relation learner enables every entity dynamically pay more attention to their preferable relations, resulting in more flexible utilization of multiple relations.\n\\item \nOur proposed A2GNN is quite general such that it can be applied to both time series and spatial-temporal forecasting tasks.\nAnd, extensive experiments have shown that our method outperforms the state-of-the-art methods on a couple of well-known benchmark datasets.\n\\end{itemize}\n\n\n\n\\section{RELATED WORK}\n\\label{related work}\nOur work is related to three lines of research: time series forecasting methods, spatial-temporal graph neural networks, and graph neural networks sparsification methods.\n\n\\subsection{Time Series Forecasting Methods}\n\nThere are plenty of works on time series forecasting problem \\citep{5,7,8,gru1,gru2,lstm}. Recently, long and short-term time-series network (LSTNet) \\citep{10} utilizes the convolution neural network (CNN) \\citep{cnn} and recurrent neural network (RNN) to extract short-term local dependency and long-term patterns for time series trends. Shih proposes a temporal pattern attention (TPA-LSTM)\\citep{11} to select relevant time series, and leverages its frequency domain information for multivariate forecasting. A Lot of studies \\citep{unfoldingtemporaldynamics-2016-aaai,restful-wu-2018,dsanet-huang-2019} also focus on multi-scale temporal information extraction. Existing methods focus more on time series information utilization for forecasting, but they neglect the implicit relation among entities. \n\n\n\n\n\\subsection{Spatial-Temporal Graph Neural Networks}\n\nA spatial-temporal forecasting task has pre-defined relation (i.e. explicit relation) by human knowledge, and traffic forecasting is a typical and hot problem of spatial-temporal forecasting as the natural geographic relation. Similar like time series task, a lot of works focus on temporal information mining, such as deep spatio-temporal residual networks (ST-ResNet) \\citep{stresnet}, spatio-temporal graph convolutional network (STGCN) \\citep{stgcn}, Graph-WaveNet \\citep{graphwavenet,wavenet}. Some works focus on natural geographic relation utilization, such as diffusion convolutional recurrent neural network (DCRNN) \\citep{dcrnn}, spatial-temporal forecasting with meta knowledge (ST-MetaNet) \\citep{stmetanet}, graph multi-attention network (GMAN) \\citep{gman}, multi-range attentive bicomponent GCN (MRA-BGCN) \\citep{mrabgcn}, spatial temporal graph neural network (STGNN) \\citep{STGNN},  Spectral temporal graph neural network (StemGNN) \\citep{StemGNN}, and etc. \n\n\n\n\n\n\n\nImplicit relation discover aroused researcher's attention. Graph convolutional neural network with data driven graph filter (DDGF) \\citep{ddgf} breaks this limitation and discovers implicit relation to replace the pre-defined relation, by calculating all pairwise correlations between nodes. However, the risk of over smoothing and computation complexity increase when taking all nodes as neighbors. \nGumbel Graph Network (GGN) \\citep{Gumbel-graph-neural-network-2019}  proposes a model-free, data-driven deep learning framework to accomplish the reconstruction of network connections, while the one time sampler will curb the efficiency of connection reconstruction.\nMultivariate time graph neural network (MTGNN)~\\citep{mtgnn} accelerates the computation efficiency in traffic forecasting by employing a sparse uni-directional graph to learn hidden spatial dependencies among variables. Nevertheless, a imprudent employment of sparse constraints at the earlier learning phase may limit the model capacity of discovering crucial relation and the assumption of uni-direction limits its application in many real scenarios. \n\n\n\n\\subsection{Graph Neural Networks Sparsification Methods}\nGraph sparsification aims at finding small subgraphs from given implicit large graphs that\nbest preserve desired properties. \nFor instance, Fast learning with graph neural networks (FastGCN) \\citep{fastgcn-2018-ICLR}  interpret graph convolutions as integral transforms of embedding functions under probability measures and uses Monte Carlo approaches to consistently estimate the integrals.\nNeuralSparse \\citep{graphsparse-2020-ICML} considers node\/edge features as parts of input and optimizes graph sparsification by supervision signals from errors made in downstream tasks.\n\n\n\\begin{figure*}[ht!]\n\\centerline{\n\\includegraphics[width=0.75\\linewidth]{photo\/framework_of_A2GNN.png}\n}\n\\caption{The framework of A2GNN. AGL is the auto graph learner for implicit graph discover. ARL is the attentional relation learner that enables each entity dynamically pay attention to different relations. $\\mathbf{m}_{v_i}$ is the learnable node embedding of node $v_i$. $\\mathbf{x}_{v_i}$ is historical observed features. \n$\\mathbf{h}^1_{v_i}$, $\\mathbf{h}^2_{v_i}$, and $\\mathbf{h}^3_{v_i}$ mean own temporal information, neighbor information by implicit relation, and neighbor information by explicit relation, respectively. $p_{v_i,1}$, $p_{v_i,2}$, and $p_{v_i,3}$ are  dynamic weight given by ARL.}\n\\label{fig:framework of A2GNN}\n\\end{figure*}\n\n\\section{PROBLEM FORMULATION}\nIn this section, we first provide a detailed problem formulation. Suppose there are $N$ forecasting entities, we use $x_t\\in R^{N\\times d}$ to stand for the values of these $N$ entities at time step $t$, and $d$ is the feature dimension. $y_{t}\\in R^{N\\times 1}$ is the feature\/variable need to forecast. The historical observations of past $t_{in}$ steps before $t$ is defined as $X=\\{x_{t-t_{in}+1},x_{t-t_{in}+2},\\cdots,x_t \\}$.\nOur goal is to build a function to predict a sequence of values $Y=\\{y_{t+1},y_{t+2},... ,y_{t+t_{out}}\\}$ for future $t_{out}$ steps.\n\nWe give formal definitions of graph-related concepts:\n\n\\textbf{\\textit{Graph: }}\nA graph is formulated as $G=(V,E)$ where $V$ is the set of nodes, and $E$ is the set of edges. We use $N=|V|$ to denote the number of nodes in the graph. In particular, in forecasting scenario, we consider each forecasting entity as a node in the graph.\n\n\\textbf{\\textit{Node neighbors: }}\nFor any node $v_i,v_j\\in V$, $(v_i,v_j)\\in E$ denote an edge between $v_i$ and $v_j$. The neighbors of $v_i$ are defined as $N_{v_i}=\\{v_k ~|~ (v_k,v_i)\\in E \\land v_k\\in V \\}$.\n\n\\textbf{\\textit{Weighted adjacency matrix: }}\nThe weighted adjacency matrix is a mathematical representation of the relation. We denote the weighted adjacency matrix as $A\\in R^{N\\times N}$, where $A_{i,j}>0$ if there is an edge between the node $v_i$ and node $v_j$, otherwise, $A_{ij}=0$. The weight $A_{i,j}$ stands for some metrics of relation between these two nodes. \n\n\n\n\\section{Proposed Framework}\nIn this section, we introduce the proposed attentional multi-graph neural network with automatic graph learning (A2GNN). \nAs shown in Figure~\\ref{fig:framework of A2GNN}, the whole framework of A2GNN consists of 5 parts: temporal encoder, auto graph learner (AGL), graph neural network (GNN), and attentional relation learner (ARL). The key parts of this study are AGL and ARL. In the end, we employ the multi-layer perception (MLP) for efficient inference instead of RNN.\n\nThe algorithm is shown in Alg. \\ref{alg:algorithm}. \nIt can be easily applied both spatial-temporal and time series forecasting tasks, and the difference between them is whether exist pre-defined relation. For better understand A2GNN, we make a brief introduction about its temporal encoder and a detailed introduction about the key parts: AGL and ARL.\n\n\n\n\n\\newcommand{\\parag}[1]{\\vspace{1mm}\\textit{#1}:\\ }\n\\parag{Temporal encoder} 1D-CNN~\\citep{tcn-2018,cnn}, RNN~\\citep{gru1,gru2,lstm}, and transformer~\\citep{attention-is-all-you-need}, can be employed to extract the temporal information for each node.\nThe representation will be used as input for the further graph neural network. \nIn this paper, we use LSTM to extract temporal information.\n\n\n\\begin{algorithm}[ht]\n\\caption{A2GNN}\n\\label{alg:algorithm}\n\\textbf{Input}: Node temporal information $X\\in R^{N\\times T_{in} \\times D}$; Pre-defined graph $A^p \\in R^{N\\times N}$(Optional).\\\\\n\\textbf{Parameter}: Node embedding $M\\in R ^ {N \\times D_m}$; Random initialized adjacent matrix $A\\in R^{N \\times N}$; Parameter of graph neural network; Parameter of auto relation learner.\\\\\n\\textbf{Output}: $\\hat{Y}\\in R^{N\\times T_{out} \\times D}$\n\\begin{algorithmic}[1]\n\\STATE{\/* Discover implicit relation *\/}\n\\IF {training} \n\\STATE $A^* = \\textsc{AGL}_{training}(A)$\n\\ELSIF{inference}\n\\STATE $A^* = \\textsc{AGL}_{inference}(A)$\n\\ENDIF\n\\STATE Extract temporal information: $S=\\textsc{LSTM}(X)$\n\\STATE Extract node own information: $H^1=\\textsc{MLP}(S)$\n\\STATE Aggregate implicit neighbor: $H^2=\\textsc{GNN}(S,A^*)$\n\\STATE Aggregate pre-defined neighbor: $H^3=\\textsc{GNN}(S,A^p)$\n\\STATE Utilize multiple relations: $Z=\\textsc{ARL}(M,H^1,H^2,H^3)$\n\\STATE \/* efficient inference *\/\n\\STATE $Y=ZW$\n\\STATE \\textbf{return} $\\hat{Y}$\n\\end{algorithmic}\n\\end{algorithm}\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Auto Graph Learner} \n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=1\\linewidth]{photo\/auto_graph_learner_process.png}\n    \\caption{The process of auto graph learner in training. }\n    \\label{fig:auto graph learner process}\n\\end{figure}\n\n\nAuto graph learner (AGL) kicks off the learning for implicit relation from a randomly initiated adjacency matrix. Towards minimizing the over smoothing risk, we use sample-based method to select potential edges. Moreover, multiple edges of each node are sampled for an efficient training.\nHowever, the vanilla sample operation is not differentiable, which hinders the back propagation for edges' weight optimization. To address this issue, we employ Gumbel-Softmax \\citep{gumbelsoftmax} method to make a differentiable sampling. There are some differences in training and inference phases and the detailed procedures are shown in below.\n\n\n\\parag{Training} Given the randomly initiated adjacency matrix $A\\in R^{N\\times N}$, the calculation steps include: \n\n1. $\\forall v_j\\in N_{v_i}$, we employ a softmax function to compute the sampling probability of each edge:\n\\begin{equation}\n    \\pi_{v_i,v_j}=\\frac{\\exp(a_{v_i,v_j})} {\\sum_{v_k\\in N_{v_i}} \\exp(a_{v_i,v_k})},\n    \\label{eqn:matrix softmax}\n\\end{equation}\nwhere $\\pi_{v_i,v_j}$ is the weight of edge $(v_i,v_j)$ after softmax, and $a_{v_i,v_j}$ is the weight of edge $(v_i,v_j)$ in adjacency matrix $A$. \n\n2. We then generate differentiable samples through Gumbel-Softmax:\n\\begin{equation}\n\t\t\t\\pi_{v_i,v_j}^{G}=\\frac{\\exp( (\\log(\\pi_{v_i,v_j})+\\epsilon_{v_j})\/\\tau )}   {\\sum_{v_k\\in N_{v_i}} \\exp( (\\log(\\pi_{v_i,v_k})+\\epsilon_{v_k})\/\\tau )},\n\\label{eqn:gumbel softmax}\n\\end{equation}\nwhere $\\pi_{v_i,v_j}^{G}$ is the weight of edge $(v_i,v_j)$ after Gumbel-Softmax.  $\\epsilon=-\\log(-\\log(s))$, with randomly generated $s$ from Uniform distribution U$(0,1)$, and $\\tau \\in {(0,+\\infty)}$ is a hyper-parameter called temperature. As the softmax temperature $\\tau$ approaches $0$, samples from the Gumbel-Softmax distribution approximate one-hot vector, which means discrete.\n\n\n3. Repeating above procedure $C$ times, we can obtain $C$ samples. Finally, we apply normalization over all samples to get the final correlations for all nodes:\n\\begin{equation}\n\t\t\ta^*_{v_i,v_j}= \\frac{\\sum_{c=1}^{C} \\pi ^{G,c}_{v_i,v_j} }   {\\sum_{c=1}^{C}\\sum_{v_k\\in N_{v_i}} \\pi^{G,c}_{v_i,v_k}},\n\\label{eqn:training rate}\n\\end{equation}\nwhere $a^*_{v_i,v_j}$ is the weight of edge $(v_i,v_j) $and will be further used in the calculation procedures in the graph neural network. $\\pi ^{G,c}$ is the $c$-th Gumbel-Softmax result.\n\n\n\\parag{Inference} Since the auto graph learner has already learned the nodes' relation by updating matrix $A$ in the training phase, we replace Gumbel-Softmax sample operation by selecting the top related $C$ neighbors $\\textsc{top}_C (N_{v_i})$ for each node $v_i$ (the edge weight is used as the metric for $\\textsc{top}$ operation). Then softmax is applied to normalize these edge weights:\n\\begin{equation}\n    a^*_{v_i,v_j}=\\frac{\\exp(a_{v_i,v_j})} {\\sum_{v_k\\in \\textsc{top}_C (N_{v_i})} \\exp(a_{v_i,v_k})}, ~~v_j\\in \\textsc{top}_C (N_{v_i})\n    \\label{eqn:evaluation matrix softmax}\n\\end{equation}\n\nThe $C$ times sampling operation ensures the rationality and efficacy of relation learning process.\nAt the begin of training, all the edges have the probability to be sampled, and the sampled edge's weight can be properly updated.\nAs time goes on, the useful edges will have larger weights through the updating with back propagation algorithm.\n\n\nThe output of AGL is $A^*\\in R^{N\\times N}$, and the element of which is $a^*_{v_i,v_j}$.\n\n\n\\subsection{Graph Neural Network}\nThe node's time series information is processed by LSTM before input into the graph neural network. Specifically, the hidden state from LSTM in each time step will be concatenated as the output. The output from LSTM block (i.e. temporal encoder) is defined as $S=\\textsc{LSTM}(X)$, and $S\\in R^{N\\times d_{enc}}$. \n\n\n\nWith the assistance of the output from LSTM, the node itself information is processed by multi-layer perceptron (MLP) like $H^1=\\textsc{MLP}(S)$.\nThe neighbor information by implicit relation is aggregated by graph neural network like \n$H^2=A^* \\cdots (A^* S W^{(1)}_2) \\cdots W^{(l)}_2$, and $l$ is the layer depth, $W$ is learnable parameter for feature transportation. For some scenarios, such as traffic, there always exists natural geographic relation. Thus, the pre-defined natural relation can also be employ by graph neural network like $H^3=A^p \\cdots (A^p S W^{(1)}_3) \\cdots W^{(l)}_3$, and $A^p$ us the pre-dedined weighted adjacency matrix.\n\nThus, after the graph neural network, for each node $v_i$, the own information $h^1_{v_i}$, the aggregated information based on implicit relation $h^2_{v_i}$, and the aggregated information based on explicit relation $h^3_{v_i}$ are available from $H^1$, $H^2$, and $H^3$, respectively.\n\n\n\n\\subsection{Attentional Relation Learner}\n\nThe representations $H^1$, $H^2$, and $H^3$ are fed into the attentional relation learner (ARL).  Unlike previous studies that always rely on static fusion of multiple relations (e.g. explicit or implicit relation), AGL enables each node dynamically rely on different relations under various circumstances. Furthermore, node itself information, which can be called own relation, is maintained through $H^1$ instead of skip connection. \n\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=1\\linewidth]{photo\/attentional_relation_learner_process.png}\n    \\caption{The process of attentional relation learner.}\n    \\label{fig:attentional relation learner process}\n\\end{figure}\n\n\nAs show in Fig. \\ref{fig:attentional relation learner process}, in ARL, each node has a random initialized embedding vector, that is $M\\in R^{N\\times d_m}$. \nEach node learns its attention coefficient on multiple representations, that is $H^1$, $H^2$, and $H^3$. In particular, in Eq. \\eqref{eqn:attentional relation coefficient}, the dot production and $\\textsc{Softmax}$ operation are employed to calculate the attention coefficient.\n\n\n\n\n\\begin{equation}\n    p_{v_i,k\\in \\{1,2,3\\}} =\\textsc{Softmax}_k\\left( \\frac{(m_{v_i}W_{query})\\cdot(h^k_{v_i}W_{key}) }{\\sqrt{d}}\\right)\n    \\label{eqn:attentional relation coefficient}\n\\end{equation}\nwhere $p_{v_i,k\\in \\{1,2,3\\}}$ is the attention coefficient scalar of node $v_i$ on the representation $h^k_{v_i}$. $m_{v_i}$ is node $v_i$'s embedding from $M$. $h^k_{v_i}$ is node $v_i$'s representation from corresponding $H^k$. $W_{query} \\in R^{d_m \\times d}$ and $W_k \\in R^{d_h \\times d}$ are learnable parameters for feature transportation. $d_m$ is the dimension of $m_{v_i}$, $d_h$ is the dimension of $h^k_{v_i}$, $d$ is the output dimension of $W_{query}$ and $W_{key}$. $\\cdot$ means dot production.\n\n\n\n\n\n\nAfter that, the attention coefficient scalar is employed to merge multiple representations for each node $v_i$ as:\n\\begin{equation}\n\t\t\tz_{v_i} = \\textsc{Concat}_{k\\in \\{1,2,3\\}}\\left(p_{v_i}(h^k_{v_i}W_{value}) \\right)\n\\label{eqn: attentional relation learner attention concat}\n\\end{equation}\nwhere $W_{value}$ is the learnable parameter. \n\n\n\\subsection{Optimizaiton}\nFor each node $v_i$, final forecasting module will use the $z_{v_i}$ to predict, that is $\\hat{y}_{v_i} = z_{v_i}W$. The Root Mean Squared Error (RMSE) is employed as the loss function.\n\n\n\n\n\\section{Experiments}\nWe evaluate the proposed A2GNN framework in $5$ datasets: Solar-energy, Traffic, Electricity, METR-LA, and PEMS-BAY.\nParticularly, the first $3$ datasets are used to prove that A2GNN can be employed to model time series forecasting problem, and they have no pre-defined relation. And then, we make further experiments on $2$ well known spatial-temporal traffic datasets compared with state-of-the-art graph neural network method, and these datasets have pre-defined relation (i.e. natural geographic relation).  Details of these datasets are introduced in appendix.\n\n\n\n\nTo  evaluate  the  model  performances,  we adopt five metrics, which are Mean Absolute Error (MAE), Root Mean  Squared  Error  (RMSE),  Mean  Absolute  Percentage  Error(MAPE), Relative Squared Error (RSE) and Empirical Correlation Coefficient (CORR). The mathematical formulas are shown in appendix.\nAll the experiments $5$ times and report the average score in order to remove the influence of randomness (e.g. instability of Gumbel-Softmax operation and randomness of model initialized parameters). More settings of our experiments are shown in appendix.\n\n\n\n\\subsection{Baseline Methods for Comparison}\nAs we mentioned above, the biggest difference between time series forecasting task and spatial-temporal forecasting task lies in whether there exists a pre-defined relation. \nAll the methods mentioned in Section Related Work are concluded. The details of these baselines are shown in the appendix.\n\n\n\n\n\n\n\\subsection{Result Comparison}\n\\subsubsection{Result Comparison on Time Series Dataset}\nWe compare the performances of the proposed A2GNN model with above-mentioned baseline methods on $3$  well-known time series forecasting datasets, and we want to prove that our method can find the implicit relation and make a better prediction compared with other previous time series forecasting methods.\n\n\n\\begin{table}[ht!]\n\\renewcommand\\arraystretch{1.05}        \n\\renewcommand\\tabcolsep{2.0pt}\n\n\\caption{Experiments on time series forecasting datasets.}\n\\begin{center}\n\\scalebox{0.66}{\n\\begin{tabular}{cc|ccc|ccc|ccc}\n\\toprule\n\\multicolumn{2}{c|}{Dataset} & \\multicolumn{3}{c|}{Solar-Energy} & \\multicolumn{3}{c|}{Traffic} & \\multicolumn{3}{c}{Electricity} \\\\ \\hline\n\n\\multicolumn{2}{c|}{ } & \\multicolumn{3}{c|}{ Horizon} & \\multicolumn{3}{c|}{ Horizon} & \\multicolumn{3}{c}{ Horizon} \\\\ \\hline\n\n\n\\hline\nMethods & Metrics & 6 & 12 & 24  & 6 & 12 & 24 & 6 & 12 & 24\\\\ \\hline\nAR & RSE$\\downarrow$  & 0.379 & 0.591 & 0.869  & 0.621 & 0.625 & 0.630  & 0.103 & 0.105 & 0.105\\\\\n & CORR$\\uparrow$  & 0.926 & 0.810 & 0.531  & 0.756 & 0.754 & 0.751  & 0.863 & 0.859 & 0.859\\\\ \\hline\nVAR-MLP & RSE$\\downarrow$  & 0.267 & 0.424 & 0.684  & 0.657 & 0.602 & 0.614  & 0.162 & 0.155 & 0.127\\\\\n & CORR$\\uparrow$  & 0.965 & 0.905 & 0.714  & 0.769 & 0.792 & 0.789 & 0.838 & 0.819 & 0.867\\\\ \\hline\nGP & RSE$\\downarrow$  & 0.328 & 0.520 & 0.797  & 0.677 & 0.640 & 0.599  & 0.190 & 0.162 & 0.127\\\\\n & CORR$\\uparrow$  & 0.944 & 0.851 & 0.597 & 0.740 & 0.767 & 0.790  & 0.833 & 0.839 & 0.881\\\\ \\hline\nRNN-GRU & RSE$\\downarrow$  & 0.262 & 0.416 & 0.485  & 0.552 & 0.556 & 0.563  & 0.114 & 0.118 & 0.129\\\\\n & CORR$\\uparrow$  & 0.967 & 0.915 & 0.882  & 0.840 & 0.834 & 0.830 & 0.862 & 0.847 & 0.865\\\\ \\hline\nLSTNet & RSE$\\downarrow$ & 0.255 & 0.325 & 0.464  & 0.489 & 0.495 & 0.497 & 0.093 & 0.100 & 0.100\\\\\n & CORR$\\uparrow$ & 0.969 & 0.946 & 0.887 & 0.869 & 0.861 & 0.858 & 0.913 & 0.907 & 0.911\\\\ \\hline\nTPA-LSTM  & RSE$\\downarrow$ & 0.234 & 0.323 & 0.438 &  0.465 & 0.464 & 0.476 & 0.091 & 0.096 & 0.100\\\\\n& CORR$\\uparrow$ &  0.974 & 0.948 & 0.908 & 0.871 & 0.871 & 0.862 & 0.933 & 0.925 & 0.913\\\\ \\hline\nMTGNN & RSE$\\downarrow$ & 0.234 & 0.310 & 0.427 & 0.475 & 0.446 & 0.453 & 0.087 & 0.091 & \\textbf{0.095}\\\\\n & CORR$\\uparrow$ &  0.972 & 0.950 & 0.903 & 0.866 & 0.879 & 0.881  & 0.931 & 0.927 & 0.923\\\\ \\hline \\hline\n A2GNN & RSE$\\downarrow$  &\\textbf{ 0.223} &\\textbf{ 0.288} & \\textbf{0.407} & \\textbf{0.427}  &\\textbf{ 0.437 }& \\textbf{0.448} &  \\textbf{0.0858} &\\textbf{ 0.0903} & 0.0970\\\\ \n(ours) & CORR$\\uparrow$ &\\textbf{ 0.976} &\\textbf{ 0.958} &\\textbf{ 0.910}  &\\textbf{ 0.890 }& \\textbf{0.885} & \\textbf{ 0.881}  & \\textbf{0.934 }& \\textbf{0.929} & \\textbf{0.930}\\\\\n\\bottomrule\n\\multicolumn{11}{l}{$\\downarrow$ means lower is better and $\\uparrow$ means higher is better} \\\\\n\\end{tabular}}\n\\vspace{-0.5cm}\n\\label{tab:one step forecasting}\n\\end{center}\n\\end{table}\n\n\nFrom the Table \\ref{tab:one step forecasting}, we can see that A2GNN achieves best performance over almost all time steps on Solar-Energy, Traffic, and Electricity data. In particular, compared to previously state-of-the-art methods, A2GNN can achieve significant improvements in terms of the RSE score on Solar-Energy dataset with $4.6\\%$, $7.3\\%$ and $4.5\\%$ error reduction when $t_{out}$ is set to $6$, $12$ and $24$, respectively. Furthermore, on Traffic dataset, the error reduction in terms of RSE is $10.1\\%$, $1.8\\%$ and $1.0\\%$ correspondingly. \n\n\n\n\n\\begin{table}[ht]\n\\renewcommand\\arraystretch{1.05}        \n\\renewcommand\\tabcolsep{2.0pt}\n\\caption{Experiments on spatial-temporal forecasting datasets.}\n\\begin{center}\n\\scalebox{0.70}{\n\n\\begin{tabular}{c|ccc|ccc|ccc}\n\\toprule\n\\multicolumn{1}{c|}{Dataset} & \\multicolumn{9}{c}{METR-LA} \\\\  \\hline\n\n\\multicolumn{1}{c}{} & \\multicolumn{3}{|c|}{Horizon 3 } & \\multicolumn{3}{|c|}{Horizon 6} & \\multicolumn{3}{|c}{Horizon 12}\\\\\n\n\\hline\nMethods\/Metrics & MAE & RMSE & MAPE & MAE & RMSE & MAPE & MAE & RMSE & MAPE \\\\ \\hline\nDCRNN & 2.77 & 5.38 & 7.30\\% & 3.15 & 6.45 & 8.80\\% & 3.60 & 7.60 & 10.50\\%   \\\\\nSTGCN & 2.88 & 5.74 & 7.62\\% & 3.47 & 7.24 & 9.57\\% & 4.59 & 9.40 & 12.70\\% \\\\\nGraph-WaveNet & 2.69 & 5.15 & 6.90\\% & 3.07 & 6.22 & 8.37\\% & 3.53 & 7.37 & 10.01\\%\\\\\nST-MetaNet & 2.69 & 5.17 & 6.91\\% & 3.10 & 6.28 & 8.57\\% & 3.59 & 7.52 & 10.63\\%\\\\\nMRA-BGCN & 2.67 & 5.12 & 6.80\\% & 3.06 & 6.17 & 8.30\\% & 3.49 & 7.30 & 10.00\\%\\\\\nGMAN & 2.77 & 5.48 & 7.25\\% & 3.07 & 6.34 & 8.35\\% & 3.40 & 7.21 & 9.72\\%\\\\\nMTGNN & 2.69 & 5.18 & 6.86\\% & 3.05 & 6.17 & 8.19\\% & 3.49 & 7.23 & 9.87\\%\\\\\nA2GNN (ours)& \\textbf{2.63} & \\textbf{4.99} & \\textbf{6.74\\%} & \\textbf{2.95} & \\textbf{5.95} &\\textbf{ 8.02}\\% & \\textbf{3.34} & \\textbf{7.00} & \\textbf{9.65}\\% \\\\ \\hline\nDataset & \\multicolumn{9}{c}{PEMS-BAY} \\\\ \\hline\nDCRNN& 1.38 & 2.95 & 2.90\\% & 1.74 & 3.97 & 3.90\\% & 2.07 & 4.74 & 4.90\\% \\\\\nSTGCN& 1.36 & 2.96 & 2.90\\% & 1.81 & 4.27 & 4.17\\% & 2.49 & 5.69 & 5.79\\% \\\\\nGraph-WaveNet& 1.30 & 2.74 & 2.73\\% & 1.63 & 3.70 & 3.67\\% & 1.95 & 4.52 & 4.63\\% \\\\\nST-MetaNet& 1.36 & 2.90 & 2.82\\% & 1.76 & 4.02 & 4.00\\% & 2.20 & 5.06 & 5.45\\% \\\\\nMRA-BGCN& 1.29 & 2.72 & 2.90\\% & 1.61 & 3.67 & 3.80\\% & 1.91 & 4.46 & 4.60\\% \\\\\nGMAN& 1.34 & 2.82 & 2.81\\% & 1.62 & 3.72 & 3.63\\% & 1.86 & 4.32 &\\textbf{ 4.31}\\% \\\\\nMTGNN& 1.32 & 2.79 & 2.77\\% & 1.65 & 3.74 & 3.69\\% & 1.94 & 4.49 & 4.53\\% \\\\\nA2GNN (ours)& \\textbf{1.28} & \\textbf{2.70} & \\textbf{2.72}\\% &\\textbf{ 1.58} & \\textbf{3.62} & \\textbf{3.61}\\% & \\textbf{1.85} & \\textbf{4.29} & 4.39\\%\\\\\n\\bottomrule  \n\n\\multicolumn{10}{c}{%\n  \\begin{minipage}{12cm} %\n    \\small In these experiments, our goal is to forecast a sequence of values $Y=\\{x_{t+1},... ,x_{t+12}\\}$. We show the scores for $x_{t+3},\\,x_{t+6},\\text{ and } x_{t+12}$ for simplicity. %\n  \\end{minipage}%\n}\\\\\n\\end{tabular}\n\\label{tab:multi step forecasting}\n}\n\\vspace{-0.7cm}\n\\end{center}\n\n\\end{table}\n\n\n\n\n\n\\begin{table}[ht]\n\\renewcommand\\arraystretch{1.05}        \n\\renewcommand\\tabcolsep{2.0pt}\n\\caption{Experiments on spatial-temporal forecasting datasets (mean scores).}\n\\center\n\\scalebox{0.7}{\n\\begin{tabular}{c|ccc|ccc|ccc}\n\\toprule\n\\multicolumn{1}{c|}{Dataset} & \\multicolumn{9}{c}{METR-LA}  \\\\ \\hline\n\n\\multicolumn{1}{c}{} & \\multicolumn{3}{|c|}{Horizon 1$\\sim$3 } & \\multicolumn{3}{|c|}{Horizon 1$\\sim$6} & \\multicolumn{3}{|c}{Horizon 1$\\sim$12} \\\\\n\n\\hline\nMethods\/Metrics & MAE & RMSE & MAPE & MAE & RMSE & MAPE & MAE & RMSE & MAPE\\\\ \\hline\nStemGNN & 2.56& 5.06 & 6.46\\% & 3.01 & 6.03 & 8.23\\% & 3.43 & 7.23 & 9.85\\% \\\\\nSTGNN & 2.62 &  4.99 &  6.55\\%  & 2.98 &  5.88  & 7.77\\% &  3.49  & 6.94 &  9.69\\% \\\\\nA2GNN & \\textbf{2.43} & \\textbf{4.44} & \\textbf{6.12\\%} & \\textbf{2.65} & \\textbf{5.05} & \\textbf{6.93\\%} & \\textbf{2.92} & \\textbf{5.83} & \\textbf{8.05\\%} \\\\ \\hline\n& \\multicolumn{9}{c}{PEMS-BAY}\\\\ \\hline\nStemGNN& 1.23 & 2.48 & 2.63 & - & - & - & - & - & - \\\\\nSTGNN&  1.17 &  2.43 &  2.34\\%  & 1.46 &  3.27 &  3.09\\%  & 1.83 &  4.20 &  4.15\\% \\\\\nA2GNN& \\textbf{1.08} & \\textbf{2.14} & \\textbf{2.23}\\% & \\textbf{1.29} & \\textbf{2.75} & \\textbf{2.81\\%} & \\textbf{1.52} & \\textbf{3.42} & \\textbf{3.42\\%} \\\\ \\bottomrule  \n\\multicolumn{10}{c}{%\n  \\begin{minipage}{12cm} %\n    \\small As STGNN and StemGNN published the mean score of all predicted horizons in their paper. Thus, we show the mean score (e.g. Horizon 1$\\sim$3 represents the mean score of horizon $1,2,3$). %\n  \\end{minipage}%\n}\\\\\n\\end{tabular}\n\\vspace{-2cm}\n\n}\n\\label{tab:multi step forecasting (mean score)}\n\\end{table}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsubsection{Result Comparison on Spatial-temporal Dataset}\nIn order to further verify the performance of our model in the traditional spatial-temporal forecasting problem, we make further experiments on $2$ well known spatial-temporal datasets in traffic. \nWe compare the performance of A2GNN with above-mentioned baseline methods, and these methods can not be employed to model time series problem without pre-defined graph. \n\n\nWe present the results on spatial-temporal forecasting tasks in Table \\ref{tab:multi step forecasting} and compares corresponding performance of A2GNN with other spatial-temporal graph neural network methods. From this table, we can see that A2GNN achieves the best performance in terms of both RMSE and MAPE over almost all steps. In particular, A2GNN reduces the RMSE by $2.2\\%$, $3.3\\%$, $4.3\\%$ on the METR-LA dataset and $3.0\\%$, $4.4\\%$ and $4.6\\%$ on PEMS-BAY dataset when horizon is set to $3$, $6$ and $12$, respectively. \n\nOur method achieves the fastest training and inference time compared with other graph based models. The details are shown in appendix.\n\n\n\n\n\n\n\n\n\\subsection{Ablation Study}\n\\label{sec:ablation}\nTo gain a better understanding of the effectiveness of A2GNN's key components, we perform ablation studies through the time series forecasting task on Soloar-Energy dataset as well as the spatial-temporal forecasting task on META-LA dataset. The settings are summarized below:\n\\begin{itemize}\n\\item \\textbf{A2GNN} is A2GNN method.\n\\item \\textbf{w\/o AGL} is A2GNN without auto graph learner.\n\\item \\textbf{w\/o $A^p$} is A2GNN without pre-defined relation.\n\\item \\textbf{w\/o ARL} represents A2GNN  attentional relation learner, which is replaced by $\\rm{concatenate}$ operation.\n\\item \\textbf{w\/o A2} represents A2GNN without auto graph learner and attentional relation learner.\n\\end{itemize}\n\n\\begin{table}[t]\n\\caption{Ablation study.}\n\\renewcommand\\arraystretch{1.05}        \n\\centering\n\\renewcommand\\tabcolsep{3.0pt}\n\\scalebox{0.7}{\n\\begin{tabular}{c|cc|cc|cc|cc|cc}\n    \\toprule\n& \\multicolumn{4}{c}{Solar-Energy}  & \\multicolumn{6}{|c}{METR-LA} \\\\\n \\hline\n & \\multicolumn{2}{c}{RSE$\\downarrow$}  & \\multicolumn{2}{|c|}{CORR$\\uparrow$} & \\multicolumn{2}{c}{MAE$\\downarrow$}  & \\multicolumn{2}{|c|}{RMSE$\\downarrow$} &   \\multicolumn{2}{c}{MAPE$\\downarrow$} \\\\ \\hline\nMethods & Valid & Test & Valid & Test & Valid & Test & Valid & Test & Valid & Test\\\\ \\hline\nw\/o AGL & 0.37 & 0.35     & 0.94 & 0.93 & 2.73 & 2.97     & 5.33 & 5.99     & 7.57\\% & 8.27\\%      \\\\\nw\/o $A^p$ & - & - & - & - & 2.75  & 2.98     & 5.29  & 5.97    & 7.53\\%  & 8.17\\%  \\\\\nw\/o ARL & 0.33 & 0.30     & 0.95 & 0.95 & 2.73 & 2.98     & 5.26 & 5.92     & 7.60\\% & 8.28\\%      \\\\ \nw\/o A2  & 0.39 & 0.37     & 0.94 & 0.93 & 2.82 & 3.05     & 5.35 & 6.04     & 7.87\\% & 8.57\\%  \\\\ \\hline\nA2GNN & \\textbf{0.32} & \\textbf{0.28} & \\textbf{0.95} & \\textbf{0.95} & \\textbf{2.71} & \\textbf{2.93}     & \\textbf{5.23} & \\textbf{5.82}     & \\textbf{7.50\\%} & \\textbf{8.14\\%}      \\\\\n \\bottomrule\n\\end{tabular}}\n\n\\label{tab:ablation study}\n\n\\end{table}\n\n\n\n\n\nTo perform the studies for time series forecasting tasks, we run experiments on the Solar-Energy dataset with $t_{out} = 12$. To perform the ablation study on spatial-temporal forecasting tasks, we report the experiments on the METR-TA dataset on all $12$ steps. In each experiment, the model is trained for $50$ epochs, and $10$ repeated runs ensure the reliability.\nTable \\ref{tab:ablation study} show the performance in terms of evaluation scores on both validation and test sets. From the table, we can see that, without auto graph learner, the performance drops drastically, which indicates that our auto graph learner plays an indispensable role in A2GNN for achieving more accurate forecasting. \nSimilarly, the attentional relation learner is also responsible for a considerable performance gain by A2GNN. \nSpecially, in spatial-temporal forecasting task, pre-defined graph\/relation by human knowledge is a further information to be employed, which can also influence the model performance.\nIn the end, the experiment without auto graph learner and attentional relation learner prove the superiority of our overall approach.\n\n\\textit{The Effect of Neighbor Amount: }\nAuto graph Learner (AGL) is designed to discover the implicit relation. In AGL, the $C$ is the key factor to control the neighbor amount for each node. Therefore, we set multiple $C$ to study the influence. As shown in Table \\ref{tab:Effect of neighbor amount}, $C$ value that is too large or too small will make the model's performance worse.\nA reasonable $C$ value will have a big improvement compared with the w\/o AGL. Especially, the implicit relation discovered by AGL have a significant for time series forecasting task.\n\n\n\\begin{table}[]\n\\caption{Effect of neighbor amount. $C$ is the number of edges for each node in AGL.}\n\\renewcommand\\arraystretch{1.05}        \n\\centering\n\\renewcommand\\tabcolsep{3.0pt}\n\\scalebox{0.75}{\n\\begin{tabular}{c|cc|cc|cc|cc|cc}\n    \\toprule\n& \\multicolumn{4}{c}{Solar-Energy}  & \\multicolumn{6}{|c}{METR-LA} \\\\\n \\hline\n & \\multicolumn{2}{c}{RSE$\\downarrow$}  & \\multicolumn{2}{|c}{CORR$\\uparrow$} & \\multicolumn{2}{|c}{MAE$\\downarrow$}  & \\multicolumn{2}{|c|}{RMSE$\\downarrow$} &   \\multicolumn{2}{c}{MAPE$\\downarrow$}\\\\\n \\hline\n $C$ & Valid & Test & Valid & Test & Valid & Test & Valid & Test & Valid & Test\\\\ \\hline\n 1 & 0.349  & 0.320  & 0.951  & 0.946  & 2.759  & 2.960  & 5.328  & 5.915  & 7.54\\%  & 8.18\\% \\\\\n 3 & 0.337  & 0.302  & 0.954  & 0.952  & 2.725  & 2.945  & 5.233  & 5.848  & 7.50\\%  & 8.16\\% \\\\\n 10 & 0.327  & 0.296  & 0.956  & 0.954 & \\textbf{2.715} & \\textbf{2.935}     & 5.233 & \\textbf{5.825}     & \\textbf{7.50\\%} & \\textbf{8.14\\%} \\\\\n 15 & \\textbf{0.322}  & \\textbf{0.289}  & \\textbf{0.958}  & \\textbf{0.957}  & 2.715  & 2.974  & 5.202  & 5.906  & 7.35\\%  & 8.03\\%\\\\\n 30 & 0.333  & 0.302  & 0.954  & 0.952  & 2.720  & 2.953  & \\textbf{5.210}  & 5.860  & 7.58\\% & 8.22\\%\\\\\n\\bottomrule\n\\end{tabular}}\n\\label{tab:Effect of neighbor amount}\n\n\\end{table}\n\n\n\n\n\n\n\n\n\\subsection{Interpretability Analysis}\n\\subsubsection{Analysis of Attentional Relation Learner}\nTo further reveal the effectiveness of the attentional relation learner, we provide intensive interpretability analysis on it (the pre-defined relation branch is disabled). The corresponding experiments are conducted on the META-LA dataset. \n\n\n\n\nTo show the relation attentions learned by attentional relation learner, we first sample some stations in the dataset, each of which corresponds to a node in the graph, and plot their attention coefficient on different relations in Figure \\ref{fig:attentional relation learner attention matrix}. From this figure, we can see that various stations pay variant attention to different relations. For example, station $16$ and station $196$ pay more attention to identity adjacency matrix, i.e. they are more concerned about their own information. \nFurthermore, Figure \\ref{fig:attentional relation learner station curve} visualizes stations with greater attention on the automatically learned adjacency matrix, i.e., station $49$ and $129$, on the left compared with stations with more concentrated attention on the identity matrix, i.e., station $16$ and $196$, on the right. It is obvious that the curves of station $49$ and $129$ are not stable with a couple of randomly occurred sudden drops. The curves of station $16$ and $196$, on the other hand, have a very clear and stable change pattern, where both the peak and the valley appear alternately and periodically. Therefore, it is much easier for station $16$ and $196$ to make accurate enough forecasting mainly based on their own information.\n\n\\begin{figure}[]\n    \\centering\n    \\includegraphics[width=0.32\\textwidth]{photo\/attentional_relation_learner_visualize.png}\n    \\caption{Visualization for the attention coefficients (over own information and neighbor information by AGL) learned by attentional relation learner for station 16, 49, 129, 196. }\n    \\label{fig:attentional relation learner attention matrix}\n\\end{figure}\n\n\\begin{figure}[]\n    \\centering\n    \\includegraphics[width=0.45\\textwidth]{photo\/station_original_curve.jpg}\n    \\caption{Station speed visualization for $49,129,16,196$. The speed curves of station $16$ and $196$ are similar to the sine waveform, while the speed curves of station $49$ and $129$ seems not stable.}\n    \\label{fig:attentional relation learner station curve}\n\\end{figure}\n\n\n\\subsubsection{Analysis of Auto Graph Learner}\nIn this section, we make a further analysis on whether the learning process of auto graph learner is reasonable or not and whether the extra useful information exists in the learned neighbors from auto graph learner. In the first place, we visualize the correlation between station $49,129$ and all other stations during the training process in figure \\ref{fig:auto graph learner learning process}. As we can see, as the training process goes on, the correlations between station $49$ and some stations, such as station $150,120$, become stronger and stronger (from light to dark in color). Moreover, we visualize station $49,129$'s neighbors with strong correlation learned from auto graph learner in figure \\ref{fig:top related node curve for station49}. As we can see, the neighbor of station $49,129$ look like a similar curve of station $49,129$. The information aggregation of neighbors will enhance the predicted results (i.e. improve the robustness) of current station.\nThus, the learned neighbors can significantly improve the forecasting of station itself.\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=0.4\\textwidth]{photo\/learning_process.jpg}\n    \\caption{The learning process of auto graph learner for station $49,129$. }\n    \\label{fig:auto graph learner learning process}\n\\end{figure}\n\n\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=0.5\\textwidth]{photo\/station_and_its_neighbor_curve.jpg}\n    \\caption{Speed visualization of station and its related neighbors learnt by auto graph learner. As we can see, the station and its implicit\/learnt neighbors have a similar trend, which will improve the robustness. }\n    \\label{fig:top related node curve for station49}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Conclusions and Future Work}\nIn this paper, we propose attentional multi-graph neural network with automatic graph learning (A2GNN). Compared with previous studies, our framework can automatically learn sparse relation by using Gumbel-Softmax with facilitating each node to dynamically pay more attention to preferred relation graphs. Experiments on a couple of real-world datasets have demonstrated the effectiveness of A2GNN on a variety of time series forecasting tasks.\n\n\n\n\n\n\n\n\n\n\n\n\n\\small\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nSince the year 2012, an important task of particle physics is to fully\nmeasure the properties of the Higgs boson with mass $M_\\text{H}\\approx\n125$\\,GeV discovered at the Large Hadron Collider\n({\\abbrev LHC}{})~\\cite{Aad:2012tfa,Chatrchyan:2012xdj}.  At the same time, the\nsearch for additional Higgs bosons, which are predicted in many extended\ntheories, is among the main missions of the {\\abbrev LHC}{} experiments. For this\npurpose, the knowledge of the corresponding production cross sections\nwith high precision is of great relevance. The latest efforts in this\ndirection are regularly summarized in the reports of the ``{\\abbrev LHC}{} Higgs\ncross section working\ngroup''\\cite{Dittmaier:2011ti,Dittmaier:2012vm,Heinemeyer:2013tqa,deFlorian:2016spz}.\n\nIn this paper, we describe the new features that have been implemented\nin version {\\tt 1.6.0{}} of the program\n\\sushi{}~\\cite{Harlander:2012pb,sushiwebpage}.  \\sushi{} is a Fortran\ncode which calculates Higgs-boson production cross sections through\ngluon fusion and bottom-quark annihilation in the Standard Model ({\\abbrev SM}),\ngeneral Two-Higgs-Doublet Models ({\\abbrev 2HDM}), the Minimal Supersymmetric\nStandard Model ({\\abbrev MSSM}) as well as its next-to-minimal extension\n({\\abbrev NMSSM}), see \\citere{Liebler:2015bka}.\\footnote{Other codes to obtain\ninclusive Higgs-boson cross sections through gluon fusion in the {\\abbrev SM}{} and beyond are described in\n\\citeres{Spira:1995mt,Bagnaschi:2011tu,Anastasiou:2011pi,Anastasiou:2009kn,\nCatani:2007vq,Catani:2008me,Ball:2013bra,Bonvini:2014jma,Bonvini:2016frm}.}\nSome of these additions to\n\\sushi{} directly improve the theoretical predictions of the cross\nsection; others are provided to allow for more sophisticated uncertainty\nestimates of these predictions. The new features are the following:\n\\begin{itemize}\n\\item \\sushi{} now includes the next-to-next-to-next-to-leading order\n  (\\nklo{3}) terms for the gluon-fusion cross section of a {\\abbrev CP}{}-even\n  Higgs boson in the heavy-top limit as described in\n  \\citeres{Anastasiou:2014lda,Anastasiou:2015ema,Anastasiou:2015yha,Anastasiou:2016cez}.\n\\item It provides the so-called soft expansion of the gluon-fusion cross\n  section around the threshold of Higgs-boson production at \n  $x\\equivM_\\phi^2\/\\hat s=1$, where $\\hat{s}$ denotes the partonic\n  center-of-mass energy and $M_\\phi$ the Higgs-boson mass. This expansion\n  is available for the cross sections in the heavy-top limit up to\n  \\nklo{3} for {\\abbrev CP}{}-even Higgs bosons. At next-to-leading order ({\\abbrev NLO}{}) and\n  next-to-{\\abbrev NLO}{} ({\\abbrev NNLO}{}), the exact $x$-dependence is still available,\n   of course, and remains the default.\n\\item In addition, \\sushi{\\_\\sushiversion}{} includes top-quark mass effects to the\n  gluon-fusion cross section of a {\\abbrev CP}{}-even Higgs boson in the\n  heavy-top limit up to {\\abbrev NNLO}{}, implemented through an expansion in\n  inverse powers of the top-quark mass as described in\n  \\citeres{Marzani:2008az,Harlander:2009my,Harlander:2009mq,%\n    Harlander:2009bw,Pak:2009dg,Pak:2009bx,Pak:2011hs}.\n  The exact top-mass dependence at lowest order can be factored out.\n  We remark that this feature is most interesting at {\\abbrev NNLO}{}, of course,\n  since at leading order ({\\abbrev LO}{}) and {\\abbrev NLO}{}, \\sushi{} also provides the full quark-mass\n  dependence.\n\\item A matching of the soft expansion to the high-energy\n  limit~\\cite{Marzani:2008az,Harlander:2009my,Harlander:2009mq},\n  i.e.\\ $x\\to 0$, is available through \\nklo{3}.\n\\item The renormalization-scale dependence of the gluon-fusion cross\n  section within an arbitrary interval is calculated in a single\n  \\sushi{} run.\n\\item The effect of \\dimension{5} operators to the gluon-fusion cross\n  section can be taken into account through \\nklo{3} {\\abbrev QCD}{} for the inclusive\n  cross section, and at {\\abbrev LO}{} and {\\abbrev NLO}{} (i.e.\\ $\\alpha_s^3$) for the Higgs\n  transverse momentum ($\\pt{}$) distribution and (pseudo)rapidity\n  distribution, respectively.\n\\item Higgs-boson production cross sections through heavy-quark\n  annihilation are implemented along the lines of\n  \\citere{Harlander:2015xur}, both for the {\\abbrev NNLO}{} {\\abbrev QCD}{} inclusive\n  cross section, as well as for more exclusive\n  cross sections up to {\\abbrev NLO}{} {\\abbrev QCD}{}.\n\\end{itemize}\n\nAll of the described features are applicable to Higgs-boson production\nin the theoretical models currently implemented in \\sushi{}, even though\nsome only work for low Higgs masses below the top-quark threshold $M_\\phi\n< 2M_\\text{t}$ or for {\\abbrev CP}{}-even Higgs bosons.\n\nOur paper is organized as follows: We start with a brief general\noverview of the code \\sushi{} in \\sct{sec:sushi}, and subsequently\npresent the new features implemented for the prediction of the\ngluon-fusion cross section in \\sct{sec:gluonfusion}. This includes a\ntheoretical description of the soft expansion, the inclusion of\n\\nklo{3} terms and the top-quark mass effects in\n\\scts{sec:softexp}--\\ref{sec:mt}. We proceed with a description of the\n``{\\scalefont{.9} RGE}{} procedure'' to determine the renormalization-scale dependence\nof the gluon-fusion cross section in \\sct{sec:scaledep}, and finally\ndescribe the implementation of an effective Lagrangian including\n\\dimension{5} operators in \\sct{sec:dim5}. The implementation of\nheavy-quark annihilation cross sections is described in\n\\sct{sec:heavyquark}. Numerical results are presented in\n\\sct{sec:numerics}; they also include a comparison of our results with\nthe most recent literature. In \\ref{app:inputfile} we\npresent a collection of example input blocks of \\sushi{}, which contain\nexample settings for the various input entries introduced in previous and the newest release.\n\n\\section{The program \\sushi{}}\n\\label{sec:sushi}\n\n\\sushi{} is a program originally designed to describe Higgs production\nin gluon fusion and bottom-quark annihilation in the {\\abbrev MSSM}{}.  It\ncollects a number of results from the literature valid through \\nklo{3}\nin the strong coupling constant, and combines them in a consistent way.\nWe subsequently discuss the present theoretical knowledge of the\ncalculation of the gluon fusion and bottom-quark annihilation cross\nsections and their inclusion in \\sushi{}.\n\nIt is well-known that {\\abbrev QCD}{} corrections to the gluon-fusion process\n$gg\\rightarrow \\phi$~\\cite{Georgi:1977gs}, mediated through heavy quarks\nin the {\\abbrev SM}{}, are very large. {\\abbrev NLO}{} {\\abbrev QCD}{} corrections are known for\ngeneral quark\nmasses~\\cite{Djouadi:1991tka,Dawson:1990zj,Spira:1995rr,Harlander:2005rq,Anastasiou:2006hc,Aglietti:2006tp}.\nIn the heavy-top limit, an effective theory can be constructed by\nintegrating out the top quark. In this case, {\\abbrev NNLO}{} corrections have\nbeen calculated a long time\nago~\\cite{Harlander:2002wh,Anastasiou:2002yz, Ravindran:2003um}.  The\n\\nklo{3} contributions were only recently obtained in\n\\citeres{Anastasiou:2014vaa,Li:2014afw,Anastasiou:2014lda,Anastasiou:2015ema,\n  Anastasiou:2016cez}, while various parts of the \\nklo{3} calculation\nhave been calculated\nindependently\\,\\cite{Hoschele:2012xc,Baikov:2009bg,Lee:2010cga,Gehrmann:2010ue,\n  Gehrmann:2010tu,Gehrmann:2011aa,Kilgore:2013gba,Duhr:2014nda,\n  Anastasiou:2015yha,Duhr:2013msa,Anastasiou:2013srw,Hoschele:2014qsa,\n  Dulat:2014mda,Anzai:2015wma,Li:2014bfa}.  Approximate \\nklo{3} results\nwere presented in\n\\citeres{Ball:2013bra,Bonvini:2014jma,deFlorian:2014vta}.  Effects of a\nfinite top-quark mass at {\\abbrev NNLO}{} were approximately taken into account\nin~\\citeres{Harlander:2009my,Harlander:2009mq,Harlander:2009bw,Pak:2009dg,Pak:2009bx,Pak:2011hs,Marzani:2008az}.\n\nMany of these effects can be taken into account in the latest\nversion of \\sushi{}; this will be discussed in detail in\n\\sct{sec:gluonfusion}. Electroweak\ncorrections~\\cite{Actis:2008ug,Aglietti:2004nj,Bonciani:2010ms} can be\nincluded as well, either in terms of the full {\\abbrev SM}{} electroweak\ncorrection factor, or restricted to the corrections mediated by light\nquarks, the latter being a more conservative estimate in certain {\\abbrev BSM}{}\nscenarios. For completeness, we note that effects beyond fixed order\nhave been addressed through soft-gluon resummation \\cite{Catani:2003zt,\n  Moch:2005ky,Idilbi:2005ni,Idilbi:2006dg,Ravindran:2006cg,Ahrens:2008nc,\n  Schmidt:2015cea,Bonvini:2016frm}, but those are not included in\n\\sushi{}.\n\nIf requested in the input file, \\sushi{} uses the {\\abbrev SM}{} results\ndescribed above also for the {\\abbrev 2HDM}{}, the {\\abbrev MSSM}{} or the {\\abbrev NMSSM}{}\nthrough the proper rescaling of the Yukawa couplings.  In supersymmetric\nmodels, also squarks induce an interaction of the Higgs boson to two\ngluons.  In the {\\abbrev MSSM}{}, the corresponding {\\abbrev NLO}{} virtual contributions,\ninvolving squarks, quarks and gluinos, are either known in an expansion\nof inverse powers of heavy {\\abbrev SUSY}{}\nmasses~\\cite{Degrassi:2010eu,Degrassi:2011vq,Degrassi:2012vt} or in the\nlimit of a vanishing Higgs mass, see\n\\citeres{Harlander:2003bb,Harlander:2004tp,Harlander:2005if,Degrassi:2008zj}.\nIn this limit, even {\\abbrev NNLO}{} corrections of stop-induced contributions\nare known, see \\citeres{Pak:2010cu,Pak:2012xr}; an approximation of\nthese effects\\,\\cite{Harlander:2003kf} is included in \\sushi{}, see\n\\citere{Bagnaschi:2014zla}.  Whereas for the {\\abbrev MSSM}{} \\sushi{} relies on\nboth expansions, for the {\\abbrev NMSSM}{} the {\\abbrev NLO}{} virtual corrections are\npurely based on an expansion in heavy {\\abbrev SUSY}{}\nmasses~\\cite{Liebler:2015bka}.\nWe note that numerical results for the exact {\\abbrev NLO}{} virtual\ncontributions involving squarks, quarks, and gluinos were presented in\n\\citeres{Anastasiou:2008rm,Muhlleitner:2010nm}, and analytic results for\nthe pure squark-induced contributions can be found in\n\\citeres{Anastasiou:2006hc,Aglietti:2006tp,Muhlleitner:2006wx}.\n\nThe associated production of a Higgs boson with bottom quarks,\n$pp\\rightarrow b\\bar b\\phi$, is of particular relevance for Higgs\nbosons, where the Yukawa coupling to bottom quarks is enhanced. This\nhappens in models with two Higgs doublets, for example, if $\\tan\\beta$,\nthe ratio of the vacuum expectation values of the two neutral Higgs\nfields, is large. \\sushi{} includes the cross section for this process\nin the so-called 5-flavor scheme, i.e.\\ for the annihilation process\n$b\\bar b\\to \\phi$. The inclusive cross section for this process is\nimplemented at {\\abbrev NNLO}{} {\\abbrev QCD}{} \\cite{Maltoni:2003pn,Harlander:2003ai}; it\nis reweighted by effective Yukawa couplings in the model under\nconsideration. \\sushi{\\_\\sushiversion}{} now also includes general heavy-quark\nannihilation cross sections\\,\\cite{Harlander:2015xur} at {\\abbrev NNLO}{} {\\abbrev QCD}{},\nwhich we will describe in \\sct{sec:heavyquark}.\n\nFor completeness we note that \\sushi{} can be linked to {\\tt\nFeynHiggs}~\\cite{Heinemeyer:1998yj,Heinemeyer:1998np,Degrassi:2002fi,Frank:2006yh}\nand {\\tt 2HDMC}~\\cite{Eriksson:2009ws} to obtain consistent sets of\nparameters in the {\\abbrev MSSM}{} or the {\\abbrev 2HDM}{}, respectively.\n\n\\sushi{} is controlled via an {\\scalefont{.9} SLHA}-style \\cite{Skands:2003cj}\ninput file. In the following, we will refer to the entries of a {\\tt\n  Block \"NAME\"} and their possible values as\n\\blockentry{NAME}{ENTRY}{=VALUE}. If more than one value is required, we\nwrite \\blockentry{NAME}{ENTRY}{=\\{VALUE1,VALUE2,$\\dots$\\}} or, when\nreferring only to one specific value,\n\\blockentry{NAME}{ENTRY,1}{=VALUE1}, etc.\nIn \\ref{app:inputfile} we include input blocks with\nexample settings for the various new input entries.\n\n\\section{Higgs production through gluon fusion}\n\\label{sec:gluonfusion}\n\nThe hadronic cross section for Higgs production in gluon fusion can be\nwritten as\n\\begin{equation}\n\\begin{split}\n\\sigma(pp\\to H+X) = \\sum_{i,j\\in\\{q,\\bar q,g\\}}\\tilde{\\phi}_i\\otimes\n\\tilde{\\phi}_j\\otimes\\hat\\sigma_{ij}\\,,\n\\label{eq:convolution}\n\\end{split}\n\\end{equation}\nwhere $\\phi_i(x,\\mu_\\text{F})=\\tilde{\\phi}_i(x,\\mu_\\text{F})\/x$ are parton\ndensities, $q$ ($\\bar q$) denotes the set of all (anti-)quarks ($q=t$\nand $\\bar q=\\bar t$ can be neglected), and $\\otimes$ is the convolution\ndefined as\n\\begin{equation}\n\\begin{split}\n(f\\otimes g)(z) \\equiv \\int_0^1{\\rm d} x_1\\int_0^1{\\rm d} x_2\n  f(x_1)g(x_2)\\delta(z-x_1x_2)\\,.\n\\end{split}\n\\end{equation}\nThe perturbative expansion of the partonic cross section,\n\\begin{equation}\n\\begin{split}\n\\hat\\sigma_{ij,\\text{\\nklo{n}}}&= \n\\sum_{l=0}^n\\hat\\sigma^{(l)}_{ij}\\,,\n\\label{eq:sigmanij}\n\\end{split}\n\\end{equation}\ncan be represented in terms of Feynman diagrams where the external\npartons couple to the Higgs bosons through a top-, bottom-, or\ncharm-quark loop (contributions from lighter quarks are\nnegligible).\n\nThe first two terms in the perturbative expansion of $\\hat\\sigma_{ij}$\n($l=0,1$ in \\eqn{eq:sigmanij}) are known for general quark mass and\nincluded in \\sushi{\\_\\sushiversion}.\\footnote{We focus on the {\\abbrev SM}{} contributions\n  here, but also {\\abbrev SUSY}{} contributions can be added in the first two\n  terms and partially even at {\\abbrev NNLO}{}, see the description in\n  \\sct{sec:sushi}.} For the top-quark contribution, the\n{\\abbrev NNLO}{} term $\\hat\\sigma^{(2)}_{ij}$ has been evaluated on the basis of\nan effective Higgs-gluon interaction vertex which results from\nintegrating out the top quark from the {\\abbrev SM}{} Lagrangian. At {\\abbrev NLO}{}, it\nhas been checked that this results in an excellent approximation of the\n{\\abbrev NLO}{} {\\abbrev QCD}{} correction factor to the {\\abbrev LO}{} cross section, even for\nrather large Higgs-boson masses.  At {\\abbrev NNLO}{}, the validity of the\nheavy-top limit for the {\\abbrev QCD}{} corrections factor was investigated\nthrough the calculation of a number of terms in an expansion around\n$M_\\text{t}^2\\gg \\hat s,M_\\phi^2$, and matching it to the high-energy limit of\n$\\hat\\sigma^{(2)}_{ij}$\\,\\cite{Harlander:2009my,Harlander:2009mq,Harlander:2009bw,Pak:2009dg,Pak:2009bx,Pak:2011hs,Marzani:2008az}.\nIt was found that the mass effects to the {\\abbrev QCD}{} correction factor are\nat the sub-percent level.\n\nRecently, also the \\nklo{3}-term $\\hat\\sigma_{ij}^{(3)}$ has become\navailable in terms of a soft expansion and assuming the heavy-top\nlimit. We will comment on its implementation in the latest release of\n\\sushi{} in \\sct{sec:n3lo}.\n\nThe exact {\\abbrev NLO}{} and the approximate higher order results for the cross\nsection are combined in \\sushi{} through the formula\n\\begin{equation}\n\\begin{split}\n\\sigma_\\text{{\\scalefont{.9} X}} &= \\sigma_\\text{{\\abbrev NLO}} +\n\\Delta_\\text{{\\scalefont{.9} X}} \\sigma^t\\,,\\qquad\\Delta_\\text{{\\scalefont{.9} X}}\n\\sigma^t \\equiv (1+\\delta_\\text{EW})\\sigma^t_\\text{{\\scalefont{.9} X}} - \\sigma_\\text{{\\abbrev NLO}}^t\\,,\n\\label{eq:sigmasushi}\n\\end{split}\n\\end{equation}\nwhere $\\sigma_\\text{{\\abbrev NLO}}$ refers to the {\\abbrev NLO}{} cross section with exact\ntop-, bottom- and charm-mass dependence, while $\\sigma^t_\\text{{\\scalefont{.9} X}}$\n({\\scalefont{.9} X}$=$\\nklo{n}, $n\\geq 1$) is obtained in the limit of a large top-quark\nmass.  Electroweak effects\\,\\cite{Actis:2008ug}, encoded in\n$\\delta_\\text{EW}$, are included by assuming their full factorization\nfrom the {\\abbrev QCD}{} effects, as suggested by \\citere{Anastasiou:2008tj} for\na {\\abbrev SM}{} Higgs boson.  In {\\abbrev BSM}{} scenarios, this assumption may be no\nlonger justified. \\sushi{} therefore provides an alternative way to\ninclude electroweak effects which is based solely on the light-quark\ncontributions to the electroweak correction factor; for details, we\nrefer the reader to \\citeres{Bagnaschi:2014zla,Harlander:2012pb}. For\nour purpose, it suffices to assume \\eqn{eq:sigmasushi}. The new release\nof \\sushi{} provides various approximations to evaluate\n$\\sigma^t_\\text{{\\scalefont{.9} X}}$, in particular through expansions in $1\/M_\\text{t}$, and\nexpansions around $\\hat s=M_\\phi^2$.\n\nIn addition to $\\sigma_\\text{{\\scalefont{.9} X}}$, which can be found in {\\tt Block\n  SUSHIggh}, \\sushi{} also outputs the individual terms of\n\\eqn{eq:sigmasushi}. The exact {\\abbrev LO}{} and {\\abbrev NLO}{} cross sections are\ncollected in {\\tt Block XSGGH}, while the $\\sigma^t_\\text{{\\scalefont{.9} X}}$ are given\nin {\\tt Block XSGGHEFF}, which also contains the electroweak correction\nterm $\\delta_\\text{EW}$, if requested.\n\nIt is understood that the \\nklo{n} terms in \\eqn{eq:sigmasushi} are\nevaluated with \\nklo{n} {\\abbrev PDF}{}s.\\footnote{Since \\nklo{3} {\\abbrev PDF}{}s are not\nyet available, we use {\\abbrev NNLO}{} {\\abbrev PDF}{}s for the evaluation of the\n\\nklo{3} cross section in this paper. The user of \\sushi{} can specify\nthe {\\abbrev PDF}{} set at each order individually.} Note that this means that,\nfor example, $\\Delta_\\text{{\\abbrev NNLO}}\\sigma^t$ is not simply the convolution\nof $\\hat\\sigma^{t,(2)}$ with {\\abbrev NNLO}{} {\\abbrev PDF}{}s, but retains a sensitivity\nto $\\hat\\sigma^{t,(1)}$. Thus, the final result for the {\\abbrev NNLO}{}\ngluon-fusion cross section obtained from \\sushi{} through\n\\eqn{eq:sigmasushi} depends on the approximation applied to the\nevaluation of both $\\hat\\sigma^{t,(2)}$ and $\\hat\\sigma^{t,(1)}$. If\nelectroweak effects are included, this even holds for \\sushi{}'s final\nresult for $\\sigma_\\text{{\\abbrev NLO}}$ due to the definition of\n$\\Delta_\\text{X}\\sigma^t$ in \\eqn{eq:sigmasushi}.\n\nIn the remainder of this section, we first discuss the soft expansion\naround the threshold of Higgs production, $\\hat{s}=M_\\phi^2$, in\n\\sct{sec:softexp}. The implementation of \\nklo{3} contributions is\ndescribed in \\sct{sec:n3lo}, and top-quark mass effects through {\\abbrev NNLO}{}\nas well as the matching to the high-energy limit in \\sct{sec:mt}. While\nthese features are only available for {\\abbrev CP}{}-even Higgs bosons (partially\nin a certain range of Higgs-boson masses $M_\\phi$ only), the analytic\ncalculation of the $\\mu_\\text{R}$ dependence of the gluon-fusion cross section\ndescribed in \\sct{sec:scaledep} is available for all Higgs bosons. The\ninclusion of \\dimension{5} operators is discussed in \\sct{sec:dim5}.\n\n\\subsection{Soft expansion}\\label{sec:softexp}\n\nThe {\\abbrev NLO}{} and {\\abbrev NNLO}{} coefficients of $\\sigma^t$ are approximated very\nwell by the first few terms\\footnote{The first $16$ terms in\n this expansion lead to an accuracy of better than $1$\\% with respect to the heavy-top\n limit with exact $x$-dependence at {\\abbrev NNLO}{}, for example. For more details see\n below.} in an expansion around the ``soft limit'', $x\\to 1$. In fact,\nthe gain of the full $\\hat s$-dependence becomes doubtful anyway when\nworking in the heavy-top limit, since the latter formally breaks down\nfor $\\hat s>4M_\\text{t}^2$, meaning $x\\lesssim 0.13$ for $M_\\text{H}=125$\\,GeV.\nApart from the exact $\\hat s$-dependence at {\\abbrev LO}{}, {\\abbrev NLO}{},\nand {\\abbrev NNLO}{}, \\sushi{\\_\\sushiversion}{} provides the soft expansion of the cross\nsection for {\\scalefont{.9} CP}-even Higgs production through order\n$(1-x)^{16}$ at these perturbative orders, and also at \\nklo{3} (for\nmore details on the latter, see \\sct{sec:n3lo}).\n\nThe precise way in which the soft expansion is applied is governed by\nthe new {\\tt Block GGHSOFT}. Each line in this block contains four\nintegers:\n\\begin{lstlisting}\nBlock GGHSOFT\n  <entry> <n1> <n2> <n3>\n\\end{lstlisting}\nFollowing \\sct{sec:sushi}, we will refer to such a line as\n\\blockentry{GGHSOFT}{<entry>}{\\tt=\\{<n1>,<n2>,<n3>\\}} in the text, and\nto the individual entries as \\blockentry{GGHSOFT}{<entry>,1}{\\tt=<n1>},\netc. The integer \\blockentry{GGHSOFT}{$n$,1}{} determines\n  whether the soft expansion is applied ({\\tt =1}) or not ({\\tt =0}) at\norder \\nklo{n}.\n  Setting \\blockentry{GGHSOFT}{$n$}{=\\{1,$N,a$\\}} evaluates the soft\n  expansion of $\\hat\\sigma^{t,(n)}_{ij}$ in the following way:\n\\begin{equation}\n  \\begin{split}\n    \\hat \\sigma^t_{ij}\\to\n    \\hat\\sigma^t_{ij,N} \\equiv x^{a}{\\cal\n      T}^x_{N}\\left(\\frac{\\Delta\\hat\\sigma^t_{ij}}{x^{a}}\\right)\\,,\n    \\label{eq:softexp}\n  \\end{split}\n\\end{equation}\nwhere ${\\cal T}^x_N$ denotes the asymptotic expansion around $x=1$\nthrough order $(1-x)^N$, and $a$ is a non-negative integer. Setting\n\\blockentry{GGHSOFT}{$n$,2}{=-1} will keep only the soft and collinear\nterms, whose $x$ dependence is given by\n\\begin{equation}\n\\begin{split}\n  \\delta(1-x)\\quad\\text{or}\\quad\\left(\\frac{\\ln^k(1-x)}{1-x}\\right)_+\\,,\\quad\n  k\\geq 0\n\\end{split}\n\\end{equation}\nby definition.  Here $(\\cdot)_+$ denotes the usual plus distribution,\ndefined by\n\\begin{equation}\n  \\begin{split}\n    \\int_z^1{\\rm d} x\\left(f(1-x)\\right)_+ g(x) = \\int_z^1{\\rm d} x\n    f(x)\\left[g(x)-g(1)\\right] + g(1)\\int_0^z{\\rm d} xf(x)\\,.\n  \\end{split}\n\\end{equation}\nThe parameters \\blockentry{GGHSOFT}{$n$}{} apply to all partonic\nsubchannels at order \\nklo{n}, and to all terms in the $1\/M_\\text{t}$\nexpansion as requested by the input {\\tt Block GGHMT}, see \\sct{sec:mt}\nbelow.\n\nThe exact $x$-dependence is obtained by setting\n\\blockentry{GGHSOFT}{$n$,1}{=0} (only available for $n\\leq 2$). The\nother two entries in \\blockentry{GGHSOFT}{$n$}{} are then irrelevant.\nThe default values for the block {\\tt GGHSOFT} through \\nklo{3} are\n\\begin{equation}\n{{\\footnotesize\n\\begin{split}\n\\fbox{\\it default:}\\quad\n&\\text{\\blockentry{GGHSOFT}{1,1}{=0}}\\,;\\quad\n\\text{\\blockentry{GGHSOFT}{2,1}{=0}}\\,;\\quad\n\\text{\\blockentry{GGHSOFT}{3}{=\\{1,16,0\\}}}\\,.\n\\end{split}}}\n\\end{equation}\nAgain, all terms of the soft expansion are available including the full\n$\\mu_\\text{F}$- and $\\mu_\\text{R}$-dependence.\n\nA sample input block reads\n\\begin{lstlisting}\nBlock GGHSOFT\n  1 0 0 0\n  2 1 16 1\n  3 1 16 1\n\\end{lstlisting}\nwhich provides the result including the exact $x$-dependence at {\\abbrev NLO}{},\nand the soft expansion through $(1-x)^{16}$ at {\\abbrev NNLO}{} and \\nklo{3}\nafter factoring out a factor of $x$ ($a=1$ in \\eqn{eq:softexp}).  We\nrecall that these settings only affect the heavy-top results\n$\\hat\\sigma^t_\\text{{\\scalefont{.9} X}}$ in \\eqn{eq:sigmasushi}; $\\sigma_\\text{{\\abbrev NLO}}$ is\nalways calculated by taking into account the full quark-mass and\n$x$-dependence. The soft expansion is available for all {\\abbrev CP}{}-even\nHiggs bosons of arbitrary masses.\n\n\\subsection{\\nklo{3} terms}\\label{sec:n3lo}\n\nRecently, the \\nklo{3} {\\abbrev QCD}{} corrections to the Higgs production cross\nsection through gluon fusion have become\navailable\\,\\cite{Anastasiou:2014lda,Anastasiou:2015ema,Anastasiou:2015yha,Anastasiou:2016cez}. More\nspecifically, the result was provided in terms of the soft expansion\nthrough order $(1-x)^{37}$ of the leading term in $1\/M_\\text{t}$ for\n$\\mu=\\mu_\\text{R}=\\mu_\\text{F}$.  We implemented this expansion through $(1-x)^{16}$;\nhigher order terms do not change the result within the associated\nuncertainty. In addition, we included the $\\mu_\\text{F}$- and $\\mu_\\text{R}$-dependent\nterms at the same order.  Experience from {\\abbrev NNLO}{} lets one expect that\nthese terms are sufficient to obtain an excellent approximation of the\n{\\abbrev QCD}{} correction factor to the {\\abbrev LO}{} cross section, at least for\nHiggs masses in the validity range of the effective theory description.\n\nThe \\nklo{3} result is accessible in \\sushi{\\_\\sushiversion}{} by setting the input\nparameter \\blockentry{SUSHI}{5}{=3}. This will evolve $\\alpha_s(M_Z)$ to\n$\\alpha_s(\\mu_\\text{R})$ at 4-loop order when calculating the cross section,\nwhere $\\mu_\\text{R}\/M_\\phi$ is defined in \\blockentry{SCALES}{1}{}. Note that\nwith this setting, the hadronic cross section will formally still suffer\nfrom an inconsistency because \\nklo{3} {\\abbrev PDF}{} sets are not yet\navailable.\nAs described in \\sct{sec:softexp}, the depth of the soft\nexpansion at \\nklo{3}, as well as the power $a$ in \\eqn{eq:softexp} can\nbe controlled through the input variables\n\\blockentry{GGHSOFT}{3}{}.\\footnote{The setting\n  \\blockentry{GGHSOFT}{3,1}{=0} is not available, of course.}\n\nFinally, we remark that, also at \\nklo{3}, the full $\\mu_\\text{R}$- and\n$\\mu_\\text{F}$-dependence is available, again accessible through the variables\n{\\tt SCALES(1)} and {\\tt SCALES(2)}, respectively. It follows from\ninvariance of the hadronic result under these scales, and only requires\nthe {\\abbrev NNLO}{} result as input, as well as the {\\abbrev QCD}{} $\\beta$ function and\nthe {\\abbrev QCD}{} splitting functions through three loops. The required\nconvolutions can be evaluated with the help of the program {\\tt\n  MT.m}\\,\\cite{Hoeschele:2013gga}, for example.\n\n\\subsection{Top-quark mass effects}\\label{sec:mt}\n\nIn versions before \\sushi{\\_\\sushiversion}{}, only the formally leading terms in\n$1\/M_\\text{t}$ were available for $\\hat\\sigma^{t,(2)}_{ij}$. However, in\norder to allow for thorough studies of the theoretical uncertainty\nassociated with the gluon-fusion cross section, \\sushi{\\_\\sushiversion}{} includes\nalso subleading terms in $1\/M_\\text{t}$ for the production of a {\\abbrev CP}{}-even\nHiggs (\\blockentry{SUSHI}{2}{}$\\in\\{${\\tt\n11,12,13}$\\}$). There are a number of options provided by\n\\sushi{\\_\\sushiversion}{} associated with this; they are controlled by the new input\n{\\tt Block GGHMT}.\n\nFirst of all,\n\\blockentry{GGHMT}{$n$}{=$P\\in\\{$0,1,$\\ldots,P_n^\\text{max}\\}$} provides\nthe expansion of $\\hat\\sigma^{t,(n)}_{ij}$ through $1\/M_\\text{t}^{P}$ (note\nthat terms with odd $P$ vanish). In addition (or alternatively), one may\ndefine the depth of the expansion individually for each partonic channel\n$\\hat\\sigma^{t,(n)}_{ij}$ through the parameters\n\\blockentry{GGHMT}{$nm$}{=$P\\in\\{$0,1,$\\ldots,P_n^\\text{max}\\}$}, where\n$ij=(gg,qg,q\\bar q,qq,qq')$ corresponds to $m=(1,2,3,4,5)$,\nrespectively. Currently, the maximal available depths of expansion\nare\\footnote{For the $q\\bar q$-channel, the maximum reduces to\n  $P_2^\\text{max}=4$, if a soft expansion beyond $N=13$ is requested.}\n$P_0^\\text{max}=P_1^\\text{max}=10$ and $P_2^\\text{max}=6$.\n\nThe default settings are\n\\begin{equation}\n{\\footnotesize\n\\begin{split}\n\\fbox{\\text{\\it default:}}\\quad \n&\\text{\\blockentry{GGHMT}{0}{=-1}}\\,;\\quad\n\\text{\\blockentry{GGHMT}{1}{=0}}\\,;\\quad\n\\text{\\blockentry{GGHMT}{2}{=0}}\\,;\\\\\n&\\text{\\blockentry{GGHMT}{1$i$}{=GGHMT(1)}}\\,,\\ i=1,\\ldots,3\\,;\\quad\n\\text{\\blockentry{GGHMT}{2$i$}{=GGHMT(2)}}\\,,\\ i=1,\\ldots,5\\,,\n\\end{split}}\n\\end{equation}%\nwhere \\blockentry{GGHMT}{0}{=-1} means to keep the full top mass\ndependence. Let us recall that these settings only affect the heavy-top\nresults $\\hat\\sigma^t_\\text{{\\scalefont{.9} X}}$ in \\eqn{eq:sigmasushi}; $\\sigma_\\text{{\\abbrev NLO}}$\nis always calculated by taking into account the full quark-mass dependence.\n\nAs an example, consider the input\n\n\\begin{lstlisting}\nBlock GGHMT\n  1    10\n  13   0\n  2    6\n  23   0\n  24   0\n  25   0\n\\end{lstlisting}\n\nwhich will cause \\sushi{\\_\\sushiversion}{} to\n\\begin{itemize}\n\\item keep the full top mass dependence at {\\abbrev LO}{}\n\\item expand the {\\abbrev NLO}{} terms $\\hat\\sigma^{t,(1)}_{gg}$ and\n$\\hat\\sigma^{t,(1)}_{qg}$ through $1\/M_\\text{t}^{10}$\n\\item expand the {\\abbrev NNLO}{} terms $\\hat\\sigma^{t,(2)}_{gg}$ and\n  $\\hat\\sigma^{t,(2)}_{qg}$ through $1\/M_\\text{t}^{6}$\n\\item keep only the terms of order $1\/M_\\text{t}^0$ for the pure quark\n  channels at {\\abbrev NLO}{} and {\\abbrev NNLO}{}.\n\\end{itemize}\nThis also shows that the variables \\blockentry{GGHMT}{$nm$}{} overrule\nthe setting of \\blockentry{GGHMT}{$n$}{} for the individual\nchannels. This may be desirable as it is known that the pure quark\nchannels show a rather bad convergence\nbehavior~\\cite{Harlander:2009my,Harlander:2009mq,Harlander:2009bw,Pak:2009dg,Pak:2009bx,Pak:2011hs},\nso one may want to include only a small number of terms for them in the\n$1\/M_\\text{t}$ expansion.  By convention, \\blockentry{GGHMT}{$n$}{} must\nalways be at least as large as the maximum of \\blockentry{GGHMT}{$nm$}{};\nif this is not the case in the input file, \\sushi{} will override the\nuser's definition of \\blockentry{GGHMT}{$n$}{} and set it to the maximum\nof all \\blockentry{GGHMT}{$nm$}{}.\n\nIn the strict heavy-top limit (i.e., $P=0$), the quality of the\napproximation improves considerably if one factors out the {\\abbrev LO}{} mass\ndependence $\\sigma^t_0$\\,\\cite{Graudenz:1992pv,Spira:1995rr} before the\nexpansion, given by\n\\vspace{-5mm}\n\\begin{equation}\n\\begin{split}\n\\sigma^t_0 &= \\frac{\\pi\\sqrt{2}G_{\\rm F}}{256}\\left(\\api{}\\right)^2\n\\tau^2\\left|1+(1-\\tau)\\arcsin^2\\frac{1}{\\sqrt{\\tau}}\\right|^2\\,,\\qquad\n\\tau = \\frac{4M_\\text{t}^2}{M_\\phi^2}\\,,\n\\label{eq:sigma0}\n\\end{split}\n\\end{equation}\nwhere $G_\\text{F}\\approx 1.16637\\cdot 10^{-5}\\,\\text{GeV}^{-2}$~\\cite{Agashe:2014kda} is\nFermi's constant. The generalization to higher orders in $1\/M_\\text{t}$\ncorresponds to\n\\vspace{-5mm}\n\\begin{equation}\n\\begin{split}\n\\hat\\sigma^{t,(n)}_{ij} = \\sigma^t_0 \\frac{{\\cal\n    T}_{P_{n,ij}}\\hat\\sigma^{t,(n)}_{ij}}{{\\cal T}_{P_n}\\sigma^t_0}\\,,\n\\label{eq:defexpmt}\n\\end{split}\n\\end{equation}\nwhere ${\\cal T}_{P}$ denotes an operator that performs an asymptotic\nexpansion through order $1\/M_\\text{t}^P$.  In a strict sense, it should be\n$P_n=P_{n,ij}$; however, \\sushi{} allows only for a global value of\n$P_n$ here, which applies to all sub-channels $ij$ and is set to\n\\blockentry{GGHMT}{$n$}{}.\n\nSetting \\blockentry{GGHMT}{-1}{=$n$} factors out the {\\abbrev LO}{} mass\ndependence through order $n$, i.e.\\\n\\begin{equation}\n\\begin{split}\n\\hat\\sigma^t_{ij} &= \\sigma^t_0\\, \\sum_{k=0}^n \\frac{{\\cal\n    T}_{P_{k,ij}}\\hat\\sigma^{t,(k)}_{ij}}{{\\cal T}_{P_k}\\sigma^t_0} +\n\\sum_{k\\geq n+1}{\\cal T}_{P_{k,ij}}\\hat\\sigma^{t,(k)}_{ij}\\,.\n\\label{eq:plainexp}\n\\end{split}\n\\end{equation}\nThis will affect all partonic channels. The default setting is\n\\begin{equation}\n{\\footnotesize\n\\begin{split}\n\\fbox{\\text{\\it default:}}\\quad& \\text{\\tt GGHMT(-1)=3}\n\\end{split}}\n\\end{equation}\nwhich means that the {\\abbrev LO}{} $M_\\text{t}$ dependence is factored out from\nall available orders.\n\nIt was observed that higher orders in $1\/M_\\text{t}$ in general spoil the\nvalidity of the expansion, since its radius of convergence is formally\nrestricted to $\\hat s<4M_\\text{t}^2$. This manifests itself in the expansion\ncoefficients containing positive powers of $\\hat s\/M_\\phi^2$. In order to\ntame the corresponding divergence as $\\hat s\\to \\infty$, it was\nsuggested to match the result to the asymptotic behavior in this limit,\nwhich is known from \\citeres{Marzani:2008az,Harlander:2009my}. Whether\nor not such a matching is performed for $\\hat\\sigma^{t,(n)}_{ij}$ is\ngoverned by the parameter \\blockentry{GGHMT}{$n\\cdot$10}{}\n(i.e.\\ \\blockentry{GGHMT}{10}{}, \\blockentry{GGHMT}{20}{}, \\ldots).  By\ndefault,\\\\[-2.5em]\n\n\\begin{equation}\n{\\footnotesize%\n\\begin{split}\n\\fbox{\\text{\\it default:}}\\quad& \\text{\\tt\n  GGHMT($n\\cdot$10)=0}\\,,\\ n=1,\\ldots,3\\,, \n\\end{split}}\n\\end{equation}\nmeaning that no matching is done; setting \\blockentry{GGHMT}{$n\\cdot$10}{=1}\nswitches the matching on for all partonic subchannels at order \\nklo{n}.\n\nAs we will find in \\sct{sec:numerics}, the matching to $x\\to 0$ is\nhelpful in approximating the full cross section even at $1\/M_\\text{t}^0$.\nThus, we provide the possibility to do this matching also at \\nklo{3},\neven though top-mass suppressed terms are not yet known at this order.\nThe form of the matching through {\\abbrev NNLO}{} has been introduced in\n\\citeres{Harlander:2009my,Harlander:2009mq}; here we adopt the same\nstrategy, generalized to \\nklo{3}:\n\\begin{equation}\n\\begin{split}\n\\hat\\sigma^{t,(n)}_{ij}(x) = \\hat\\sigma^{t,(n)}_{ij,N}(x)\n&+ \\,\\sigma^t_0 \\sum_{l=1}^{n-1}A^{(n,l)}_{ij}\\left[ \\ln \\frac{1}{x} -\n  \\sum_{k=1}^N\\frac{1}{k}(1-x)^k  \\right]^l\\\\&\n+ (1-x)^{N+1}\\,\\left[\\sigma^t_0 B_{ij}^{(n)} -\n  \\hat\\sigma^{t,(n)}_{ij,N}(0)\n  \\right]\\,,\n\\label{eq:match}\n\\end{split}\n\\end{equation}\nwhere $\\sigma^t_0 B^{(0)}_{ij} = \\hat\\sigma^{t,(0)}_{ij,N}(0)=0$, and\n$\\hat\\sigma^{t,(n)}_{ij,N}(x)$ denotes the soft expansion of the cross\nsection through order $(1-x)^N$, see \\eqn{eq:softexp}.  The coefficients\n$B^{(1)}_{ij}$ and $A^{(2,1)}_{ij}$ are given in numerical form in\n\\citeres{Marzani:2008az,Marzani:diss,Harlander:2009my},\\footnote{The\n  notation for $A^{(2,1)}_{ij}$ is $A^{(2)}_{ij}$ in that paper.}  while\n$A^{(3,2)}_{gg}$ can be found in \\citere{Marzani:diss} (where it is\ncalled $C_\\text{A}^3{\\cal C}^{(3)}$).  For the unknown coefficients\nthrough \\nklo{3}, {\\it we assume} \n\\begin{equation}\n  \\begin{split}\n\\sigma^t_0 B^{(n)}_{ij} &=\n\\hat\\sigma^{t,(n)}_{ij,N}(0)\\quad\\text{for}\\quad n\\geq 2\\,,\\qquad\nA^{(3,1)}_{ij}=0\\,.\n  \\end{split}\n\\end{equation}\nThe technical consequence of the matching procedure implemented in\n\\sushi{} is that it {\\it requires} the cross section to be expressed in\nterms of the soft expansion, i.e., one needs to set\n\\blockentry{GGHSOFT}{$n$,1}{=1} if the \\nklo{n} cross section is requested.\n\nThe effect of the matching at \\nklo{3} is shown in \\fig{fig:sgg3match}:\nthe soft expansion tends to a constant towards $x\\to 0$ by construction,\nand cannot reproduce the $\\ln^2x$-behavior of the exact result. The\nmerging of the two limits is very smooth and suggests that the matched\ncurve is not too far from the full result. Of course, the fact that some\ncoefficients in \\eqn{eq:match} are unknown introduces a theoretical\nuncertainty. However, we observe a change in the final cross section of\nonly about 0.5\\% when setting $A^{(3,1)}_{gg}=A^{(3,2)}_{gg}$ for a\n{\\abbrev SM}{} Higgs, for example.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.47\\textwidth]{figures\/N3LOmatch.pdf}\n\\end{center}\n\\vspace{-0.7cm}\n\\caption{Partonic cross section $\\hat{\\sigma}_{gg}^{t,(3)}\/\\sigma^t_0$\n  in $10^4$ according to \\eqn{eq:match} as a function of\n  $x=M_\\phi^2\/\\hat{s}$ with and without matching to the high-energy\n  limit. The order of the soft expansion applied in both cases is\n  $(1-x)^{16}$.}\n\\label{fig:sgg3match}\n\\end{figure}\n\nIt remains to say that all terms in $1\/M_\\text{t}$ are available including\nthe full $\\mu_\\text{F}$- and $\\mu_\\text{R}$-dependence. As in earlier versions of\n\\sushi{}, these parameters are accessible through the input parameters\n{\\tt SCALES(1)} and {\\tt SCALES(2)}.  Since top-quark mass effects\nare not known for the \\nklo{3} cross section, all settings\nof {\\tt GGHMT} involving $n=3$ except from\n\\blockentry{GGHMT}{30}{} have no effect in the current version\n\\sushi{\\_\\sushiversion}{}. The inclusion of $1\/M_\\text{t}$ terms is only\navailable for $M_\\phi<2M_\\text{t}$ and the matching to the\nhigh-energy limit only in a mass range $M_\\phi \\in\n\\left[100\\,\\text{GeV},300\\,\\text{GeV}\\right]$.\n\n\\subsection{Renormalization scale dependence}\\label{sec:scaledep}\n\nThe renormalization scale ($\\mu_\\text{R}$) dependence of the partonic cross\nsection can be written as\n\\begin{equation}\n\\begin{split}\n\\hat\\sigma_{ij} = \\sum_{n\\geq\n  0}\\sum_{l=0}^n\\left(\\api{(\\mu_\\text{R})}\\right)^{n+2}\n\\hat\\kappa^{(n,l)}_{ij}(\\mu_0)\\,l_\\text{R0}^l\\,,\n\\label{eq:murdepparton}\n\\end{split}\n\\end{equation}\nwhere $l_\\text{R0}=2\\,\\ln(\\mu_\\text{R}\/\\mu_0)$, and $\\mu_0$ is an arbitrary reference\nscale. The coefficients $\\hat\\kappa^{(n,l)}_{ij}(\\mu_0)$ are explicitly\ncontained in \\sushi{} (for $\\mu_0=M_\\phi$).  The dependence of the\ncross section on $\\mu_\\text{R}$ can be studied with \\sushi{} by varying the\ninput parameter \\blockentry{SCALES}{1}{}, which contains the numerical\nvalue for $\\mu_\\text{R}\/M_\\phi$. \\sushi{} will then insert this value into\n\\eqn{eq:murdepparton} and convolve the resulting partonic cross section\nover the {\\abbrev PDF}{}s. A decent picture of the $\\mu_\\text{R}$ dependence may require\nto perform this ``standard procedure'' ten times or more.\n\n\\sushi{\\_\\sushiversion}{} provides a considerably faster way to obtain the $\\mu_\\text{R}$\ndependence of the cross section by convolving the\n$\\hat\\kappa^{(n,l)}_{ij}(\\mu_0)$ with the {\\abbrev PDF}{}s {\\it before} varying\n$\\mu_\\text{R}$,\n\\begin{equation}\n  \\begin{split}\n    \\kappa^{(n,l)}(\\mu_0)=\n    \\hat\\kappa^{(n,l)}_{ij}(\\mu_0)\\otimes \\tilde\\phi_i\\otimes \\tilde\\phi_j\\,.\n    \\label{eq:kappahad}\n  \\end{split}\n\\end{equation}\nWe will refer to this as the ``{\\scalefont{.9} RGE} procedure''.  Due to the\nrenormalization group equation\\footnote{The power $n+3$\n  takes into account the fact that the {\\abbrev LO}{} cross section is of order\n  $\\alpha_s^2$.}\n\\begin{equation}\n\\begin{split}\n\\deriv{}{\\mu_\\text{R}^2}\\sigma_\\text{\\nklo{n}} = \\order{\\alpha_s^{n+3}}\n=\\deriv{}{\\mu_\\text{R}^2}\\hat\\sigma_{ij,\\text{\\nklo{n}}}\\,,\n\\label{eq:rginv}\n\\end{split}\n\\end{equation}\nwhich holds both at the partonic and the hadronic level, it suffices to\ncalculate the coefficients $\\kappa^{(n,l)}(\\mu_0)$ for $l=0$ and $n\\leq\n2$ if the \\nklo{3} result is requested. \\sushi{\\_\\sushiversion}{} does this by\ninitially assuming\n$\\mu_0\/M_\\phi=\\mu_\\text{R}\/M_\\phi=$~\\blockentry{SCALES}{1}{} in\n\\eqn{eq:murdepparton}.  All other coefficients are then determined via\nthe {\\abbrev QCD}{} $\\beta$ function, defined through\n\\begin{equation}\n\\begin{split}\n\\deriv{}{\\mu_\\text{R}^2}\\alpha_s(\\mu_\\text{R}) = \\alpha_s(\\mu_\\text{R})\\beta(\\alpha_s)\\,,\n\\qquad \\beta(\\alpha_s) = - \\api{}\\sum_{n\\geq 0}\n\\left(\\api{}\\right)^n\n\\beta_n\\,.\n\\label{eq:betafun}\n\\end{split}\n\\end{equation}\nExplicitly, one finds\n\\begin{equation}\n\\begin{split}\n  \\kappa^{(1,1)} &= 2\\,\\beta_0\\,\\kappa^{(0,0)}\\,,\\qquad \\kappa^{(2,2)} =\n  \\frac{3}{2}\\,\\beta_0\\,\\kappa^{(1,1)}\\,,\\qquad \\kappa^{(2,1)} =\n  2\\,\\beta_1\\,\\kappa^{(0,0)} +\n  3\\,\\beta_0\\,\\kappa^{(1,0)}\\,,\\\\ \\kappa^{(3,3)} &=\n  \\frac{4}{3}\\beta_0\\,\\kappa^{(2,2)}\\,,\\qquad \\kappa^{(3,2)} =\n  \\frac{3}{2}\\,\\beta_1\\,\\kappa^{(1,1)} +\n  2\\,\\beta_0\\,\\kappa^{(2,1)}\\,,\\\\ \\kappa^{(3,1)} &=\n  2\\,\\beta_2\\,\\kappa^{(0,0)} + 3\\,\\beta_1\\,\\kappa^{(1,0)} +\n  4\\,\\beta_0\\,\\kappa^{(2,0)}\\,.\n\\label{eq:murrec}\n\\end{split}\n\\end{equation}\nInserting these coefficients into the hadronic analog of\n\\eqn{eq:murdepparton}, it is possible to obtain the hadronic cross\nsection at any value of $\\mu_\\text{R}$ without any further numerical\nintegration. Since the $\\mu_\\text{R}$ dependence is typically much larger than\nthe $\\mu_\\text{F}$ dependence for gluon fusion, this feature of \\sushi{} saves a\nsignificant amount of computing time when aiming for an estimate of the\ntheoretical uncertainty of the cross section.\n\nThus, in addition to the usual output file {\\tt <outfile>}, running\n\\sushi{\\_\\sushiversion}{} with the standard command\\\\[-2em]\n\n{\\footnotesize\n\\begin{verbatim}\n.\/bin\/sushi <infile> <outfile>\n\\end{verbatim}}\n\n\\vspace*{-1em}\nwill produce an additional file {\\tt <outfile>\\_murdep} which contains\nthe gluon-fusion cross section for several values of $\\mu_\\text{R}$ in the form\n\\begin{equation}\n\\begin{array}{rrrrr}\n  \\mu_\\text{R}\\text{\/GeV}\\quad\n  & \\sigma_\\text{{\\abbrev LO}}\\text{\/pb}\\quad\n  & \\sigma_\\text{{\\abbrev NLO}}\\text{\/pb}\\quad\n  & \\sigma_\\text{{\\abbrev NNLO}}\\text{\/pb}\\quad\n & \\sigma_\\text{\\nklo{3}}\\text{\/pb}\n\\end{array}\n\\end{equation}\nwhere all cross sections are evaluated following\n\\eqn{eq:sigmasushi}, i.e.\\ they potentially contain quark-mass\neffects, {\\abbrev SUSY}{} corrections, and\/or electroweak effects.  The values\nof $\\mu_\\text{R}$ to be scanned over can be set in {\\tt <infile>}\nthrough\n\\begin{lstlisting}\nBlock SCALES\n    1  <mu0mh>\n  102  <min0>  <max0>  <N>\n\\end{lstlisting}\nwhich will cause \\sushi{\\_\\sushiversion}{} to evaluate the cross section at $N+1$\nequidistant points for $\\log\\mu_\\text{R}$ between $\\log\\mu_\\text{min}$ and\n$\\log\\mu_\\text{max}$, meaning\\footnote{{\\tt <N>}=$N$, {\\tt\n    <min0>}=$\\mu_\\text{min}\/\\mu_0$, {\\tt <max0>}=$\\mu_\\text{max}\/\\mu_0$,\n  {\\tt <mu0mh>}=$\\mu_0\/M_\\phi$.}\n\\begin{equation}\n  \\begin{split}\n    \\mu_\\text{R}\n    &=\n    \\mu_\\text{min}\\left(\\frac{\\mu_\\text{max}}{\\mu_\\text{min}}\\right)^{i\/N}\\,,\\qquad\n    i\\in\\{0,1,\\ldots,N\\}\\,.\n  \\end{split}\n  \\label{eq:mursampling}\n\\end{equation}\nIn addition, \\sushi{\\_\\sushiversion}{} includes a theoretical error estimate on the\ninclusive cross section into the standard output file {\\tt <outfile>},\ngiven as the maximal and minimal deviation (in pb) within the interval\n$\\mu_\\text{R}\\in [\\mu_1,\\mu_2]$ from the value at $\\mu_\\text{R}=\\mu_0$, using the\nsampled values of $\\mu_\\text{R}$ defined in \\eqn{eq:mursampling}, and the cross\nsections at the two boundaries $\\mu_\\text{R}=\\mu_1$ and $\\mu_2$. The interval is\nspecified as\n\\blockentry{SCALES}{101}{=\\{$\\mu_{1}\/\\mu_0$,$\\mu_2\/\\mu_0$\\}} (recall\nthat $\\mu_0\/M_\\phi=$\\blockentry{SCALES}{1}{}); it defaults to\n$[\\mu_1,\\mu_2]=[\\mu_0\/2,2\\mu_0]$.\n\nWe remark that this feature works at all perturbative orders through\n\\nklo{3}, for any settings in the blocks {\\tt GGHMT} or {\\tt GGHSOFT},\nand for any model under consideration. The only restriction is that all\nparameters except for the strong coupling constant need to be defined\non-shell. If this is not the case, \\sushi{\\_\\sushiversion}{} will not produce {\\tt\n  <outfile>\\_murdep}. Note that, due to \\eqn{eq:sigmasushi}, the\nprocedure implemented in \\sushi{\\_\\sushiversion}{} is a slightly refined version of\nthe one described above. In particular, this implies that the {\\abbrev NNLO}{}\n$\\mu_\\text{R}$ dependence is {\\it exact}, since it is fully determined by the\nexact {\\abbrev NLO}{} cross section $\\sigma_\\text{{\\abbrev NLO}}$. On the other hand, the\nrenormalization-scale dependence at \\nklo{3} derived from the {\\scalefont{.9}\n  RGE} procedure inherits whatever approximations were made (or not\nmade) at {\\abbrev NNLO}{}.  Thus, the results obtained through the standard and\nthe {\\scalefont{.9} RGE} procedure are usually not identical. For example, if\none keeps the full $x$-dependence at {\\abbrev NNLO}{}, one also obtains the full\n$x$-dependence of the $\\mu_\\text{R}$-terms at \\nklo{3} with the {\\scalefont{.9} RGE}\nprocedure, while the standard procedure would only provide them in the\nsoft expansion.\n\n\\subsection{Effective Lagrangian - \\dimension{5} operators}\\label{sec:dim5}\n\nLet us start from a particular well-defined theory {\\abbrev TH}{}; in the\ncurrent version of \\sushi{}, this could be the {\\abbrev SM}{}, a general {\\abbrev 2HDM}{},\nthe {\\abbrev MSSM}{}, or the {\\abbrev NMSSM}{}. We may now include additional gauge\ninvariant \\dimension{5} operators to {\\abbrev TH}{} which couple the neutral\nHiggs bosons of {\\abbrev TH}{} to gluons in the following\nway\\footnote{{\\abbrev CP}{}-even and -odd scalars, which couple\nthrough \\dimension{5} operators only, can also be studied, see the description after \\eqn{eq:yukfactors}.}:\n\\begin{equation}\n\\begin{split}\n{\\cal L} &= {\\cal L}_\\text{{\\abbrev TH}} +\n\\sum_{i=1}^{N_1}\\frac{\\alpha_s}{12\\pi v}c_{5,1i}\\,H_{1i}G^a_{\\mu\\nu}G^{a,\\mu\\nu} +\n\\sum_{i=1}^{N_2}\\frac{\\alpha_s}{8\\pi v}c_{5,2i}\\,H_{2i}G^a_{\\mu\\nu}\\tilde\nG^{a,\\mu\\nu}\\,.\n\\label{eq:leff}\n\\end{split}\n\\end{equation}\nHere, ${\\cal L}_\\text{{\\abbrev TH}}$ is the Lagrangian of the initial theory\n{\\abbrev TH}{}, $G^a_{\\mu\\nu}$ is the gluonic field strength tensor with\ncolor index $a$ and Lorentz indices $\\mu$ and $\\nu$, and $\\tilde\nG_{\\mu\\nu}^a\\equiv \\varepsilon_{\\mu\\nu\\rho\\sigma}G^{a,\\rho\\sigma}$ is\nits dual ($\\varepsilon^{0123}=+1$). As usual, $\\alpha_s$ is the strong\ncoupling constant and $v$ the {\\abbrev SM}{} Higgs-boson vacuum expectation\nvalue, which we express in terms of Fermi's constant\n$v=1\/\\sqrt{\\sqrt{2}G_F}$.  $N_1$ and $N_2$ are the numbers of {\\scalefont{.9}\n  CP}-even and {\\scalefont{.9} CP}-odd Higgs bosons of the theory,\nrespectively. The particles themselves are generically denoted by\n$H_{1i}$ and $H_{2i}$ (cf.\\ also Table\\,\\ref{tab:htype} below).\n\nThe $c_{5,ni}$ denote dimensionless Wilson coefficients which are\nunderstood as perturbative series in $\\alpha_s$:\n\\begin{equation}\n c_{5,ni} = \\sum_{k=0}^3 \\left(\\frac{\\alpha_s}{\\pi}\\right)^k c_{5,ni}^{(k)}\\,.\n\\end{equation}\nThe normalization is such that $c_{5,ni}^{(0)}=1$ corresponds to the\n{\\abbrev LO}{} contribution of an infinitely heavy up-type quark $u'$ with\n{\\abbrev SM}{}-like couplings.\\footnote{``{\\abbrev SM}-like'' refers to the interaction\n  Lagrangian ${\\cal L}_\\text{int} = -(m_{u'}\/v)H_{1i}\\bar{u'} u'$ for a\n  {\\scalefont{.9} CP}-even, and ${\\cal L}_\\text{int} =\n  -i(m_{u'}\/v)H_{2i}\\bar{u'}\\gamma_5 u'$ for a {\\scalefont{.9} CP}-odd Higgs\n  boson.} The {\\abbrev NLO}{} term for a {\\scalefont{.9} CP}-even Higgs in this case\nwould be $c_{5,11}^{(1)}=\\tfrac{11}{4}$, etc. In a theory that obeys\nnaturalness, on the other hand, the order of magnitude of the Wilson\ncoefficients would be $c_{5,ni} = \\order{v\/\\Lambda}$, where $\\Lambda$ is\na scale of physics beyond the {\\abbrev SM}{}.\n\nThe basic structures for the implementation of the effective Lagrangian\nin \\eqn{eq:leff} have already been present in earlier versions of\n\\sushi{}. The reason for this is that the very same operators result\nfrom integrating out the top quark or heavy squarks and gluinos from\n${\\cal L}_\\text{{\\abbrev TH}}$. In fact, the {\\abbrev NNLO}{} corrections due to top\nquarks, as well as the {\\abbrev NLO}{} corrections due to top, stop, and gluino\nare evaluated on the basis of these \\dimension{5}\noperators.\n\nThus, \\sushi{\\_\\sushiversion}{} does not implement any new results; it simply re-uses\npreviously available {\\tt functions} and {\\tt subroutines} in order\nto extend the gluon-fusion amplitudes to take into account the effect of\nthe additional terms in \\eqn{eq:leff}.  The numerical values for the\ncoefficients $c_{5,ni}$ in \\eqn{eq:leff} are specified through the newly\nintroduced {\\tt Block DIM5}.\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|ccc|}\n\\hline\n{\\tt <htype>} & {\\abbrev SM} & {\\abbrev 2HDM}\/{\\abbrev MSSM} & {\\abbrev NMSSM} \\\\\n\\hline\n11 & $H$ & $h$ & $H_1$\\\\\n12 & $-$ & $H$ & $H_2$\\\\\n13 & $-$ & $-$ & $H_3$\\\\\n21 & $A$ & $A$ & $A_1$\\\\\n22 & $-$ & $-$ & $A_2$\\\\\n\\hline\n\\end{tabular}\n\\caption[]{\\label{tab:htype} Assignment of the \\sushi{} input parameter\n\\blockentry{SUSHI}{2}{=<htype>} to the type of Higgs boson in the\nvarious models. A dash ($-$) means that the assignment is not\nmeaningful; it will lead to a fatal error in \\sushi{}. }\n\\end{center}\n\\end{table}\n\nFor example, within the {\\abbrev MSSM}{},\n\\begin{lstlisting}\nBlock DIM5\n   11   1.00000000E-04    # c5h0\n   12   4.00000000E-05    # c5H0\n   21  -3.00000000E-07    # c5A0\n\\end{lstlisting}\ncorresponds to $c_{5,11}^{(0)} \\equiv c_{5,h}^{(0)}=10^{-4}$,\n$c_{5,12}^{(0)} \\equiv c_{5,H}^{(0)}=4\\cdot 10^{-5}$, and\n$c_{5,21}^{(0)} \\equiv c_{5,A}^{(0)}=-3\\cdot 10^{-7}$.  Note that\n\\sushi{} calculates the cross section of only one particular type of\nHiggs boson per run (defined in \\blockentry{SUSHI}{2}{}),\nsee \\citere{Harlander:2012pb}. Correspondingly, only the pertinent entry in\n{\\tt Block DIM5} will have an effect on the result, the other\nentries will be ignored. The corrections at higher orders are\nspecified by setting \\blockentry{DIM5}{<k><ni>}{} for\ncoefficients $c_{5,ni}^{(k)}$ with $k\\geq 1$.  At {\\abbrev NLO}{} the\ncontribution of an infinitely heavy up-type quark $u'$ is thus\nreproduced by setting \\blockentry{DIM5}{11}{=1} and\n\\blockentry{DIM5}{111}{=2.75}.\n\nThe scale dependence of the \\dimension{5} Wilson coefficient can be\nderived from the non-renormalization of the trace anomaly\nterm\\,\\cite{Crewther:1972kn,Chanowitz:1972vd,Chanowitz:1972da,Collins:1976yq},\n\\begin{equation}\n  \\begin{split}\n    \\mu^2\\deriv{}{\\mu^2} \\beta(\\alpha_s) G_{\\mu\\nu}G^{\\mu\\nu} \\equiv 0\\,,\n  \\end{split}\n\\end{equation}\nwhere $\\beta(\\alpha_s)$ is given in \\eqn{eq:betafun}. Since also\n$\\alpha_sc_{5,1i}G_{\\mu\\nu}G^{\\mu\\nu}$ must be scale invariant, this immediately\nleads to~\\cite{Spira:1995mt,Brooijmans:2016vro}\n\\begin{equation}\n  \\begin{split}\n    c_{5,1i}(\\mu_\\text{R})=c_{5,1i}(\\mu_{\\phi})\n    \\frac{(\\beta\/\\alpha_s)|_{\\mu_\\text{R}}}{(\\beta\/\\alpha_s)|_{\\mu_{\\phi}}}\\,.\n    \\label{eq:c5evolve}\n  \\end{split}\n\\end{equation}\nPerturbatively, we can write this as\n\\begin{equation}\n  c_{5,1i}(\\mu_\\text{R})=\n  \\sum_{n\\geq 0}\\sum_{l=0}^n\\left(\\frac{\\alpha_s(\\mu_\\text{R})}{\\pi}\\right)^n\n  c_{5,1i}^{(n,l)}(\\mu_{\\phi})l_{R\\phi}^l\\,,\n\\label{eq:pertc5} \n\\end{equation}\nwith $l_{R\\phi}=2\\ln(\\mu_\\text{R}\/\\mu_{\\phi})$,\n\\begin{equation}\n  c_{5,1i}^{(n,0)} = c_{5,1i}^{(n)}(\\mu_{\\phi})\\,,\\quad\n  c_{5,1i}^{(n,n)} = 0\\qquad \\forall\\ n\n\\end{equation}\nand, through {\\abbrev NNLO}{},\n\\begin{equation}\n\\begin{split}\n  &c_{5,1i}^{(2,1)}=\\beta_0c_{5,1i}^{(1,0)}-\\beta_1c_{5,1i}^{(0,0)}\\,,\\\\&\nc_{5,1i}^{(3,2)}=\\beta_0(\\beta_0c_{5,1i}^{(1,0)}-\\beta_1c_{5,1i}^{(0,0)})\\,,\\quad\nc_{5,1i}^{(3,1)}=2(\\beta_0c_{5,1i}^{(2,0)}-\\beta_2c_{5,1i}^{(0,0)})\\,.\n\\end{split}\n\\end{equation}\nSetting \\blockentry{DIM5}{0}{=1} makes \\sushi{} evolve the Wilson\ncoefficient perturbatively, i.e.\\ according to \\eqn{eq:pertc5}; this\nis the default. On the other hand, one can also employ\n\\eqn{eq:c5evolve} for the evolution by setting \\blockentry{DIM5}{0}{=2},\nsimilar to the implementation in {\\tt HIGLU}~\\cite{Spira:1995mt}.\nThe evolution can also be switched off\n(i.e.\\ $c_{5,1i}(\\mu_\\text{R})=c_{5,1i}(\\mu_{\\phi})$) by setting \\blockentry{DIM5}{0}{=0}.\nThe {\\scalefont{.9} RGE} procedure described in the previous\nsection is only applicable for \\blockentry{DIM5}{0}{=1}.\n\\sushi{} will assume the Wilson coefficient provided in the input {\\tt\nBlock DIM5} to be renormalized at $\\mu_{\\phi}=M_\\phi$. The corresponding\nvalues at $\\mu_\\text{R}$ (where $\\mu_\\text{R}$ is given in\n\\blockentry{SCALES}{1}{}) are output in {\\tt Block DIM5OUT}.\n\nMoreover, the inclusion of \\dimension{5} operators is not compatible with\nthe inclusion of $1\/M_\\text{t}$ terms, i.e.\\ \\sushi{} stops if \\blockentry{GGHMT}{1}{$\\neq$0} or\n\\blockentry{GGHMT}{2}{$\\neq$0}. The {\\abbrev LO}{} dependence including quark-mass\neffects must not be factored out, i.e. \\sushi{} only accepts\nthe setting \\blockentry{GGHMT}{-1}{=-1}, in order not to reweight\nthe \\dimension{5} operator contributions with top-quark mass effects.\n\nWe note that through the {\\tt Block FACTORS}, which existed also in\nearlier versions, \\sushi{} allows to alter the couplings of the Higgs\nboson to quarks and squarks. Thus, for example additional factors\n$\\kappa_t$ and $\\kappa_b$ for the Higgs-boson coupling to top and bottom\nquarks can be chosen. In case of the {\\abbrev SM}{} the corresponding Lagrangian\nthen takes the following form for the {\\abbrev CP}{}-even Higgs boson $H_{11}=H$\n\\begin{equation}\n \\mathcal{L}_\\text{{\\abbrev TH}} \\ni -\\kappa_t\\sqrt{2}\\frac{M_\\text{t}}{v}t\\bar t H -\\kappa_b\\sqrt{2}\\frac{M_\\text{b}}{v}b\\bar b H\\,.\n \\label{eq:yukfactors}\n\\end{equation}\nIt is therefore easily possible to perform an analysis as presented in\n\\citere{Grojean:2013nya} in \\sushi{}, where the dependence of the\ngluon-fusion cross section on $\\kappa_t$ and $c_{5,1i}$ is discussed.\nWe will later also focus on this dependence for a very boosted Higgs\ntaking into account the bottom-quark induced contribution in addition.\nMoreover, by setting the couplings to quarks and gauge bosons to\nzero through the settings in {\\tt Block FACTORS} and \\blockentry{SUSHI}{7}{=0}, respectively,\nalso {\\abbrev CP}{}-even or -odd scalars beyond the implemented models can be studied.\nWe will demonstrate this option by providing inclusive cross sections for a scalar with\na mass of $750$\\,GeV at the $13$\\,TeV {\\abbrev LHC}{} in \\sct{sec:numdim5}.\n\n\\section{Heavy-quark annihilation}\\label{sec:heavyquark}\n\nIn this section we shortly comment on the implementation of the total\ninclusive {\\abbrev NNLO}{} Higgs-production cross sections through heavy-quark\nannihilation, ${\\scriptstyle Q}'\\bar{\\scriptstyle Q}\\to \\phi$, as described in\n\\citere{Harlander:2015xur}. Its activation is through the presence of\nthe {\\tt Block QQH} in the input file, which has the following form:\n\\begin{lstlisting}\nBlock QQH\n         1      <parton1>\n         2      <parton2>\n        11      <v*y>\n        12      <mu>\n\\end{lstlisting}\nHere, {\\tt <parton1>}$\\in\\{1,\\dots,5\\}$ denotes the initial-state quark\nflavor ${\\scriptstyle Q}'$, and {\\tt <parton2>}$\\in\\{-1,\\dots,-5\\}$ the initial-state\nanti-quark flavor $\\bar {\\scriptstyle Q}$. {\\tt <v*y>} is the $\\Q'\\bar \\Q\\phi$ coupling in the\n$\\overline{\\mbox{\\abbrev MS}}$ scheme at scale {\\tt <mu>}$=\\mu$\/GeV, normalized such that the\n{\\abbrev SM}{} value of the $q\\bar qH$ coupling is {\\tt <v*y>}$=m_q(\\mu)$\/GeV.\nFor further details regarding the implementation in \\sushi{\\_\\sushiversion}{} and\nresults we refer to \\citere{Harlander:2015xur}.\n\nIf the {\\tt Block QQH} is provided, \\sushi{} will not calculate the\ngluon-fusion cross section. The calculation of heavy-quark annihilation\ncross sections is also compatible with cuts on the (pseudo)rapidity or\ntransverse momentum of the Higgs boson up to $\\mathcal{O}(\\alpha_s^3)$, controlled\nthrough the settings in {\\tt Block DISTRIB}.\nAlso $\\pt{}$ distributions (\\blockentry{DISTRIB}{1}{=1}) can be requested.\nSince all quarks\nare assumed massless in this approach, the underlying theory is chirally\nsymmetric. Therefore the results for a scalar and a pseudo-scalar Higgs\nare identical and the setting of \\blockentry{SUSHI}{2}{} is irrelevant.\nNote also that the collision of an up-type quark with a down-type\nanti-quark (or vice versa) implies that $\\phi$ carries an electric\ncharge.  The only model dependence of the $\\Q'\\bar \\Q\\phi$ cross section as\ncalculated by \\sushi{} is through the setting of the Yukawa coupling in\n\\blockentry{QQH}{11}, such that a calculation in the {\\abbrev SM}{}-mode is\nsufficient (\\blockentry{SUSHI}{1}{=0}), unless the Higgs mass should be\nobtained from some external code like {\\tt FeynHiggs}.\n\nOther parameters of the $\\Q'\\bar \\Q\\phi$ calculation are determined by the same\ninput values as they are used for the $b\\bar b\\phi$ cross section when no input\n{\\tt Block QQH} is present. In particular, the perturbative order of\n$\\Q'\\bar \\Q\\phi$ is controlled through \\blockentry{SUSHI}{6}{=$n$}, where\n$n=1,2,3$ results in the {\\abbrev LO}{}, {\\abbrev NLO}{}, or {\\abbrev NNLO}{} prediction,\nrespectively, and the renormalization and factorization scales (relative\nto $M_\\phi$) are defined through \\blockentry{SCALES}{11}{} and\n\\blockentry{SCALES}{12}{}, respectively.\n\n\\section{Numerical results}\\label{sec:numerics}\n\nThis section demonstrates the newly implemented features of \\sushi{\\_\\sushiversion}{}\nwith the help of exemplary numerical results. We start with a discussion\nof the convergence of the soft expansion at individual perturbative\norders up to \\nklo{3}, proceed with top-quark mass effects in the\neffective field-theory approach, move to the {\\scalefont{.9} RGE} procedure to\ndetermine the renormalization-scale dependence, before we use these\nfeatures to provide a prediction for the cross section of the {\\abbrev SM}{}\nHiggs boson. Finally, we study the effect of higher dimensional\noperators to the transverse momentum~$\\pt{}$ of the {\\abbrev SM}{} Higgs boson\nand provide inclusive cross sections for a {\\abbrev CP}{}-even scalar with a\nmass of $750$\\,GeV. For numerical results concerning heavy-quark annihilation, we refer the\nreader to \\citere{Harlander:2015xur}.\n\nIf not stated otherwise, the setup for the numerical evaluations is as\nfollows: The {\\abbrev LHC}{} center-of-mass energy is set to $\\sqrt{s}=13$\\,TeV,\nand the {\\abbrev SM}{} Higgs mass to $M_\\text{H}=125$\\,GeV. We employ {\\tt\nPDF4LHC15}~\\cite{Butterworth:2015oua,Dulat:2015mca,Harland-Lang:2014zoa,\nBall:2014uwa,Gao:2013bia,Carrazza:2015aoa,Carrazza:2015hva}\nas parton distribution functions\n({\\abbrev PDF}{}), where the {\\tt (n)nlo\\_mc} Monte Carlo is used by default, and\nthe {\\tt (n)nlo\\_100} Hessian sets if noted.  Since \\nklo{3} {\\abbrev PDF}{} sets\nare not available, we use the {\\abbrev NNLO}{} set also for the evaluation of the\n\\nklo{3} terms. Nevertheless, in the \\nklo{3} calculation, we evolve\n$\\alpha_s$ at $4$-loop level; using 3-loop running of $\\alpha_s$\ninstead, the final prediction of the cross section for a {\\abbrev SM}{} Higgs\nboson changes at the level of $10^{-5}$. The remaining input follows the\nrecommendation of the {\\abbrev LHC}{} Higgs cross section working group,\nsee~\\citere{Denner:2047636}. The on-shell charm-quark mass is set to\n$m_c^{\\rm OS} = 1.64$\\,GeV, which is the upper edge of the range given\nin \\citere{Denner:2047636}.  The central scale choice for the\nrenormalization and factorization scale is $\\mu_\\text{R}=\\mu_\\text{F}=M_\\text{H}\/2$.\n\nNote that the results of\n\\scts{sec:numsoftexp}--\\ref{sec:bestandrenorm} are obtained for\na {\\abbrev SM}{} Higgs boson. However, \\sushi{\\_\\sushiversion}{} allows to take into account\nthe effects of \\nklo{3} contributions in the heavy-top limit and\n$1\/M_\\text{t}$ terms to the {\\abbrev NNLO}{} contributions for any {\\abbrev CP}{}-even Higgs\nboson in the implemented models, as long as the mass of the Higgs boson\nunder consideration is sufficiently light, i.e.\\ below $2M_\\text{t}{}$.\nEffects of \\dimension{5} operators (see \\sct{sec:dim5} and\n\\ref{sec:numdim5}), on the other hand, can be taken into account for any\nof the neutral Higgs bosons of the implemented models and\n{\\abbrev CP}{}-even and -odd scalars, which couple through \\dimension{5} operators\nonly.\n\n\\subsection{Soft expansion up to \\nklo{3}}\n\\label{sec:numsoftexp}\n\nIn this section, we study the behavior of the expansion around the\n``soft limit'', $x\\rightarrow 1$, for the gluon-fusion cross section,\nsee also \\sct{sec:softexp}. For the sake of clarity, top-quark\nmass effects beyond {\\abbrev LO}{} will be neglected in this section, although\nthe {\\abbrev LO}{} cross section including the full top-quark mass dependence is\nfactored out to all orders (i.e.\\ we set \\blockentry{GGHMT}{-1}{=3}, see\n\\sct{sec:mt}).\nIn order to discuss the convergence of the soft expansion, we define the\nquantity\n\\begin{equation}\n \\left(\\frac{\\delta \\sigma}{\\sigma}\\right)^{\\text{\\nklo{n}}} =\n \\frac{\\sigma^t_{\\text{\\nklo{n}},N,a}}{\\sigma^t_{\\text{\\nklo{n-1}}}}-1\\quad\\text{\n   with } n\\geq 1 \\,,\n\\label{eq:delsigbysig}\n\\end{equation}\nwhere $\\sigma^t_\\text{\\nklo{n}}$ has been introduced in\n\\eqn{eq:softexp}.  Through $\\order{\\alpha_s^{n+1}}$, the exact\n$x$-dependence is taken into account. In the highest-order terms,\ni.e.\\ the terms of order $\\order{\\alpha_s^{n+2}}$ in\n$\\sigma^t_{\\text{\\nklo{n}},N,a}$, the soft expansion is applied\naccording to \\eqn{eq:softexp} up to order $(1-x)^N$ with $N\\leq\n16$. All studies in this subsection were performed without matching the\ncross section to the result at $x\\to 0$, i.e., we set\n\\blockentry{GGHMT}{$n\\cdot$10}{=0} for $n=1,2,3$.\n\nAt infinite order of the soft expansion, the value of the parameter $a$\nin \\eqn{eq:softexp} is obviously irrelevant.  If only a finite number of\nterms in the expansion is available, the dependence of the result on the\nparameter $a$ has been studied in detail in\n\\citere{Anastasiou:2016cez}. It was shown that the soft expansion seems\nto converge particularly well for small, non-negative values of $a$. The\ndifferences among the final results for different values of $a$ are smaller\nat higher orders, as we demonstrate subsequently. One\nobserves that the $\\mu_\\text{F}$-dependent terms of $\\hat\\sigma$ at {\\abbrev NLO}{} are\npolynomial in $x$, which means that they are {\\it identical} to their\nsoft expansion for $a=0$ once it is taken to sufficiently high order ($N=3$, to be\nspecific). This is no longer true with the choice $a > 0$. Let us add\nthat, since the $\\mu_\\text{R}$-dependent terms at {\\abbrev NLO}{} are proportional to\n$\\delta(1-x)$, they are the same whether the soft expansion is applied\nor not.\n\n\\begin{figure}\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics[width=0.47\\textwidth]{figures\/norm_scale05_NLO.pdf} &\n\\includegraphics[width=0.47\\textwidth]{figures\/norm_scale05_NNLO.pdf}  \\\\[-0.4cm]\n (a) & (b)\n\\end{tabular}\n\\end{center}\n\\vspace{-0.7cm}\n\\caption{ (a) Convergence of the {\\abbrev NLO}{} cross section as a function of\n$N$ for $a=0,1,2,3$ in \\eqn{eq:softexp}; (b) Convergence of the\n{\\abbrev NNLO}{} cross section as a function of $N$ for $a=0,1,2,3$ in\n\\eqn{eq:softexp}.  In both figures the colors depict $a=0$ (red),\n$a=1$ (blue), $a=2$ (green), $a=3$ (black). The black, dashed line\ncorresponds to the exact result in the heavy-top limit. \nThe results are obtained for a {\\abbrev SM}{} Higgs\nwith $M_\\text{H}=125$\\,GeV at the $\\sqrt{s}=13$\\,TeV {\\abbrev LHC}{}.}\n\\label{fig:softexp}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics[width=0.47\\textwidth]{figures\/norm_scale05_N3LO.pdf} &\n\\includegraphics[width=0.47\\textwidth]{figures\/norm_scale05_N3LO_gg.pdf}  \\\\[-0.4cm]\n (a) & (b)\n\\end{tabular}\n\\end{center}\n\\vspace{-0.7cm}\n\\caption{ (a) Convergence of the \\nklo{3} cross section as a\nfunction of $N$ for $a=0,1,2,3$ in \\eqn{eq:softexp}; (b) Convergence\nof the gg channel of the \\nklo{3} as a function of $N$ for\n$a=0,1,2,3$ in \\eqn{eq:softexp}.  A zoom for larger values of $N$ is\nprovided in the upper right corner of the figures.  In both figures\nthe colors depict $a=0$ (red), $a=1$ (blue), $a=2$ (green), $a=3$\n(black). The results are obtained for a {\\abbrev SM}{} Higgs\nwith $M_\\text{H}=125$\\,GeV at the $\\sqrt{s}=13$\\,TeV {\\abbrev LHC}{}. }\n\\label{fig:softexp2}\n\\end{figure}\n\n\\figs{fig:softexp}\\,(a) and (b) show the convergence of the soft\nexpansion at {\\abbrev NLO}{} and {\\abbrev NNLO}{}, respectively.  Both figures also\ninclude the result without soft expansion as dashed black line,\ni.e.\\ where $\\sigma^t_{\\text{\\nklo{n}},N,a}$ is replaced by\n$\\sigma^t_{\\text{\\nklo{n}}}$ in \\eqn{eq:delsigbysig}. At {\\abbrev NLO}{}, the\ncase $a=0$ appears to be clearly preferable; for larger value of $a$,\nthe soft expansion is further away from the exact\n$x$-dependence. For $N\\geq 9$, the deviation for $a=0$ is less\nthan $2.5$\\%.\\footnote{Note that this refers to the absolute {\\abbrev NLO}{}\n{\\it correction} term in pb; with respect to the total cross\nsection, this translates into an approximation which is better than\n$1.6$\\%.} It decreases down to $1.3$\\% at $N=16$, while the result\nfor $a=1$ is still more than $7$\\% off.  \n\nAt {\\abbrev NNLO}, convergence of the soft expansion appears to be a bit faster,\nwith no significant impact of the terms higher than $(1-x)^6$ both for\n$a=0$ and $a=1$. For $N\\geq 9$, the result for $a=0$ ($a=1$)\napproximates the exact $x$-dependence of the correction term to better\nthan $5$\\% ($2$\\%) (translating into about $0.9$\\% ($0.3$\\%) for the total cross\nsection).\n\n\\fig{fig:softexp2}~(a) depicts the convergence of the soft expansion for\nthe cross section at \\nklo{3}. Above $N=11$, the spread among the\ncurves for $a=0,1,2,3$ is of the order of $3$\\% of\n$\\delta\\sigma\/\\sigma$, which means about $0.1$\\% of the total cross\nsection. For completeness, the same plot for the dominant $gg$ channel\nalone is shown in \\fig{fig:softexp2}~(b). Note that in this case, we\nonly include the $gg$ channel also in the denominator of\n\\eqn{eq:delsigbysig}. At lower orders of the soft expansion, the curve\nfor $a=0$ behaves less smoothly compared to $a\\geq 1$; at sufficiently\nhigh orders though, all results can be considered consistent with each\nother at the level of accuracy indicated above.\n\n\\subsection{Top-quark mass effects through {\\abbrev NNLO}{} and matching to the high-energy limit}\n\\label{sec:topquarkmass}\n\nIn this section we comment on top-quark mass effects beyond the\nheavy-top limit, which can be taken into account in \\sushi{} up to\n$1\/M_\\text{t}^{10}$ at {\\abbrev LO}{} and {\\abbrev NLO}{} and up to $1\/M_\\text{t}^6$ at {\\abbrev NNLO}{}. As\nalready pointed out in \\sct{sec:mt}, a naive expansion of the\npartonic cross section in $1\/M_\\text{t}$ breaks down. Thus, in this section,\nwe apply the matching to the high-energy limit as described in\n\\sct{sec:mt}, i.e.\\ we set\n\\blockentry{GGHMT}{$n\\cdot$10}{=1} for $n=1,2,3$.\n\nRecall that the matching procedure of\n\\citeres{Harlander:2009mq,Harlander:2009my} requires the soft expansion\nof the partonic cross section. Thus before discussing the relevance of\nthe top-quark mass effects, it is necessary to study the convergence of\nthe soft expansion also for these terms.  For the result at {\\abbrev NLO}{} we\ncan compare to the result in the heavy-top limit, but also to the exact\ntop-quark mass dependence; the difference between these two results is\nabout $1$\\%.  At {\\abbrev NNLO}{}, on the other hand, only a comparison to the\nheavy-top limit is possible.  The results are shown in\n\\fig{fig:softexpmt}, including terms through $1\/M_\\text{t}^8$ at {\\abbrev NLO}{}, and\nthrough $1\/M_\\text{t}^4$ at {\\abbrev NNLO}{} (for the $gg$ and the $qg$ channels\nalso $1\/M_\\text{t}^6$ terms are implemented in \\sushi{} but provide a\nnegligible contribution, see \\fig{fig:mtterms} below).\nFollowing \\eqn{eq:delsigbysig}, we keep the exact\n$x$-dependence one order below to allow for a better comparison with\nthe figures of \\sct{sec:numsoftexp}. At {\\abbrev NLO}{}, one observes a nice convergence of\nthe soft expansion to the exact result, provided $a=0$. Terms beyond\n$(1-x)^{10}$ have only negligible effects on the final result in this\ncase. At {\\abbrev NNLO}{}, convergence of the soft expansion is significantly\nslower, but the available number of terms in this expansion seems\nsufficient for a prediction of the mass effects with permille level\naccuracy, provided that $a=0$ is indeed the most reliable choice for the\nparameter defined in \\eqn{eq:softexp}.  \\fig{fig:softexpmt2} shows\nthe \\nklo{3} result with matching to the high-energy limit as\ndescribed in \\sct{sec:mt}. The convergence of the soft expansion as a\nfunction of $N$ is slightly worse compared to the result without\nmatching, but shows a similar behavior as the results at {\\abbrev NLO}{} and\n{\\abbrev NNLO}{} depicted in \\fig{fig:softexpmt}. The correction\nat $N=16$ is comparable to the result without matching, see\n\\fig{fig:softexp2}.\n\n\\begin{figure}\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics[width=0.47\\textwidth]{figures\/norm_scale05_mt_NLO.pdf} &\n\\includegraphics[width=0.47\\textwidth]{figures\/norm_scale05_mt_NNLO.pdf}  \\\\[-0.4cm]\n (a) & (b)\n\\end{tabular}\n\\end{center}\n\\vspace{-0.7cm}\n\\caption{\n(a) Convergence of the {\\abbrev NLO}{} cross section as a function of $N$ for\n$a=0,1,2,3$ in \\eqn{eq:softexp} with top-quark mass effects up to $1\/M_\\text{t}^8$;\n(b) Convergence of the {\\abbrev NNLO}{} cross section as a function of $N$ for\n$a=0,1,2,3$ in \\eqn{eq:softexp} with top-quark mass effects up to $1\/M_\\text{t}^4$.\nIn both figures the colors depict $a=0$ (red), $a=1$ (blue), $a=2$ (green),\n$a=3$ (black). The black, dashed line corresponds to\nthe exact result in the heavy-top limit, the black, dot-dashed line\nto the exact result with full top-quark mass dependence (only known at {\\abbrev NLO}{}).\nThe results are obtained for a {\\abbrev SM}{} Higgs with $M_\\text{H}=125$\\,GeV\nat the $\\sqrt{s}=13$\\,TeV {\\abbrev LHC}{}.\n}\n\\label{fig:softexpmt}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics[width=0.47\\textwidth]{figures\/norm_scale05_mt_N3LO.pdf} &\n\\includegraphics[width=0.47\\textwidth]{figures\/norm_scale05_mt_N3LO_gg.pdf}  \\\\[-0.4cm]\n (a) & (b)\n\\end{tabular}\n\\end{center}\n\\vspace{-0.7cm}\n\\caption{ (a) Convergence of the \\nklo{3} cross section as a\nfunction of $N$ for $a=0,1,2,3$ in \\eqn{eq:softexp}; (b) Convergence\nof the $gg$ channel of the \\nklo{3} as a function of $N$ for\n$a=0,1,2,3$ in \\eqn{eq:softexp}.  A zoom for larger values of $N$ is\nprovided in the upper right corner of the figures.  In both figures\nthe colors depict $a=0$ (red), $a=1$ (blue), $a=2$ (green), $a=3$\n(black). In contrast to \\fig{fig:softexp2} the \\nklo{3} result\nis matched to the high-energy limit. \nThe results are obtained for a {\\abbrev SM}{} Higgs\nwith $M_\\text{H}=125$\\,GeV at the $\\sqrt{s}=13$\\,TeV {\\abbrev LHC}{}.}\n\\label{fig:softexpmt2}\n\\end{figure}\n\nLet us now discuss the top-quark mass effects at different orders\n$1\/M_\\text{t}^P$ in more detail, while applying the soft expansion through\n$(1-x)^{16}$ with $a=0$ (see \\eqn{eq:softexp}).  The result is presented\nin \\fig{fig:mtterms}, where the relative difference\n\\begin{equation}\n  \\begin{split}\n    \\left(\\frac{\\delta\\sigma}{\\sigma}\\right)_{M_\\text{t}}\n    =\\frac{\\sigma_{P}}%\n    {\\sigma_\\text{htl}}\n    -1\n    \\label{eq:delsigmt}\n  \\end{split}\n\\end{equation}\nto the heavy-top limit at the corresponding perturbative order is shown.\nAt {\\abbrev NLO}{}, $\\sigma_{P}$ is obtained by including terms of order\n$1\/M_\\text{t}^P$ in the partonic cross section and matching it to the $x\\to0$\nlimit (i.e.\\ \\blockentry{GGHMT}{1}{=$P$}, \\blockentry{GGHMT}{10}{=1},\n\\blockentry{GGHSOFT}{1}{=\\{1,16,0\\}}), while $\\sigma_\\text{htl}$ is the\nheavy-top limit at {\\abbrev NLO}{}\n(i.e.\\ \\blockentry{GGHMT}{1}{=}\\blockentry{GGHMT}{10}{=0},\n\\blockentry{GGHSOFT}{1}{=\\{0,0,0\\}}). In both cases, the value for the\ncross section provided by \\sushi{} in \\blockentry{XSGGHEFF}{1}{} is used.\n\nAt {\\abbrev NNLO}{}, we use \\eqn{eq:sigmasushi} which corresponds to the \\sushi{}\noutput \\blockentry{SUSHIggh}{1}, neglecting bottom- and charm-quark, and\nelectroweak effects\n(\\blockentry{FACTORS}{1}{=}\\blockentry{FACTORS}{3}{=}\\blockentry{SUSHI}{7}{=0}).\nFurthermore, we make sure that only the genuine {\\abbrev NNLO}{} effects of the\n$1\/M_\\text{t}$ terms are shown, by fixing the approximation used at\n$\\order{\\alpha_s^3}$; specifically, we set \\blockentry{GGHMT}{1}{=6},\n\\blockentry{GGHMT}{10}{=1}, and \\blockentry{GGHSOFT}{1}{=\\{1,16,0\\}}, both for\n$\\sigma_P$ and $\\sigma_\\text{htl}$. For the $\\mathcal{O}(\\alpha_s^4)$-terms, we apply\nthe analogous settings of the {\\abbrev NLO}{} case described above. I.e., we\ninclude terms of order $1\/M_\\text{t}^P$ in $\\sigma_P$ (modulo the restriction\nto $1\/M_\\text{t}^4$ for the pure quark channels, see above), and match them\nto the $x\\to 0$ limit, while we apply the usual heavy-top limit\nfor $\\sigma_\\text{htl}$.\n\nThe results are shown in \\fig{fig:mtterms}, together with the relative\ndifference of the {\\it exact} {\\abbrev NLO}{} cross section to its heavy-top\nlimit (black dashed).\nThe points at $P=0$ illustrate the effect of using the soft expansion\ncombined with matching to the result at $x=0$, as opposed to keeping the\nfull $x$ dependence (without matching). Both at {\\abbrev NLO}{} and {\\abbrev NNLO}{}, this\neffect is obviously larger than the genuine $1\/M_\\text{t}$-terms.  This\nunderlines that, as long as one works in a heavy-top approximation,\nwhich is strictly valid only for $x>M_\\phi^2\/(4M_\\text{t}^2)$, the full\n$x$-dependence is not necessarily an improvement w.r.t.\\ the soft\nexpansion, in particular if additional information like the $x\\to 0$\nlimit is available.\n\nBoth at {\\abbrev NLO}{} and {\\abbrev NNLO}{}, the $1\/M_\\text{t}$ terms exhibit\na nice convergence behavior. However, the observation at {\\abbrev NLO}{} is that,\nwhile the $1\/M_\\text{t}^0$ result almost exactly reproduces the full mass\ndependence after matching to the high-energy limit and employing the soft expansion,\nincluding higher-order mass effects moves the approximation\n{\\it away} from the exact result. Thus, we cannot expect that their\ninclusion at {\\abbrev NNLO}{} leads to an improved result w.r.t.\\ the heavy-top\nlimit.  Nevertheless, we believe that their overall behavior allows to\nderive an upper bound on the top-mass effects to the heavy-top limit of\nthe order of\n$1$\\%\\,\\cite{Harlander:2009my,Harlander:2009mq,Pak:2009dg}.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.47\\textwidth]{figures\/step2_scale05_mt_rel.pdf}\n\\end{center}\n\\vspace{-0.7cm}\n\\caption{ Relevance of $1\/M_\\text{t}^P$ terms to the cross section in the\nheavy-top limit in percent at {\\abbrev NLO}{} (black) up to $P=10$ and at {\\abbrev NNLO}{}\n(red) up to $P=6$ with respect to the exact heavy-top limit at the\ncorresponding order.  The black, dashed line corresponds to the exact\n{\\abbrev NLO}{} result. The results are obtained for a {\\abbrev SM}{} Higgs\nwith $M_\\text{H}=125$\\,GeV at the $\\sqrt{s}=13$\\,TeV {\\abbrev LHC}{}.  }\n\\label{fig:mtterms}\n\\end{figure}\n\n\\subsection{Cross section prediction for the {\\abbrev SM}{} Higgs boson and scale dependence}\n\\label{sec:bestandrenorm}\n\nHaving discussed the top-quark mass terms to the {\\abbrev NLO}{} and {\\abbrev NNLO}{}\ncross section in the heavy-top limit and the convergence of the soft\nexpansion, we can finally provide a prediction for the cross section of\nthe {\\abbrev SM}{} Higgs boson including its scale uncertainty. In this section\nwe make use of the Hessian {\\abbrev PDF}{} sets {\\tt PDF4LHC15\\_(n)nlo\\_100}.\nFollowing the arguments of the preceding sections, the best prediction\nof \\sushi{} is obtained with the following settings: use the\nperturbative result through \\nklo{3}, i.e.\\ set\n\\blockentry{SUSHI}{5}{=3}; at each order of the effective-theory result,\napply the soft expansion through $(1-x)^{16}$ with $a=0$, i.e.\\ set\n\\blockentry{GGHSOFT}{$n$}{=\\{1,16,0\\}} for $n\\in\\{1,2,3\\}$; take into\naccount top-quark mass terms to the predictions of the {\\abbrev NLO}{} and\n{\\abbrev NNLO}{} cross sections in the heavy-top limit through the settings\n\\blockentry{GGHMT}{$n$}{=4} for $n\\in\\{1,2\\}$, i.e.\\ $1\/M_\\text{t}^4$ terms\nare taken into account at {\\abbrev NLO}{} and {\\abbrev NNLO}{};\nmatch to the high-energy limit $x\\rightarrow 0$ at {\\abbrev NLO}{}, {\\abbrev NNLO}{},\nand \\nklo{3}, i.e.\\ set \\blockentry{GGHMT}{$n\\cdot$10}{=1} for\n$n=1,2,3$. The choice of $a=0$ is motivated through the reproduction\nof the correct scale dependence at {\\abbrev NLO}{} and the observations in\n\\sct{sec:numerics}. Also note that for all predictions in the effective\nfield-theory approach, we factor out the full top-quark mass dependence,\ni.e.\\ \\blockentry{GGHMT}{-1}{=3}.  Finally, we include the electroweak\ncorrection factor according to \\eqn{eq:sigmasushi}, i.e.\\ we set\n\\blockentry{SUSHI}{7}{=2}.  The exact {\\abbrev NLO}{} cross section of\n\\eqn{eq:sigmasushi} contains the contributions from the three heaviest\nquarks: top, bottom, and charm.  The numbers can be reproduced with the\ninput file {\\tt SM-N3LO\\_best.in} in the {\\tt example}-folder of the\n\\sushi{\\_\\sushiversion}{} distribution.\n\n\\newpage\nWith this setup, we obtain\n\\vspace{-7mm}\n\\begin{equation}\n\\begin{split}\n \\text{{\\abbrev NNLO}{}}:                 &\\qquad \\sigma = 43.55\\,\\text{pb}\\pm 4.44\\,\\text{pb}(\\mu_\\text{R})\\,,\\\\\n +\\text{\\nklo{3}}:               &\\qquad \\sigma = 45.20\\,\\text{pb}\\pm 1.61\\,\\text{pb}(\\mu_\\text{R})\\,,\\\\\n +\\text{$1\/M_\\text{t}$ effects at {\\abbrev NLO}{} and {\\abbrev NNLO}{}}\\,\\, &\\\\\n +\\text{matching ($x\\to 0$) at {\\abbrev NLO}{}, {\\abbrev NNLO}{} and \\nklo{3}}:&\\qquad \\sigma = 45.80\\,\\text{pb}\\pm 1.87\\,\\text{pb}(\\mu_\\text{R})\\,,\\\\\n +\\text{electroweak corrections}:&\\qquad \\sigma = 48.28\\,\\text{pb}\\pm 1.97\\,\\text{pb}(\\mu_\\text{R})\\,, \n \\label{eq:res}\n\\end{split}\n\\end{equation}\nwhere the uncertainty $\\pm \\Delta(\\mu_\\text{R})$ only takes into account the\nrenormalization-scale dependence. Here, $\\Delta(\\mu_\\text{R})$ is the maximum\ndeviation of the cross section within the interval $\\mu_\\text{R}\/M_\\text{H}\n\\in[1\/4,1]$ from the value at $\\mu_\\text{R}=M_\\text{H}\/2$. Each line of\n\\eqn{eq:res}, including the uncertainty, has been obtained in a single\nrun of \\sushi{}, which takes a few seconds on a modern desktop\ncomputer. The final result is perfectly consistent within its\nuncertainties with the prediction $48.58\\,\\text{pb}\\pm 1$\\,pb($\\mu_\\text{R}$) given\nin \\citere{Anastasiou:2016cez} and the result $48.1\\,\\text{pb}\\pm 2.0$\\,pb\n(without resummation) employing the Cacciari-Houdeau Bayesian approach~\\cite{Cacciari:2011ze}\nto estimate higher unknown orders presented in \\citere{Bonvini:2016frm}.\nWe note that the result of\n\\citere{Anastasiou:2016cez} was computed with the {\\abbrev NNLO}{} {\\abbrev PDF}{} set at\nall orders, whereas we employ the {\\abbrev NLO}{} {\\abbrev PDF}{} set for the {\\abbrev NLO}{} terms\nin \\eqn{eq:sigmasushi}.  If we employ {\\tt PDF4LHC15\\_nnlo\\_100} instead\nat all orders, we obtain $48.37$\\,pb.  Other uncertainties need to be\nadded as described in \\citeres{deFlorian:2016spz,Anastasiou:2016cez}.\n\n\\begin{figure}\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics[width=0.47\\textwidth]{figures\/hessian_scale_analytic.pdf} &\n\\includegraphics[width=0.47\\textwidth]{figures\/hessian_scale_num.pdf}  \\\\[-0.4cm]\n (a) & (b)\n\\end{tabular}\n\\end{center}\n\\vspace{-0.7cm}\n\\caption{ (a) {\\abbrev LO}{} (red, dotted), {\\abbrev NLO}{} (green, dashed), {\\abbrev NNLO}{}\n(blue, dot-dashed) and \\nklo{3} (black, solid) gluon-fusion cross\nsection in pb (see \\eqn{eq:sigmasushi}) as a function of\n$\\mu_\\text{R}\/M_\\text{H}$ (obtained in a single run); (b) Best prediction cross\nsection in pb as a function of $\\mu_\\text{F}\/M_\\text{H}$ (together with\n$\\mu_\\text{R}=M_\\text{H}\/2$) (blue) and $\\mu_\\text{F}\/M_\\text{H}=\\mu_\\text{R}\/M_\\text{H}$ (red) and\n$\\mu_\\text{R}\/M_\\text{H}$ (together with $\\mu_\\text{F}=M_\\text{H}\/2$) (black). Each curve\nis shown twice, once for $a=0$ (solid) and $a=1$ (dashed) in the soft\nexpansion at \\nklo{3}. The dotted, thin black line depicts the \\nklo{3}\nresult from (a). Both figures are obtained for a {\\abbrev SM}{} Higgs\nwith $M_\\text{H}=125$\\,GeV at the $\\sqrt{s}=13$\\,TeV {\\abbrev LHC}{}. }\n\\label{fig:scaledep}\n\\end{figure}\n\nRunning the input file {\\tt SM-N3LO\\_best.in} also generates a file\nincluding the renormalization-scale dependence. Its content is shown in\n\\fig{fig:scaledep}~(a). The dependence clearly\nreduces successively from {\\abbrev NLO}{} to \\nklo{3}.  Note that at\neach order we follow \\eqn{eq:sigmasushi} and thus include the\nelectroweak correction factor beyond {\\abbrev LO}{}.  The flat behavior around\n$\\mu_\\text{R}=M_\\text{H}\/2$ leads to a highly asymmetric scale variation around the\ncentral value, suggesting a symmetrization of the corresponding\nuncertainty band as done in \\eqn{eq:res}.  As explained in\n\\sct{sec:scaledep}, the $\\mu_\\text{R}$ dependence obtained through the {\\scalefont{.9}\n  RGE} procedure at \\nklo{n} is as precise as the calculation at\n\\nklo{n-1}, while in the standard procedure (by manually varying\n\\blockentry{SCALES}{1}{}), its precision is determined by the \\nklo{n}\ncalculation.  We show the result of the standard procedure in\n\\fig{fig:scaledep}~(b) (black lines). In addition, the $\\mu=\\mu_\\text{F}$\ndependence for $\\mu_\\text{R}=M_\\text{H}\/2$ (blue) and the combined $\\mu=\\mu_\\text{F}=\\mu_\\text{R}$\ndependence (red) are shown. In each case, the solid and dashed line\ncorresponds to setting $a=0$ and $a=1$ in \\eqn{eq:softexp},\nrespectively.  The differences between these two cases, as well as\nbetween the standard and the {\\scalefont{.9} RGE} procedure are small, except for small\nvalues of $\\mu$. We also observe that the behavior at low values of $\\mu$ in\n\\fig{fig:scaledep}~(b) is dependent on the soft expansion and the\nmatching performed at {\\abbrev NLO}{} and {\\abbrev NNLO}{}.\nHowever, within the interval $\\mu\\in\\left[M_\\text{H}\/4,M_\\text{H} \\right]$\nwhich we use for the uncertainty determination, the agreement is good.\n\n\\subsection{Dimension~$5$ operators}\n\\label{sec:numdim5}\n\nIn order to study the effect of the \\dimension{5} operators, it is\nhelpful to consider the fraction of events where the {\\abbrev SM}{} Higgs boson is\nproduced at transverse momenta above a certain value~$\\pt{}^\\text{cut}$. We define\n\\begin{equation}\n\\begin{split}\nR(\\pt{}^\\text{cut})\n=\\frac{1}{\\sigma^\\text{tot}}\\sigma(\\pt{}^\\text{cut})\\quad\\text{with}\\quad\n\\sigma(\\pt{}^\\text{cut})\n=\\int_{\\pt{}>\\pt{}^\\text{cut}}\n{\\rm d}\\pt{}\\frac{{\\rm d}\\sigma}{{\\rm d}\\pt{}}\\,,\n\\label{eq:ptrat}\n\\end{split}\n\\end{equation}\nwhere $\\sigma\\equiv \\sigma_{ni}(c_{5,ni})$ denotes the cross section for\nthe production of a Higgs boson $H_{ni}$ within the theory defined by\n\\eqn{eq:leff}, and follow the numerical setup described at the beginning\nof \\sct{sec:numerics}.  However, we do not take into account charm-quark\nand electroweak contributions and choose a $\\pt{}$-dependent\nrenormalization and factorization scale for the result presented in\n\\fig{fig:corrfactors}.\nIf not stated otherwise, the relative Yukawa couplings to top- and\nbottom quarks are set to one, i.e.\\ we discuss the specific model\n{\\abbrev TH}{} with additional \\dimension{5} operator.\nIn the subsequent {\\abbrev NLO}{} analysis, we set $c_{5}^{(1)}=\\tfrac{11}{4}c_{5}^{(0)}$,\ni.e.\\ our \\dimension{5} operator assumes the same (rescaled) {\\abbrev NLO}{}\ncorrection as for the top-quark induced Wilson coefficient.\n\nThe ratio $R(\\pt{}^\\text{cut})$ of \\eqn{eq:ptrat} is shown in\n\\fig{fig:r-11} for the {\\abbrev SM}{} Higgs boson as a function of\n(a)~$\\pt{}^\\text{cut}$ for various values of $c_{5,H}^{(0)}$, and (b)~$c_{5,H}^{(0)}$\nfor various values of $\\pt{}^\\text{cut}$.  Similarly, \\fig{fig:r-21}\nshows the ratio for a {\\abbrev CP}{}-odd Higgs boson with mass $125$\\,GeV. For\n\\fig{fig:r-11}~(a) and \\fig{fig:r-21}~(a), $\\sigma^\\text{tot}$ is chosen\nsuch that each $R(\\pt{}^{\\text{cut}})$ is normalized to its {\\abbrev NLO}{}\ninclusive cross section.  For \\fig{fig:r-11}~(b) and \\fig{fig:r-21}~(b),\n$\\sigma^{\\text{tot}}=\\sigma(\\pt{}^{\\text{cut}})$ for $c_5=0$ to ensure\nthat all curves start at one.  The minima, which are clearly visible\naround $\\pt{}^{\\text{cut}}=50$\\,GeV, are induced by the negative\ninterference with the bottom-quark induced contributions to gluon\nfusion, which turns into a positive interference for higher values of\n$\\pt{}^{\\text{cut}}$. Accordingly, these minima affect also the\ndependence on $c_5$ in \\fig{fig:r-11}~(b) and\n\\fig{fig:r-21}~(b), i.e.\\ the lowest curve is obtained for a value of $\\pt{}^{\\text{cut}}$\naround $40$\\,GeV. Apart from the impact on the\ninclusive cross section, the point-like\ninteraction encoded in the coefficient $c_5$ thus distorts the shape of\nthe $\\pt{}$ distributions with respect to the loop-induced massive top-\nand bottom-quark contributions, as expected.\n\n\\begin{figure}\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics[width=0.47\\textwidth]{figures\/c11.pdf} &\n\\includegraphics[width=0.47\\textwidth]{figures\/p11.pdf}  \\\\[-0.4cm]\n (a) & (b)\n\\end{tabular}\n\\end{center}\n\\vspace{-0.7cm}\n\\caption{\n(a) Ratio of $R(\\pt{}^\\text{cut})$ with different $c_{5,H}^{(0)}$ (see figure)\nand $R(\\pt{}^\\text{cut})$ with $c_{5,H}=0$ as a function of $\\pt{}^\\text{cut}$ in GeV;\n(b) Ratio of $R(\\pt{}^\\text{cut})$ and $R(\\pt{}^\\text{cut})$ with $c_{5,H}=0$\nas a function of $c_{5,H}^{(0)}$ for different $\\pt{}^\\text{cut}$ (see figure).\nIn both figures we set $c_{5,A}^{(1)}=\\tfrac{11}{4}c_{5,A}^{(0)}$.\nBoth figures are obtained for a {\\abbrev CP}{}-even {\\abbrev SM}{} Higgs\nwith $M_\\text{H}=125$\\,GeV at the $\\sqrt{s}=13$\\,TeV {\\abbrev LHC}{}.}\n\\label{fig:r-11}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics[width=0.47\\textwidth]{figures\/c21.pdf} &\n\\includegraphics[width=0.47\\textwidth]{figures\/p21.pdf}  \\\\[-0.4cm]\n (a) & (b)\n\\end{tabular}\n\\end{center}\n\\vspace{-0.7cm}\n\\caption{\n(a) Ratio of $R(\\pt{}^\\text{cut})$ with different $c_{5,A}^{(0)}$ (see figure)\nand $R(\\pt{}^\\text{cut})$ with $c_{5,A}=0$ as a function of $\\pt{}^\\text{cut}$ in GeV;\n(b) Ratio of $R(\\pt{}^\\text{cut})$ and $R(\\pt{}^\\text{cut})$ with $c_{5,A}=0$\nas a function of $c_{5,A}^{(0)}$ for different $\\pt{}^\\text{cut}$ (see figure).\nIn both figures we set $c_{5,A}^{(1)}=\\tfrac{11}{4}c_{5,A}^{(0)}$.\nBoth figures are obtained for a {\\abbrev CP}{}-odd Higgs\nwith $m_A=125$\\,GeV at the $\\sqrt{s}=13$\\,TeV {\\abbrev LHC}{}.\n}\n\\label{fig:r-21}\n\\end{figure}\n\nFollowing the study performed in \\citere{Grojean:2013nya}, we now work\nout the dependence of the cross section with a minimal cut on $\\pt{}$ on\nthe factors $\\kappa_t$ and $c_{5,H}^{(0)}$ for the {\\abbrev SM}{} Higgs\nboson\\footnote{Our $c_{5,H}^{(0)}$ corresponds to $\\kappa_g$ in\n\\citere{Grojean:2013nya}.}.  In addition, we include the dependence on\nthe bottom-quark induced contribution through the factor $\\kappa_b$,\nsince the latter is non-negligible for $\\pt{}^\\text{cut}<200$\\,GeV.  For\nthis study we also choose $\\pt{}$-dependent renormalization and\nfactorization scales $\\mu_\\text{R}=\\mu_\\text{F}=\\sqrt{M_\\text{H}^2+\\pt{}^2}\/2$, which is\npossible through the setting \\blockentry{SCALES}{3}{=1}.  We define\n$\\tilde{\\sigma}(\\pt{}^\\text{cut})$, which just includes the top-quark\ninduced contribution, i.e.\\ we set $\\kappa_t=1$ and $c_{5,H}=\\kappa_b=0$,\nand then perform a fit of\n\\begin{equation}\n \\frac{\\sigma(\\pt{}^\\text{cut})}{\\tilde{\\sigma}(\\pt{}^\\text{cut})}\n =(\\kappa_t+c_{5,H}^{(0)})^2+\\delta\\kappa_tc_{5,H}^{(0)}+\\epsilon (c_{5,H}^{(0)})^2\n +\\delta_{bt}\\kappa_b\\kappa_t+\\delta_{bg}\\kappa_bc_{5,H}^{(0)}+\\epsilon_b\\kappa_b^2\\,, \n\\end{equation}\nwhere we set $c_{5,H}^{(1)}=\\tfrac{11}{4}c_{5,H}^{(0)}$ and\n$\\delta$ and $\\epsilon$ are defined identically to\n\\citere{Grojean:2013nya}.\nIn addition, however, we include the\nbottom-quark induced contribution, which is understood as pure\ncorrection entering through $\\delta_{bg}$, $\\delta_{bt}$, and\n$\\epsilon_{b}$.  The values for $\\delta$ and $\\epsilon$ coincide at the\npercent level with the values of Table~$1$ in \\citere{Grojean:2013nya},\nwhere for completeness we note that our calculation also includes the\n$qq$ induced contribution to gluon fusion. For our numerical setup we\nshow the dependence of the five correction factors on the lower cut\n$\\pt{}^\\text{cut}$ in \\fig{fig:corrfactors}.\n\n\\begin{figure}[htp]\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics[width=0.47\\textwidth]{figures\/deleps_ptmin.pdf} &\n\\includegraphics[width=0.47\\textwidth]{figures\/delepswithb_ptmin.pdf}  \\\\[-0.4cm]\n (a)  & (b)\n\\end{tabular}\n\\end{center}\n\\vspace{-0.7cm}\n\\caption{(a) Correction factors $\\delta$, $\\epsilon$ as a function of the\nlower cut $\\pt{}^\\text{cut}$ in GeV and in addition (b) $\\delta_{bg}$, $\\delta_{bt}$ and $\\epsilon_{b}$\nas a function of $\\pt{}^\\text{cut}$. Both figures are obtained for a {\\abbrev SM}{}\nHiggs with $M_\\text{H}=125$\\,GeV at the $\\sqrt{s}=13$\\,TeV {\\abbrev LHC}{}.}\n\\label{fig:corrfactors} \n\\end{figure}\n\nAs can be seen in \\fig{fig:corrfactors}~(a), the larger the lower cut\n$\\pt{}^\\text{cut}$, the more the degeneracy between $\\kappa_t$ and\n$c_{5,H}$, which are indistinguishable in the inclusive cross section, is\nbroken.  On the other hand \\fig{fig:corrfactors}~(b) points out that for\nlow $\\pt{}^\\text{cut}<200$\\,GeV bottom-quark induced contributions\nshould also be taken into account.  The interferences of the latter with\nthe top-quark induced contributions on the one hand and with the\neffective coupling $c_{5,H}$ on the other hand, encoded in $\\delta_{bt}$\nand $\\delta_{bg}$, are identical only for low $\\pt{}^\\text{cut}$.  We\nnote that the cross section prediction for the {\\abbrev SM}{} Higgs boson of\ncourse should include the full correction by bottom quarks given by\n$\\delta_{bt}$ and $\\epsilon_b$.  For completeness we partially also\nreproduced Fig.~$2$ of \\citere{Grojean:2013nya}, which illustrates the\ndisentanglement of the degeneracy between $\\kappa_t$ and $c_{5,H}$.\n\nAs a last example we discuss the calculation of the gluon-fusion\ncross section for an arbitrary scalar, which couples to gluons\nthrough an effective operator $c_{5}^{(0)}=1$ only. Motivated\nby the background deviation in the diphoton channel at $750$\\,GeV\nin both {\\abbrev LHC}{} experiments~\\cite{CMS:2016owr,ATLAS750}, we choose\nthe mass of the scalar to be $m_X=750$\\,GeV. We pick an input\nfile for the {\\abbrev SM}{}, set the {\\abbrev SM}{} Higgs-boson mass to $M_\\text{H}{}=750$\\,GeV,\ninclude a \\dimension{5} operator through \\blockentry{DIM5}{11}{=1},\nbut set the {\\abbrev SM}{} Higgs-boson couplings\nto quarks and gauge bosons to zero in {\\tt Block FACTORS} and \nthrough \\blockentry{SUSHI}{7}{=0}. The results are shown\nin \\tab{tab:750}. We include the renormalization scale\nuncertainty $\\pm \\Delta(\\mu_\\text{R})$, which was obtained simultaneously. \nAgain $\\Delta(\\mu_\\text{R})$ is the maximum\ndeviation of the cross section within the interval $\\mu_\\text{R}\n\\in[1\/4,1]m_X$ and $\\mu_\\text{R} \\in[1\/2,2]m_X$ for\nthe central scale choices $\\mu_\\text{R}=\\mu_\\text{F}=m_X\/2$ and\n$\\mu_\\text{R}=\\mu_\\text{F}=m_X$, respectively. For this\npurpose the Wilson coefficient is evolved perturbatively,\ni.e. \\blockentry{DIM5}{0}{=1}. At \\nklo{3} the\nsoft expansion is performed up to $(1-x)^{16}$ with $a=0$.\nThe matching to the high-energy limit, $x\\to 0$, is not applied.\nSimilar to the {\\abbrev SM}{} Higgs boson we observe a good convergence of\nthe perturbative series with a renormalization scale\nuncertainty of less than $\\pm 1.3$ and $\\pm 2.9$\\% at \\nklo{3} {\\abbrev QCD}{}\nfor the central scale choices $\\mu_\\text{R}=\\mu_\\text{F}=m_X\/2$ and $\\mu_\\text{R}=\\mu_\\text{F}=m_X$,\nrespectively.\n\n\\begin{table}[h]\n\\begin{center}\n\\begin{tabular}{|c|cc|}\n\\hline\n$\\sigma(gg\\to X)$ [fb] & $\\mu_\\text{R}=\\mu_\\text{F}=m_X\/2$ & $\\mu_\\text{R}=\\mu_\\text{F}=m_X$ \\\\\n\\hline\n{\\abbrev LO}{}      & $246.2  \\pm 52.8$ & $185.8  \\pm 36.0$\\\\\n{\\abbrev NLO}{}     & $368.7  \\pm 43.1$ & $316.3  \\pm 39.1$\\\\\n{\\abbrev NNLO}{}    & $410.0  \\pm 19.1$ & $384.9  \\pm 24.0$\\\\\n\\nklo{3}   & $414.6  \\pm 5.4$  & $407.2  \\pm 11.7$\\\\\n\\hline\n\\end{tabular}\n\\caption[]{\\label{tab:750} Inclusive gluon-fusion cross section in fb\nfor a {\\abbrev CP}{}-even scalar with mass $m_X=750$\\,GeV, which couples to gluons through $c_{5}^{(0)}=1$ only.\nThe results are given at different orders \\nklo{k}, $k=0,1,2,3$, in {\\abbrev QCD}{}\nfor the $\\sqrt{s}=13$\\,TeV {\\abbrev LHC}{} for two renormalization\nand factorization scale choices.\nThe depicted uncertainty is the renormalization-scale uncertainty $\\pm \\Delta(\\mu_\\text{R})$.}\n\\end{center}\n\\end{table}\n\n\\section{Conclusions}\n\nWe presented the new features implemented in version {\\tt 1.6.0}\nof the code \\sushi{}.  Aside from the implementation of heavy-quark\nannihilation, many new features aim at the improvement of the\ngluon-fusion cross-section prediction and its associated uncertainty\nestimate. In particular, \\sushi{} now provides the soft expansion around\nthe threshold of Higgs production and the matching to the high-energy\nlimit for {\\abbrev CP}{}-even Higgs bosons, at {\\abbrev NLO}{}, {\\abbrev NNLO}{} and\n\\nklo{3} {\\abbrev QCD}{}. Top-quark mass effects beyond the usual infinite top-mass\nlimit can be taken into account at {\\abbrev NLO}{} and {\\abbrev NNLO}{}.  We investigated\nthe relevance of these effects for a {\\abbrev SM}{}-like Higgs boson with a mass\nof $125$\\,GeV and provide a prediction of the corresponding gluon-fusion\ncross section at the {\\abbrev LHC}{} with a center-of-mass energy of $13$\\,TeV.\nBoth for {\\abbrev CP}{}-even and -odd Higgs bosons, \\sushi{} now calculates the\nrenormalization-scale uncertainty simultaneously to the calculation of\nthe gluon-fusion cross section at the central scale.  Moreover, the\neffects of \\dimension{5} operators can be studied in any model currently\nsupported by \\sushi{}.  We showed how the degeneracy between the\ntop-quark mass contribution and a point-like \\dimension{5} operator\ncontribution can be broken at large values of the transverse momentum of\na Higgs boson with mass $125$\\,GeV. The implementation of arbitrary\n\\dimension{5} operators is also particularly suited for the study of new\n{\\abbrev CP}{}-even and -odd scalars beyond the implemented models. We showed the\nconvergence of the perturbative series for the inclusive gluon-fusion\ncross section of a scalar with mass $750$\\,GeV at the $13$\\,TeV {\\abbrev LHC}{}.\n\nOur description and the subsequent appendix include examples how the user can control the new features through\nthe setting of blocks in the input file of \\sushi{}. Example input files are\ncontained in the {\\tt example}-folder of the current \\sushi{} release to be found\nat \\cite{sushiwebpage}.\n\n\\section*{Acknowledgments}\n\nRVH would like to thank {\\scalefont{.9} DFG} for financial support.  SL\nacknowledges support by the {\\scalefont{.9} SFB} 676 ``Particles, Strings and\nthe Early Universe''. Many of the calculations presented in this paper\nhave been performed on the {\\scalefont{.9} FUGG} cluster at Bergische\nUniversit\\\"at Wuppertal.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:Intro}\n\n\n\\subsection{Background}\n\\label{sec:Bkgrnd}\n\n\\begin{enumerate}\n\\item \\label{Bkgrndi} In \\cite{Q}*{\\S1} Quillen proved his Theorem~B\n  which states that, for a functor $f\\colon \\cat X \\to \\cat Z$ and an\n  object $Z \\in \\cat Z$, the rather simple \\emph{over category}\n  $\\commacat{f}{Z}$ is a \\emph{homotopy fiber} of $f$ if $f$ has a\n  certain \\emph{property $B_{1}$}.\n\\item \\label{Bkgrndii} This was generalized in \\cite{DKS}*{\\S6} where\n  it was shown that increasingly weaker \\emph{properties} $B_{n}$\n  ($n>1$) allowed for increasingly less simple description of these\n  homotopy fibers as \\emph{$n$-arrow overcategories}\n  $\\subcommacat{f}{n}{Z}$.\n\\item \\label{Bkgrndiii} It was also noted that a sufficient condition\n  for a functor $f\\colon \\cat X \\to \\cat Z$ to have property $B_{n}$\n  was that the category $\\cat Z$ had a certain property $C_{n}$.\n\\end{enumerate}\n\n\\subsection{The current paper}\n\\label{sec:CurPap}\n\nOur main results in this paper are the following.\n\\begin{enumerate}\n\\item \\label{CurPapi} We show that for a zigzag $\\zigzag{f}{\\cat\n    X}{\\cat Z}{\\cat Y}{g}$ between categories in which $f$ has\n  property $B_{n}$ \\eqref{Bkgrndii} (and in particular if $\\cat Z$ has\n  property $C_{n}$) \\eqref{Bkgrndiii}, its \\emph{homotopy pullback}\n  admits a description similar to the one mentioned in\n  \\ref{sec:Bkgrnd} namely as a \\emph{$n$-arrow pullback category\n    $\\subcommacat{f\\cat X}{n}{g\\cat Y}$}.  Moreover its\n  \\emph{pullback} comes with a monomorphism into the homotopy pullback\n  and hence is itself a homotopy pullback if the monomorphism is a\n  weak equivalence.\n\\item \\label{CurPapii} We then deduce from this similar results for\n  zigzags in the categories of \\emph{relative categories} and\n  \\emph{$k$-relative categories} ($k>1$) and do this not only with\n  respect to their \\emph{Reedy structure}, but also with respect to\n  their \\emph{Rezk structure} which turns them into models for the\n  theories of \\emph{$(\\infty,1)$- and $(\\infty,k)$-categories}\n  respectively.\n\\item \\label{CurPapiii} We also note that a sufficient condition for a\n  category, a relative category or a $k$-relative category to have\n  \\emph{property $C_{3}$} is that it admits what we will call a\n  \\emph{strict $3$-arrow calculus}.\n\\item \\label{CurPapiv} Our main tool for proving all this consists of\n  the \\emph{sharp maps} of Hopkins and Rezk \\cite{R2} which, because\n  of their fibration-like properties we prefer to call\n  \\emph{fibrillations}.  They are the dual of Hopkins' \\emph{flat\n    maps} which have similar cofibration-like properties and which we\n  therefore call \\emph{cofibrillations}.  These cofibrillations do not\n  play any role in the current paper, except for a surprise appearance\n  in \\ref{def:strcalculus}(iii)$'$.\n\\end{enumerate}\n\n\\subsection{The genesis of the current paper}\n\\label{sec:genesis}\n\nThe original version of this paper consisted of only the results\nmentioned in \\ref{CurPapi} and the corresponding part of\n\\ref{CurPapiii}.  That was exactly what we needed in \\cite{BK3} where\nit enabled us to reduce the proof that certain pullbacks were\nhomotopy pullbacks to a rather straightforward calculation.  However\nour proof of these results was rather ad hoc and not very\nsatisfactory.\n\nFortunately two things happened.\n\\begin{enumerate}\n\\item \\label{genesisi} We discovered a manuscript of Charles Rezk\n  \\cite{R2} in which he studied the \\emph{sharp maps} of Mike Hopkins.\n  These maps seemed to be exactly what we needed.  Just like\n  fibrations, they could be used in a \\emph{right proper} model\n  category to obtain pullbacks which were homotopy pullbacks (which by\n  the way made us call them by the more suggestive name of\n  \\emph{fibrillations}).\n\\item \\label{genesisii} Moreover when subsequently we took a closer\n  look at the lemma of Quillen \\cite{Q}*{\\S1} which started it all and\n  which he used to prove his Theorem~B, we noted that this lemma was\n  essentially just an elegant way of constructing fibrillations, some\n  of which were exactly the ones we needed.\n\\end{enumerate}\n\nCombining all this with some simple properties of fibrillations we\nthen not only obtained a much better proof of the above mentioned\nresult \\ref{CurPapi}, but also realized that with relatively little\neffort this proof could be extended to a proof of similar results for\n\\emph{relative categories} and the \\emph{$k$-relative categories} of\n\\cite{BK2}, which as result the current manuscript.\n\n\\subsection{Acknowledgements}\n\\label{sec:Ack}\n\nWe would like to thank Phil Hirschhorn, Mike Hopkins and especially\nDan Dugger for several useful conversations.\n\n\n\\setcounter{tocdepth}{1}\n\\tableofcontents\n\n\\section{An overview}\n\\label{sec:Oview}\n\nThe paper consists of three parts, each of which consists of three\nsections.\n\n\\boldcentered{Part I: Categorical preliminaries}\n\nIn sections 2, 3 and 4 we discuss the various models and relative\ncategories involved and some of the relative functors between them.\n\\subsection{The categories involved}\n\\label{sec:CatInvlv}\n\n\\begin{enumerate}\n\\item \\label{CatInvlvi} The categories $\\mathbf{Rel}^{k}\\mathbf{Cat}$ ($k \\ge 0$) of small\n  \\emph{$k$-relative categories} of \\cite{BK2}.\n\n  For $k=1$ this is just the category $\\mathbf{RelCat}$ of small\n  \\emph{relative categories} of \\cite{BK1}.\n\n  For $k=0$ this is the category of small \\emph{maximal} relative\n  categories (i.e.\\ in which all maps are weak equivalences).  As\n  $\\mathbf{Rel}^{0}\\mathbf{Cat}$ thus is isomorphic, although not equal, to the category\n  $\\mathbf{Cat}$ of small categories, we \\emph{will often denote $\\mathbf{Rel}^{0}\\mathbf{Cat}$ by\n  $\\widehat{\\mathbf{Cat}}$}.\n\\item \\label{CatInvoviii} The categories $\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ and\n  $\\mathrm{s}^{k}\\cat S$ ($k\\ge 0$) which are respectively the categories\n  of \\emph{$k$-simplicial diagrams} in $\\widehat{\\mathbf{Cat}}$ and the categories of\n  small \\emph{$k$-simplicial spaces}, i.e.\\ $(k+1)$-simplicial sets.\n\\end{enumerate}\n\n\\subsection{The functors involved}\n\\label{sec:FuncInvov}\n\n\\begin{enumerate}\n\\item \\label{FuncInvovi} The \\emph{$k$-simplicial nerve functors}\n  $\\mathrm{s}^{k}\\N\\colon \\mathbf{Rel}^{k}\\mathbf{Cat} \\to \\mathrm{s}^{k}\\cat S$ ($k\\ge 0$) of\n  \\cite{BK2}, which for $k=0$ is just he \\emph{classical nerve\n    functor $\\N$}.\n\\item \\label{FuncInfovii} The functors $\\N_{*}\\colon \\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}\n  \\to \\mathrm{s}^{k}\\cat S$ ($k\\ge 0$) induced by $\\N$.\n\\item The (unique) functors $w_{*}\\colon \\mathbf{Rel}^{k}\\mathbf{Cat} \\to\n  \\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ ($k\\ge 0$) such that $\\N_{*}w_{*} = \\mathrm{s}^{k}\\N$.\n\\end{enumerate}\n\n\\subsection{The model and relative structures}\n\\label{sec:ModRlSt}\n\n\\begin{enumerate}\n\\item \\label{ModRlSti} We endow $\\cat S = \\mathrm{s}^{0}\\cat S$ with the\n  classical model structure and $\\widehat{\\mathbf{Cat}}$ with the Quillen equivalent\n  Thomason structure \\cite{T2}.\n\\item \\label{ModRlStii} We endow $\\mathrm{s}^{k}\\cat S$ and\n  $\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ ($k\\ge 1$) with the resulting Reedy model structure.\n\\item \\label{ModRlStiii} We endow $\\mathbf{RelCat}$ and $\\mathbf{Rel}^{k}\\mathbf{Cat}$ ($k>1$)\n  respectively with the Quillen equivalent Reedy \\emph{model}\n  structure which \\cite{BK1} is lifted from the Reedy structure on\n  $\\mathrm{s}\\cat S$ and the homotopy equivalent \\emph{relative} Reedy\n  structure which \\cite{BK2} is lifted from the Reedy structure on\n  $\\mathrm{s}^{k}\\cat S$.\n\\item \\label{ModRlStiv} In the application of our main result we will\n  consider the categories $\\mathbf{Rel}^{k}\\mathbf{Cat}$ and $\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ ($k\\ge 1$)\n  as models for the theory of $(\\infty,k)$-categories by endowing them\n  with a \\emph{Rezk} structure which has more weak equivalences than\n  the Reedy structure and we will denote those categories so endowed\n  by $\\mathrm{L}\\mathbf{Rel}^{k}\\mathbf{Cat}$ and $\\mathrm{L}\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$.\n\\end{enumerate}\n\n\\subsection{Properties of the functors involved}\n\\label{sec:FncPrp}\n\nThe functors considered in \\ref{sec:FuncInvov} above all have the same\nnice properties as the classical nerve functors (and we will therefore\nrefer to them as \\emph{abstract nerve functors}).\n\nIn particular they all are \\emph{relative functors} which are\n\\emph{homotopy equivalences}.\n\n\\boldcentered{Part II: Homotopy pullbacks and potential homotopy\n  pullbacks}\n\n\n\\subsection{}\n\\label{sec:empty}\n\nIn section \\ref{sec:HmtpyPlbk} we will define homotopy pullbacks of\nzigzags in a model category in a more general fashion than is usually\ndone, but that enables us to extend the definition to certain\nsaturated relative categories.\n\\begin{enumerate}\n\\item \\label{emptyi} In a model category we define a homotopy pullback\n  of a zigzag as \\emph{any} object which is weakly equivalent to its\n  image under a \\emph{``homotopically correct'' homotopy limit functor}.\n\\item \\label{emptyii} In a saturated relative category we then define\n  a homotopy pullback of a zigzag of a \\emph{any} object which is\n  weakly equivalent to its image under what we will call a \\emph{weak\n    homotopy limit functor} which is a functor which has only some of\n  the properties of the above (i) homotopy limit functors.\n\\item \\label{emptyiii} Our main result then is a \\emph{Global\n    equivalence lemma} which states that\n  \\begin{itemize}\n  \\item \\emph{if $f\\colon \\cat C \\to \\cat D$ is a homotopy equivalence\n    between saturated relative categories,} then \\emph{$\\cat C$ admits\n  weak homotopy limit functors iff $\\cat D$ does, and in that case $f$\n  preserves homotopy pullbacks.}\n  \\end{itemize}\n\\item \\label{emptyiv} This lemma not only\n  \\begin{itemize}\n  \\item implies that any two weak homotopy limit functors on the same\n    category yield the same notion of homotopy pullbacks of zigzags, and\n  \\item enables us, in view of the fact that model categories admit\n    such weak homotopy limit functors, to prove their existence\n    elsewhere.\n  \\end{itemize}\n  but also plays a role in the proof of our main result in section\n  \\ref{sec:PrfThm}.\n\\end{enumerate}\n\n\\subsection{}\n\\label{sec:onep6}\n\nIn section \\ref{sec:PotHmPb} we discuss, for a zigzag $\\zigzag{f}{\\cat\n  X}{\\cat Z}{\\cat Y}{g}$ in $\\mathbf{Rel}^{k}\\mathbf{Cat}$ or $\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ ($k\\ge\n0$) the \\emph{$n$-arrow pullback object} $\\subcommacat{f\\cat\n  X}{n}{g\\cat Y}$ which was mentioned in \\ref{sec:CurPap} above.\n\n\\subsection{}\n\\label{sec:onep7}\n\nIn section \\ref{sec:PrpBnCn} we recall the notions of \\emph{property\n  $B_{n}$} and \\emph{property $C_{n}$} for $\\widehat{\\mathbf{Cat}}$ which were\nmentioned in \\ref{sec:Bkgrnd} above and then extend them to the\ncategories $\\mathbf{Rel}^{k}\\mathbf{Cat}$ and $\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ for $k\\ge 1$.\n\n\\boldcentered{Part III: The main results and their proof}\n\n\n\\subsection{}\n\\label{sec:onep8}\n\nIn section \\ref{sec:MainRlt} we formulate our main results, Theorems\n\\ref{thm:MpPropBn}, \\ref{thm:SuffCn}, \\ref{thm:QFibQfib} and\n\\ref{thm:MpBn}.\n\nWe also prove Theorem \\ref{thm:SuffCn} and, assuming Theorem\n\\ref{thm:MpPropBn}, Theorems \\ref{thm:QFibQfib} and \\ref{thm:MpBn} as\nwell, but defer the proof of Theorem \\ref{thm:MpPropBn} itself until\nsection \\ref{sec:PrfThm}.\n\n\\begin{enumerate}\n\\item \\label{onep8i} Theorem \\ref{thm:MpPropBn} states that, for a\n  zigzag $\\zigzag{f}{\\cat X}{\\cat Z}{\\cat Y}{g}$ in $\\mathbf{Rel}^{k}\\mathbf{Cat}$ or\n  $\\mathrm{s}^{k}\\mathbf{Cat}$ ($k\\ge 0$), for which the map $f\\colon \\cat X \\to\n  \\cat Z$ has property $B_{n}$ or the object $\\cat Z$ has property\n  $C_{n}$ \\eqref{sec:onep7}, the \\emph{$n$-arrow pullback object}\n  $\\subcommacat{f\\cat X}{n}{g\\cat Y}$ \\eqref{sec:onep6} is actually a\n  \\emph{homotopy pullback} \\eqref{sec:empty} of that zigzag.\n\\item \\label{onep8ii} Theorem \\ref{thm:MpPropBn} states that a\n  sufficient condition on an object $\\cat Z \\in \\mathbf{Rel}^{k}\\mathbf{Cat}$ ($k\\ge 0$)\n  in order that it has property $C_{3}$ is that $\\cat Z$ admits what\n  we will call a \\emph{strict $3$-arrow calculus}.\n\\item \\label{onep8iii} If $k=1$, then this (ii) is in particular the\n  case if $\\cat Z$ is a \\emph{partial model category}, i.e.\\ a\n  relative category which has the \\emph{two out of six} property and\n  admits a \\emph{$3$-arrow calculus}.\n\n  These partial model categories have the property that \\cite{BK3}, if\n  $\\mathbf{RelCat}$ is endowed with the \\emph{Rezk} structure\n  \\eqref{ModRlStiv}, then\n  \\begin{itemize}\n  \\item every partial model category is Reedy equivalent to a fibrant\n    object in $\\mathrm{L}\\mathbf{RelCat}$ \\eqref{ModRlStiv}, and\n  \\item every fibrant object in $\\mathrm{L}\\mathbf{RelCat}$ is Reedy equivalent to a\n    partial model category.\n  \\end{itemize}\n  Consequently one has the following result for\n  $(\\infty,1)$-categories.\n\\item \\label{onep8iv} Theorem \\ref{thm:QFibQfib} that states that, for\n  every zigzag $\\zigzag{f}{\\cat X}{\\cat Z}{\\cat Y}{g}$ in $\\mathbf{RelCat}$\n  between partial model categories, the \\emph{$3$-arrow pullback\n    object} $\\subcommacat{f\\cat X}{3}{g\\cat Y}$ is a \\emph{homotopy\n    pullback} of this zigzag not only in $\\mathbf{RelCat}$, but also in\n  $\\mathrm{L}\\mathbf{RelCat}$, i.e.\\ in $(\\infty,1)$-categories.\n\\item \\label{onep8v} Finally in Theorem \\ref{thm:MpBn} we state a\n  similar result for $\\mathrm{L}\\mathbf{Rel}^{k}\\mathbf{Cat}$ ($k>1$), i.e.\\ for\n  $(\\infty,k)$-categories, which however is much weaker than that\n  Theorem \\ref{thm:QFibQfib}, as we have no model structure on\n  $\\mathbf{Rel}^{k}\\mathbf{Cat}$ for $k>1$ nor an analog for partial model categories.\n\\end{enumerate}\n\n\\subsection{}\n\\label{sec:onep9}\n\nIn section \\ref{sec:HopRzF} we review the \\emph{fibrillations} of\nHopkins and Rezk which can be defined in any \\emph{relative category\n  with pullbacks} and describe the following three lemmas which we\nneed in section \\ref{sec:PrfThm} to prove Theorem \\ref{thm:MpPropBn}.\n\n\\begin{enumerate}\n\\item \\label{onep9i} A \\emph{Quillen fibrillation lemma} which is a\n  reformulation and a slight strengthening of the lemma that Quillen\n  used to prove his Theorem~B, and which enables us to obtain the\n  needed fibrillations in $\\widehat{\\mathbf{Cat}}$.\n\n  If, for an object $\\cat D \\in \\widehat{\\mathbf{Cat}}$ and a not necessarily\n  relative functor $F\\colon \\cat D \\to \\widehat{\\mathbf{Cat}}$, $\\Gr F \\in \\widehat{\\mathbf{Cat}}$\n  denotes its Grothendieck construction and $\\pi\\colon \\Gr F \\to \\cat\n  D \\in \\widehat{\\mathbf{Cat}}$ denotes the associated projection functor, then this\n  lemma states that\n  \\begin{itemize}\n  \\item the projection functor $\\pi\\colon \\Gr F \\to \\cat D \\in\n    \\widehat{\\mathbf{Cat}}$ is a \\emph{fibrillation} iff the functor $F\\colon \\cat D\n    \\to \\widehat{\\mathbf{Cat}}$ is a \\emph{relative} functor, i.e.\\ sends every map\n    of $\\cat D$ to a weak equivalence in $\\widehat{\\mathbf{Cat}}$.\n  \\end{itemize}\n\\item \\label{onep9ii} A \\emph{fibrillation lifting lemma} which\n  enables us to obtain from the fibrillation in $\\widehat{\\mathbf{Cat}}$ obtained in\n  (i) above corresponding fibrillations in the categories\n  $\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ for $k\\ge 1$.\n\\item \\label{onep9iii} A \\emph{Hopkins-Rezk fibrillation lemma} which\n  enables us to use the fibrillations of (ii) above to obtain the\n  desired homotopy pullback in the categories $\\mathrm{s}^{k}\\mathbf{Cat}$ ($k\\ge\n  0$), i.e.\\ it states that\n  \\begin{itemize}\n  \\item for every pullback square in a right proper model category\n    \\begin{displaymath}\n      \\xymatrix{\n        {\\cat U} \\ar[r] \\ar[d]\n        & {\\cat Y} \\ar[d]\\\\\n        {\\cat V} \\ar[r]\n        & {\\cat Z}\n      }\n    \\end{displaymath}\n    in which the map $\\cat C \\to \\cat Z$ is a fibrillation, the object\n    $\\cat U$ is a homotopy pullback of the zigzag $\\cat V \\to \\cat Z\n    \\gets \\cat Y$.\n  \\end{itemize}\n\\end{enumerate}\n\n\\subsection{}\n\\label{sec:onep10}\n\nIn section \\ref{sec:PrfThm} we finally prove Theorem\n\\ref{thm:MpPropBn}.  We prove this first for the categories\n$\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ ($k\\ge 0$) by using the above \\eqref{sec:onep9}\n\\emph{three lemmas} and then lift these results to the categories\n$\\mathbf{Rel}^{k}\\mathbf{Cat}$ ($k\\ge 1$) by means of the \\emph{Global equivalence lemma}\nwhich was mentioned in \\ref{emptyiii} above.\n\nActually for $\\mathbf{RelCat}$ we did not have to use this fourth lemma as we\ncould have obtained this also using only the first three.\n\n\\part{Categorical Preliminaries}\n\n\n\\section{Relative categories}\n\\label{sec:RelCat}\n\n\\subsection{Summary}\n\\label{sec:RlCtSum}\nWe start with a brief review of\n\\begin{enumerate}\n\\item \\emph{relative categories} and \\emph{relative functors} between them,\n\\item \\emph{saturated}, \\emph{maximal} and \\emph{minimal} relative\n  categories,\n\\item a \\emph{homotopy relation} between relative functors,\n\\item \\emph{strict $3$-arrow calculi} on relative categories,\n\\end{enumerate}\nand\n\\begin{resumeenumerate}{5}\n\\item the associated \\emph{simplicial nerve functor} to simplicial\n  spaces, i.e.\\ bisimplicial sets.\n\\end{resumeenumerate}\n\n\\boldcentered{Relative Categories}\n\n\\subsection{Definition}\n\\label{def:relcat}\n\n\\begin{enumerate}\n\\item \\label{relcati} A \\textbf{relative category} is a pair $(\\cat C,\n  w\\cat C)$ (usually denoted by just $\\cat C$) consisting of an\n  \\textbf{underlying category} $\\cat C$ (sometimes denoted by\n  $\\und\\cat C$) and a subcategory $w\\cat C \\subset \\cat C$ which is\n  \\emph{only} required to contain \\emph{all} the objects of $\\cat C$\n  (and hence their identity maps).\n\\item \\label{relcatii} The maps or $w\\cat C$ are called \\textbf{weak\n    equivalences} and two objects of $\\cat C$ are called\n  \\textbf{weakly equivalent} if they can be connected by a finite\n  zigzag of weak equivalences.\n\\item \\label{relcatiii} By a \\textbf{functor} $f\\colon \\cat C \\to \\cat\n  D$ between two relative categories we will mean just a functor\n  $f\\colon \\und\\cat C \\to \\und\\cat D$ (i), and\n\\item \\label{relcativ} such a functor will be called a\n  \\textbf{relative functor} if it preserves weak equivalences, i.e.\\\n  if $f(w\\cat C) \\subset w\\cat D$.\n\\end{enumerate}\n\n\\boldcentered{Saturated, maximal and minimal relative categories}\n\n\n\n\\subsection{Definition}\n\\label{def:saturated}\n\n\\begin{enumerate}\n\\item \\label{saturi} A relative category $\\cat C$ will be called\n  \\textbf{saturated} if a map in $\\cat C$ is a weak equivalence iff\n  its image in $\\Ho\\cat C$ (i.e.\\ the category obtained from $\\cat C$\n  by formally inverting all the weak equivalences) is an isomorphism.\n\\end{enumerate}\nThis is in particular the case if\n\\begin{resumeenumerate}{2}\n\\item \\label{saturii} $\\cat C$ is a \\textbf{maximal} relative\n  category, i.e.\\ \\emph{all} maps of $\\cat C$ are weak equivalences\n\\end{resumeenumerate}\nor if\n\\begin{resumeenumerate}{3}\n  \\item \\label{saturiii} $\\cat C$ is a \\textbf{minimal} relative\n    category, i.e.\\ the identity maps re the \\emph{only} weak\n    equivalences.\n\\end{resumeenumerate}\n\n\n\\boldcentered{A homotopy relation between relative functors}\n\n\n\\subsection{Definition}\n\\label{def:HomRel}\n\nLet $\\boldsymbol 1^{w}$ denote the maximal relative category\n(Df.~\\ref{saturii}) which  consists of two objects $0$ and $1$ and\none map $0\\to 1$ between them.  Then $\\boldsymbol 1^{w}$ gives rise to the\nfollowing notions.\n\\begin{enumerate}\n\\item \\label{HomReli} Given two relative functors $f,g\\colon \\cat C\n  \\to \\cat D$ between relative categories, a \\textbf{strict homotopy}\n  $h\\colon f \\to g$ between them will be a natural weak equivalence,\n  i.e.\\ a map\n  \\begin{displaymath}\n    h\\colon \\cat C\\times\\boldsymbol 1^{w} \\longrightarrow \\cat D\n  \\end{displaymath}\n  such that $h(c,0) = fc$ and $h(c,1) = gc$ for every object or map $c\n  \\in \\cat C$.\n\\item \\label{HomRelii} Two relative functors $\\cat C \\to \\cat D$ then\n  are called \\textbf{homotopic} if they can be connected by a finite\n  zigzag of strict homotopies, and\n\\item \\label{HomReliii} a functor $f\\colon \\cat C \\to \\cat D$ is\n  called a \\textbf{homotopy equivalence} if $f$ is a relative functor\n  and if there exists a relative functor $f'\\colon \\cat D \\to \\cat C$\n  (called a \\textbf{homotopy inverse} of $f$) such that the\n  compositions $f'f$ and $ff'$ are homotopic to $1_{\\cat C}$ and\n  $1_{\\cat D}$ respectively.\n\\end{enumerate}\n\n\n\\boldcentered{$3$-arrow calculi}\n\n\\subsection{Definition}\n\\label{def:calculus}\n\nA relative category $\\cat C$ is said to admit a \\textbf{$3$-arrow\n  calculus} if there exist subcategories $\\cat U$ and $\\cat V \\subset\nw\\cat C$ which behave like the categories of the \\emph{trivial\n  cofibrations} and \\emph{trivial fibrations} in a model category in\nthe sense that\n\\begin{enumerate}\n\\item \\label{calculusi} for every map $u \\in \\cat U$, its pushouts in\n  $\\cat C$ exist and are again in $\\cat U$,\n\\item \\label{calculusii} for every map $v \\in \\cat V$, it's pullbacks\n  in $\\cat C$ exist and are again in $\\cat C$, and\n\\item \\label{calculusiii} the maps $w \\in w\\cat C$ admit a\n  \\emph{functorial} factorization $w = vu$ with $u \\in \\cat U$ and $v\n  \\in \\cat V$.\n\\end{enumerate}\n\n\\subsection{Remark}\n\\label{rem:easier}\n\nIt should be noted that the conditions \\ref{calculusi} and (ii) are\nstronger than the ones in \\cite{DK}*{8.2} and \\cite{DHKS}*{36.1}.\nHowever we prefer them as they are easier to use and are likely to be\nautomatically satisfied.\n\n\\subsection{Remark}\n\\label{rem:whystrict}\n\nIn \\cite{DK}*{8.2} and \\cite{DHKS}*{36.1} $3$-arrow calculi were\ndefined in the presence of the \\emph{two out of three} and the\n\\emph{two out of six} properties respectively with the result that\n\\emph{a $3$-arrow calculus on $(\\cat C,w\\cat C)$ automatically\n  restricted to a $3$-arrow calculus on $(w\\cat C, w\\cat C)$}.\n\nAs we do not want to assume the presence of the two out of\nthree or the two out of six property, however, we define a notion of\n\\emph{strict $3$-arrow calculi} as follows.\n\n\n\\boldcentered{Strict $3$-arrow calculi}\n\n\\subsection{Definition}\n\\label{def:strcalculus}\n\nA \\textbf{strict $3$-arrow calculus} on a relative category $(\\cat\nC,w\\cat C)$ will be a $3$-arrow calculus (Df.~\\ref{def:calculus})\nwhich restricts to a $3$-arrow calculus on $(w\\cat C, w\\cat C)$.\n\nAnother way of saying this by adding in Df.~\\ref{def:calculus} to the\nconditions (i) and (ii) the conditions that\n\\begin{enumerate}\n\\item [(i)$'$] every pushout of a map in $w\\cat C$ along a map in\n  $\\cat U$ is again in $w\\cat C$, and\n\\item [(ii)$'$] every pullback of a map in $w\\cat C$ along a map in\n  $\\cat V$ is again in $w\\cat C$.\n\\end{enumerate}\n\nWith other words the maps in $\\cat U$ and $\\cat V$ behave like trivial\ncofibrations and trivial fibrations in a \\emph{proper} model category.\n\nA more compact way of saying all this is that\n\\begin{enumerate}\n\\item [(iii)$'$] a strict $3$-arrow calculus on a relative category is\n  a functorial factorization of its weak equivalences into a\n  \\emph{trivial cofibrillation} followed by a \\emph{trivial fibrillation}\n\\end{enumerate}\nwhere fibrillations are as defined in Df.~\\ref{def:fibril} below and\ncofibrillations are defined dually.\n\n\n\\boldcentered{The simplicial nerve functor}\n\n\\subsection{Definition}\n\\label{def:smpnv}\n\nLet $\\mathbf{RelCat}$ denote the category of (small) \\emph{relative\n  categories} (Df.~\\ref{def:relcat}) and let $\\mathrm{s}\\cat S$ denote the\ncategory of \\textbf{simplicial spaces}, i.e.\\ \\emph{bisimplicial}\nsets.\n\nFurthermore, for every integer $p \\ge 0$, let $\\cat p$ denote the\n$p$-arrow category\n\\begin{displaymath}\n  0 \\longrightarrow \\cdots \\longrightarrow p\n\\end{displaymath}\nand let $\\cat p^{v}$ and $\\cat p^{w}$ denote respectively the\n\\emph{minimal} and \\emph{maximal} relative categories\n(Df.~\\ref{ModRlStiii} and (ii)) which have $\\cat p$ as their\nunderlying category.\n\nThe \\textbf{simplicial nerve functor} then is the functor\n\\begin{displaymath}\n  \\mathrm{s}\\N\\colon \\mathbf{RelCat} \\longrightarrow \\mathrm{s}\\cat S\n\\end{displaymath}\nwhich sends each object $\\cat C \\in \\mathbf{RelCat}$ to the bisimplicial set\nwhich in bidimension $(p,q)$ consists of the maps\n\\begin{displaymath}\n  \\cat p^{v}\\times \\cat q^{w} \\longrightarrow \\cat C \\in \\mathbf{RelCat}\n\\end{displaymath}\n\n\\section{The categories and functors involved}\n\\label{sec:CtFncInv}\n\n\n\\subsection{Summary}\n\\label{sec:CtFncSumm}\n\nWe now describe the categories and functors which we need in the paper\nand note that\n\\begin{enumerate}\n\\item \\label{CtFncSummi} these categories come with obvious notions of\n  \\emph{strict homotopies} and \\emph{homotopy equivalences} (as in\n  Df.~\\ref{rem:easier}),\n\\item \\label{CtFncSummii} these functors \\emph{preserve} these strict\n  homotopies and homotopy equivalences, and\n\\item \\label{CtFncSummiii} these functors all have \\emph{left\n    adjoints} which are \\emph{left inverses}.\n\\end{enumerate}\n\nIn section \\ref{sec:AbsNv} we then endow these categories with model\nor relative structures which\n\\begin{resumeenumerate}{4}\n\\item \\label{CtFncSummiv} are compatible with these homotopy\n  equivalences in the sense that \\emph{every homotopy equivalence is a\n  weak equivalence}, and\n\\item \\label{CtFncSummv} turn the functors between these categories\n  into relative functors which are \\emph{homotopy equivalences}.\n\\end{resumeenumerate}\n\n\\boldcentered{$k$-relative categories}\n\n\\subsection{Definition}\n\\label{def:krel}\n\nWe extend the definition of $k$-relative categories for $k \\ge 1$ of\n\\cite{BK2}*{2.3} to include also the case $k=0$ as follows.\n\nA \\textbf{$k$-relative category} ($k>0$) will be a $(k+2)$-tuple\n\\begin{displaymath}\n  \\cat C = (a\\cat C, v_{1}\\cat C, \\ldots, v_{k}\\cat C, w\\cat C)\n\\end{displaymath}\nconsisting of an \\textbf{ambient category} $a\\cat C$ and subcategories\n\\begin{displaymath}\n  v_{1}\\cat C, \\ldots, v_{k}\\cat C \\text{ and } w\\cat C\n  \\subset a\\cat C\n\\end{displaymath}\nthat each contain all the objects of $a\\cat C$ and form a\ncommutative diagram with $2k$ arrows of the form\n\\begin{displaymath}\n  \\xymatrix@C=0.5em{\n    & {w\\cat C} \\ar[dl] \\ar[dr]\\\\\n    {v_{1}\\cat C} \\ar[dr]\n    & {{\\cdot\\;\\cdot\\;\\cdot}}\n    & {v_{k}\\cat C} \\ar[dl]\\\\\n    & {a\\cat C}\n  }\n\\end{displaymath}\nand where $a\\cat C$ is subject to the conditions that\n\\begin{enumerate}\n\\item \\label{kreli} every map in $a\\cat C$ is a finite composition of\n  maps in the $v_{i}\\cat C$ ($1 \\le i \\le k$), and\n\\item \\label{krelii} every relation in $a\\cat C$ is the consequence of\n  the commutativity relations in $v_i\\cat C$ ($1\\leq i\\leq n$) and the commutativity in $a\\cat C$ of diagrams of the form\n  \\begin{displaymath}\n    \\xymatrix{\n      {\\cdot} \\ar[r]^{x_{1}} \\ar[d]_{y_{1}}\n      & {\\cdot} \\ar[d]^{y_{2}}\\\\\n      {\\cdot} \\ar[r]_{x_{2}}\n      & {\\cdot}\n    }\n  \\end{displaymath}\n  in which $x_{1}, x_{2} \\in v_{i}\\cat C$ and $y_{1}, y_{2} \\in\n  v_{j}\\cat C$ and $1\\leq i<j\\leq n$.\n\\end{enumerate}\n\nThe maps of $w\\cat C$ are called \\textbf{weak equivalences} and a\n\\textbf{relative functor} $f\\colon \\cat C \\to \\cat D$ between two\n$k$-relative categories will be a functor $f\\colon a\\cat C \\to a\\cat\nD$ such that\n\\begin{displaymath}\n  f(w\\cat C) \\subset w\\cat D\n  \\qquad\\text{and}\\qquad\n  f(v_{i}\\cat C) \\subset v_{i}\\cat D\n  \\qquad \\text{for all $i \\le i < k$.}\n\\end{displaymath}\n\n\\subsection{Remark}\n\\label{rem:onerel}\n\nIf $\\cat C$ is a $1$-relative category, then $a\\cat C = v_{1}\\cat C$\nand hence\n\\begin{itemize}\n\\item \\emph{$1$-relative categories are the same as relative\n    categories} (Df.~\\ref{def:relcat}).\n\\end{itemize}\n\n\\subsection{Remark}\n\\label{rem:zerorel}\n\nIf $\\cat C$ is a $0$-relative category, then $a\\cat C = w\\cat C$ and\nhence\n\\begin{itemize}\n\\item $0$-relative categories are the same as maximal relative\n  categories (Df.~\\ref{saturii}).\n\\end{itemize}\n\n\\boldcentered{Strict $3$-arrow calculi and $k$-relative categories}\n\n\\subsection{Definition}\n\\label{def:Str3ar}\n\nA \\textbf{strict $3$-arrow calculus} on a $k$-relative category $\\cat\nC$ is a $3$-arrow calculus (Df.~\\ref{def:calculus}) on $(a\\cat C,w\\cat\nC)$ which restricts to a $3$-arrow calculus on\n\\begin{displaymath}\n  (w\\cat C,w\\cat C)\\qquad\\text{and}\\qquad (v_{i}\\cat C,w\\cat C)\n  \\qquad \\text{for all $i \\le i \\le k$.}\n\\end{displaymath}\n\n\\boldcentered{The $k$-simplicial nerve functor}\n\n\\subsection{Definition}\n\\label{def:kSmpNv}\n\nLet $\\mathbf{Rel}^{k}\\mathbf{Cat}$ ($k\\ge 0$) denote the category of (small) $k$-relative\ncategories (Df.~\\ref{def:krel}) and let $\\mathrm{s}^{k}\\cat S$ denote the\ncategory of (small) \\textbf{$k$-simplicial spaces}, i.e.\\\n$(k+1)$-simplicial sets.\n\nFurthermore, for every integer $p \\ge 0$ let $\\abs{\\cat p} \\subset\n\\cat p$ (Df.~\\ref{def:smpnv}) denote the subcategory which consists of\nthe objects and their identity maps only and let\n\\begin{displaymath}\n  \\text{$\\cat p^{w}$ and\n  $\\cat p^{v_{i}}$ ($1 \\le i \\le k$) $\\in \\mathbf{Rel}^{k}\\mathbf{Cat}$}\n\\end{displaymath}\nbe such that\n\\begin{gather*}\n  a\\cat p^{w} = v_{i}\\cat p^{w} = w\\cat p^{w} = \\cat p\n  \\qquad \\text{($1 \\le i \\le k$), and}\\\\\n  a\\cat p^{v_{i}} = v_{i}\\cat p^{v_{i}} = \\cat p\n  \\quad\\text{and}\\quad\n  v_{j}\\cat p^{v_{i}} = w\\cat p^{v_{i}} = \\abs{\\cat p}\n  \\quad\\text{for $j \\ne i$.}\n\\end{gather*}\n\nThe \\textbf{$k$-simplicial nerve functor} then is the functor\n\\begin{displaymath}\n  \\mathrm{s}^{k}\\N\\colon \\mathbf{Rel}^{k}\\mathbf{Cat} \\longrightarrow \\mathrm{s}^{k}\\cat S\n\\end{displaymath}\nwhich each object $\\cat C \\in \\mathbf{Rel}^{k}\\mathbf{Cat}$ to the $(k+1)$-simplicial set\nwhich in multidimension $(p_{k}, \\ldots, p_{1}, q)$ consists of the\nmaps\n\\begin{displaymath}\n  \\cat p_{k}^{v_{k}} \\times \\cdots \\times \\cat p_{1}^{v_{1}}\\times\n  \\cat q^{w} \\longrightarrow \\cat C \\in \\mathbf{Rel}^{k}\\mathbf{Cat}\n\\end{displaymath}\n\n\\subsection{Remark}\n\\label{rem:agree}\n\nIf $k=1$, then (Rk.~\\ref{rem:onerel}) $\\mathbf{Rel}^{k}\\mathbf{Cat} = \\mathbf{RelCat}$\n(Df.~\\ref{def:smpnv}) and hence in that case Df.~\\ref{def:kSmpNv}\nagrees with Df.~\\ref{def:smpnv}.\n\n\\subsection{Notation}\n\\label{not:kzero}\n\nIf $k=0$, then\n\\begin{enumerate}\n\\item \\label{kzeroi} the category $\\mathbf{Rel}^{0}\\mathbf{Cat}$ is, in view of\n  Rk.~\\ref{rem:zerorel}, isomorphic, although not identical, with the\n  category $\\mathbf{Cat}$ of (small) categories and we will therefore\n  \\begin{itemize}\n  \\item \\emph{often denote $\\mathbf{Rel}^{0}\\mathbf{Cat}$ by $\\widehat{\\mathbf{Cat}}$.}\n  \\end{itemize}\n\\item \\label{kzeroii} $\\mathrm{s}^{0}\\cat S = \\cat S$, the category of\n  (small) simplicial sets, and\n\\item \\label{kzeroiii} the functor $\\mathrm{s}^{0}\\N\\colon \\mathbf{Rel}^{0}\\mathbf{Cat} \\to\n  \\mathrm{s}^{0}\\cat S$ is just the \\textbf{classical nerve functor} which\n  we will therefore denote by\n  \\begin{displaymath}\n    \\N\\colon \\widehat{\\mathbf{Cat}} \\longrightarrow \\cat S\n  \\end{displaymath}\n\\end{enumerate}\n\nClosely related to the $k$-simplicial nerve functor are the following\ntwo functors.\n\n\n\\boldcentered{The levelwise nerve functor}\n\n\\subsection{Definition}\n\\label{def:lvlnrv}\n\nLet $k\\ge 0$.  Then the \\textbf{levelwise nerve functor}\n\\begin{displaymath}\n  \\N_{*}\\colon \\mathrm{s}^{k}\\widehat{\\mathbf{Cat}} \\longrightarrow \\mathrm{s}^{k}\\cat S\n\\end{displaymath}\nwill be the functor in which $\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ denotes the category\nof $k$-simplicial diagrams in $\\widehat{\\mathbf{Cat}}$ and $\\N_{*}$ is dimensionwise\napplication of the classical nerve functor $\\N$\n(Nt.~\\ref{rem:onerel}).\n\n\\boldcentered{The higher equivalence functor}\n\n\\subsection{Definition}\n\\label{def:HighEq}\n\nFor every integer $k \\ge 0$ we denote by\n\\begin{displaymath}\n  w_{*}\\colon \\mathbf{Rel}^{k}\\mathbf{Cat} \\longrightarrow \\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}\n\\end{displaymath}\nthe \\textbf{higher equivalence functor} which, for every $k$-fold\ndimension $p_{*} = (p_{k},\\ldots,p_{1})$ sends each object $\\cat C \\in\n\\mathbf{Rel}^{k}\\mathbf{Cat}$ to the object $w_{p_{*}}\\cat C \\in \\widehat{\\mathbf{Cat}}$ which has as its\nobjects the maps\n\\begin{displaymath}\n  \\cat p_{k}^{v_{k}} \\times \\cdots \\times \\cat p_{1}^{v_{1}}\n  \\longrightarrow \\cat C \\in \\mathbf{Rel}^{k}\\mathbf{Cat}\n\\end{displaymath}\nand as maps the natural weak equivalences between them.\n\n\\intro The above functors have the following properties.\n\\subsection{Proposition}\n\\label{prop:RlCtsmpS}\n\n\\emph{The following diagram commutes}\n\\begin{displaymath}\n  \\vcenter{\n    \\xymatrix{\n      {\\mathbf{Rel}^{k}\\mathbf{Cat}} \\ar[rr]_-{\\mathrm{s}^{k}\\N} \\ar[dr]_{w_{*}}\n      && {\\mathrm{s}^{k}\\cat S}\\\\\n      & {\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}} \\ar[ur]_{\\N_{x}}\n    }\n  }\n  \\qquad\\qquad\n  k\\ge 0\n\\end{displaymath}\n\n\\begin{proof}\n  This follows directly from the definitions.\n\\end{proof}\n\n\\subsection{Proposition}\n\\label{prop:SmpCtAdj}\n\n\\begin{em}\n  For every integer $k \\ge 0$ the functors\n  \\begin{enumerate}\n  \\item $\\N_{*}\\colon \\mathrm{s}^{k}\\widehat{\\mathbf{Cat}} \\to \\mathrm{s}^{k}\\cat S$,\n  \\item $\\mathrm{s}^{k}\\N\\colon \\mathbf{Rel}^{k}\\mathbf{Cat} \\to \\mathrm{s}^{k}\\cat S$, and\n  \\item $w_{*}\\colon \\mathbf{Rel}^{k}\\mathbf{Cat} \\to \\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$\n  \\end{enumerate}\n  have a left adjoint which is a left inverse.\n\\end{em}\n\n\\begin{proof}\n  \\leavevmode\n  \\begin{enumerate}\n  \\item It clearly suffices to prove this for the functor $\\N$.  But\n    this was already done long ago in \\cite{GZ}*{II,~4.7}.\n  \\item This was shown in \\cite{BK2}*{4.3}.\n  \\item This follows from Pr.~\\ref{prop:RlCtsmpS} above and the\n    following result.  \\qedhere\n  \\end{enumerate}\n\\end{proof}\n\n\\subsection{Proposition}\n\\label{prop:LfAdjInvComp}\n\nIf, for two functors $g_{1}\\colon \\cat A \\to \\cat B$ and $g_{2}\\colon\n\\cat B \\to \\cat C$, both $g_{2}$ and $g_{2}g_{1}$ have left adjoints\nwhich are left inverses, then so does $g_{1}$.\n\n\\begin{proof}\n  Consider the diagram\n  \\begin{displaymath}\n    \\xymatrix{\n      {\\cat A} \\ar@<-0.25ex>[r]_{g_{1}} \\ar@<-0.75ex>[r];[]_{f_{1}}\n      & {\\cat B} \\ar@<-0.25ex>[r]_{g_{2}} \\ar@<-0.75ex>[r];[]_{f_{2}}\n      & {\\cat C} \\ar `u[ll] `[ll]_{f_{12}} [ll]\n    }\n  \\end{displaymath}\n  in which $f_{2}$ and $f_{12}$ are left adjoints of $g_{2}$ and\n  $g_{2}g_{1}$ and\n  \\begin{displaymath}\n    f_{2}g_{2} = 1,\\qquad\n    f_{12}g_{2}g_{1} = 1\n    \\qquad\\text{and}\\qquad\n    f_{1} = f_{12}g_{2}.\n  \\end{displaymath}\n  Then $f_{1}g_{1} = f_{12}g_{2}g_{1} = 1$ and it thus remains to show\n  that $f_{1}$ is a left adjoint of $g_{1}$ and this one does by\n  noting that every pair of objects $A \\in \\cat A$ and $B \\in \\cat B$\n  gives rise to a natural sequence of identity maps and isomorphisms\n  \\begin{displaymath}\n    \\vcenter{\n    \\xymatrix@C=1em@R=1.5ex{\n      {\\cat A(f_{1}B,A)} \\ar@{=}[d]\n      && {\\cat B(B,g_{1}A)} \\ar@{=}[d]\\\\\n      {\\cat A(f_{12}g_{2}B,A)} \\ar@{}[r]|{\\approx}\n      & {\\cat C(g_{2}B, g_{2}g_{1}A)} \\ar@{}[r]|{\\approx}\n      & {\\cat B(f_{2}g_{2}B,g_{1}A)}\n    }\n    }  \\qedhere\n  \\end{displaymath}\n\\end{proof}\n\n\\intro\nIt remains to deal with the results that were promised in \n\\ref{CtFncSummi} and (ii).\n\\boldcentered{Homotopy relations on $\\mathbf{Rel}^{k}\\mathbf{Cat}$, $\\mathrm{s}^{k}\\cat S$ and\n    $\\mathrm{s}^{k}\\mathbf{Cat}$}\n\n\\subsection{Definition}\n\\label{def:StrHmtp}\n\nOne can define \\textbf{strict homotopies} and \\textbf{homotopy\n  equivalences} in the categories $\\mathbf{Rel}^{k}\\mathbf{Cat}$, $\\mathrm{s}^{k}\\cat S$ and\n$\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ as in Df.~\\ref{def:HomRel} by choosing an ``obvious\nanalog'' of $\\boldsymbol 1^{w}$ as follows.\n\\begin{enumerate}\n\\item \\label{StrHmtpi} For $\\mathbf{Rel}^{k}\\mathbf{Cat}$ this will be the object $\\boldsymbol\n  1^{w} \\in \\mathbf{Rel}^{k}\\mathbf{Cat}$ (Df.~\\ref{def:kSmpNv}),\n\\item \\label{StrHmtpii} for $\\mathrm{s}^{k}\\cat S$ this will be the\n  standard multisimplex $\\Delta[0,\\ldots,0,1] \\in \\mathrm{s}^{k}\\cat S$,\n  and\n\\item \\label{StrHmtpiii} for $\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ this will be the\n  object $\\boldsymbol 1^{w}_{x} \\in \\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ such that, for every\n  $k$-fold dimension $p_{*} = (p_{k}, \\ldots, p_{1})$, $\\boldsymbol\n  1^{w}_{p_{*}} = \\boldsymbol 1^{w} \\in \\widehat{\\mathbf{Cat}}$.\n\\end{enumerate}\n\n\\subsection{Proposition}\n\\label{prop:PrsStHmt}\n\n\\begin{em}\n  The functors $\\mathrm{s}^{k}\\N$, $w_{*}$ and $\\N_{*}$\n  \\begin{enumerate}\n  \\item send strictly homotopic maps to strictly homotopic maps\n  \\end{enumerate}\n  and hence\n  \\begin{resumeenumerate}{2}\n    \\item send homotopy equivalences to homotopy equivalences.\n  \\end{resumeenumerate}\n\\end{em}\n\n\\begin{proof}\n  \\leavevmode\n  \\begin{enumerate}\n  \\item The classical nerve functor $\\N\\colon \\widehat{\\mathbf{Cat}} \\to \\cat S$ has\n    the property that (\\cite{La}*{\\S2} and \\cite{Le}), for every two\n    maps $f,g\\colon \\cat A \\to \\cat B \\in \\widehat{\\mathbf{Cat}}$\n    \\begin{itemize}\n    \\item $\\N$ sends every strict homotopy between $f$ and $g$ to a\n      strict homotopy between $\\N f$ and $\\N g$, and\n    \\item conversely, every strict homotopy in $\\cat S$ between $\\N f$\n      and $\\N g$ is obtained in this fashion.\n    \\end{itemize}\n  \\item This readily implies that $\\N_{*}$ has the same properties.\n  \\item The proof for $\\mathrm{s}^{k}\\N$ is just a higher dimensional\n    version of the proof for $k=1$ in \\cite{BK1}*{7.5}.\n  \\item A proof for $w_{*}$ then is obtained by combining\n    Pr.~\\ref{prop:RlCtsmpS} with (ii) and (iii) above.\\qedhere\n  \\end{enumerate}\n\\end{proof}\n\n\\section{Abstract nerve functors}\n\\label{sec:AbsNv}\n\n\\subsection{Summary}\n\\label{sec:AbsNvSm}\n\nWe now endow the categories $\\mathbf{Rel}^{k}\\mathbf{Cat}$, $\\mathrm{s}^{k}\\cat S$ and\n$\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ ($k\\ge 0$) with a \\emph{relative} or \\emph{model}\nstructure which will turn the functors $\\mathrm{s}^{k}\\N$, $w_{*}$ and\n$\\N_{*}$ into what we will call \\emph{abstract nerve functors}, i.e.\\\nfunctors which, in addition to the properties that wee mentioned in\nPr.~\\ref{prop:SmpCtAdj} and Pr.~\\ref{prop:PrsStHmt}, have the property\nthat\n\\begin{enumerate}\n\\item \\label{AbsNvSmi} in the categories involved all \\emph{homotopy\n    equivalences} are \\emph{weak equivalences}, and\n\\item \\label{AbsNvSmii} the functors between them are \\emph{homotopy\n    equivalences}.\n\\end{enumerate}\n\nWe do this in two different ways.  In the first we endow the\ncategories $\\mathbf{Rel}^{k}\\mathbf{Cat}$, $\\mathrm{s}^{k}\\cat S$ and $\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ ($k\\ge\n1$) with \\emph{Reedy} structures.  In the second we endow them with\n\\emph{Rezk} structures which have more weak equivalences and which\nturn these categories into models for the theory of\n$(\\infty,k)$-categories.\n\n\\boldcentered{Abstract nerve functors}\n\n\\subsection{Definition}\n\\label{def:AbsNvFn}\n\nA functor $f\\colon \\cat C \\to \\cat D$ between saturated relative\ncategories (Df.~\\ref{def:saturated}) will be called an\n\\textbf{abstract nerve functor} if it has the following four\nproperties:\n\\begin{enumerate}\n\\item \\label{AbsNvFni} The relative categories $\\cat C$ and $\\cat D$\n  come with \\emph{strict homotopies} for which the associated\n  \\emph{homotopy equivalences} are \\emph{weak equivalences}.\n\\item \\label{AbsNvFnii} The functor $f$ is a relative functor which is\n  a \\emph{homotopy equivalence}.\n\\item \\label{AbsNvFniii} The functor $f$ sends strictly homotopic maps\n  to strictly homotopic maps and hence sends \\emph{homotopy\n    equivalences} to \\emph{homotopy equivalences}.\n\\item \\label{AbsNvFniv} The functor $f$ has a \\emph{left adjoint} which\n  is a \\emph{left inverse}.\n\\end{enumerate}\n\n\n\\boldcentered{The Reedy structures}\n\n\\subsection{Definition}\n\\label{def:ReeStrct}\n\nWe endow\n\\begin{enumerate}\n\\item \\label{ReeStrcti} the category $\\cat S$ with the usual model\n  structure,\n\\item \\label{ReeStrctii} the category $\\widehat{\\mathbf{Cat}}$ with the \\emph{Quillen\n  equivalent} Thomason structure \\cite{T2},\n\\item \\label{ReeStrctiii} the categories $\\mathrm{s}^{k}\\cat S$ and\n  $\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ ($k\\ge 1$) with the resulting Reedy model\n  structures,\n\\item \\label{ReeStrctiv} the category $\\mathbf{RelCat}$ with the \\emph{Quillen\n  equivalent} Reedy model structure of \\cite{BK1}*{6.1} which was\nlifted from the Reedy structure on $\\mathrm{s}\\cat S$, and\n\\item \\label{ReeStrctv} the categories $\\mathbf{Rel}^{k}\\mathbf{Cat}$ ($k>1$) with the\n  \\emph{homotopy equivalent} relative Reedy structures of\n  \\cite{BK2}*{3.3} which were lifted from the Reedy structures on\n  $\\mathrm{s}^{k}\\cat S$.\n\\end{enumerate}\n\n\\subsection{Proposition}\n\\label{prop:AbsNvFncs}\n\n\\begin{em}\n  If the categories $s^{k}\\cat S$, $\\mathbf{Rel}^{k}\\mathbf{Cat}$ and\n  $\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ ($k\\ge 0$) are endowed with the relative\n  structures of \\ref{def:ReeStrct}, then the functors $\\N_{*}$,\n  $\\mathrm{s}^{k}\\N$ and $w_{*}$ are abstract nerve functors.\n\\end{em}\n\n\\begin{proof}\n  In view of Pr.~\\ref{prop:SmpCtAdj} and \\ref{prop:PrsStHmt} it\n  suffices to show that\n  \\begin{enumerate}\n  \\item the functors $N_{*}$, $\\mathrm{s}^{k}\\N$ ($k\\ge 1$) and $w_{*}$ are\n    homotopy equivalences, and\n  \\item the homotopy equivalences in $\\mathrm{s}^{k}\\cat S$, $\\mathbf{Rel}^{k}\\mathbf{Cat}$ and\n    $\\mathrm{s}^{k}\\mathbf{Cat}$ ($k\\ge 0$) are weak equivalences.\n  \\end{enumerate}\n\n  To prove (i) we note:\n  \\begin{lettered}\n  \\item As Dana Latch \\cite{La} has shown that the functor $\\N$ is a\n    homotopy equivalence it readily follows that so are the functors\n    $\\N_{*}$.\n  \\item That the functors $\\mathrm{s}^{k}\\N$ are homotopy equivalences was\n    shown in \\cite{BK2}*{3.4}.\n  \\item That the functors $w_{*}$ are homotopy equivalences then\n    follows a), b) and Pr.~\\ref{prop:RlCtsmpS} and the observation\n    that homotopy equivalences have the two out of three property.\n  \\end{lettered}\n  To prove (ii) we note\n  \\begin{resumelettered}{4}\n  \\item As the homotopy equivalences in $\\cat S$ are weak\n    equivalences, it readily follows that so are the homotopy\n    equivalences in $\\mathrm{s}^{k}\\cat S$.\n  \\item That the homotopy equivalences in $\\mathbf{Rel}^{k}\\mathbf{Cat}$ and\n    $\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ are also weak equivalences then follows from d)\n    and the fact that the functors $\\mathrm{s}^{k}\\N$ and $\\N_{*}$ preserve\n    homotopy equivalences and reflect weak equivalences.\\qedhere\n  \\end{resumelettered}\n\\end{proof}\n\n\n\\boldcentered{The Rezk structures}\n\n\\subsection{Definition}\n\\label{def:RzkStrct}\n\nIn \\cite{R1} Charles Rezk constructed a left Bousfield localization of\nthe Reedy structure on $\\mathrm{s}\\cat S$ and showed it to be a model for\nthe theory of $(\\infty,1)$-categories.\n\nFurthermore it was noted in \\cite{B} (and a proof thereof can be found\nin \\cite{Lu}*{\\S1}) that iteration of Rezk's construction yields for\nevery integer $k>1$, a left Bousfield localization of the Reedy\nstructure on $\\mathrm{s}^{k}\\cat S$ which is a model for the theory of\n$(\\infty,k)$-categories.\n\nWe therefore will denote\n\\begin{enumerate}\n\\item \\label{RzkStrcti} by $\\mathrm{L}\\mathrm{s}^{k}\\cat S$ ($k\\ge 1$) the\n  category $\\mathrm{s}^{k}\\cat S$ with this (iterated) Rezk structure,\n\\item \\label{RzkStrctii} by $\\mathrm{L}\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ ($k\\ge 1$) the\n  induced \\cite{H}*{3.3.20} \\emph{Quillen equivalent} Rezk model\n  structure, and\n\\item \\label{RzkStrctiii} by $\\mathrm{L}\\mathbf{Rel}^{k}\\mathbf{Cat}$ the \\emph{Quillen} or\n  \\emph{homotopy equivalent} Rezk structure lifted \\cite{BK2}*{4.2}\n  from the Rezk structure on $\\mathrm{s}^{k}\\cat S$ (or the Quillen\n  equivalent Rezk structure on $\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$) which categories\n  therefore are all models for the theory of $(\\infty,k)$-categories.\n\\end{enumerate}\n\n\\subsection{Proposition}\n\\label{prop:AllAbsnv}\n\n\\begin{em}\n  If the categories $\\mathrm{s}^{k}\\cat S$, $\\mathbf{Rel}^{k}\\mathbf{Cat}$ and\n  $\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ are endowed with the Rezk structures of\n  \\ref{def:RzkStrct}, then the functors $\\mathrm{s}^{k}\\N$, $w_{*}$ and\n  $\\N_{*}$ are abstract nerve functors.\n\\end{em}\n\n\\begin{proof}\n  This follows from Pr.~\\ref{prop:AbsNvFncs} and the fact that the\n  Rezk structures have more weak equivalences than the Reedy ones.\n\\end{proof}\n\n\\part{Homotopy pullback and potential homotopy pullback}\n\n\n\\section{Homotopy pullbacks}\n\\label{sec:HmtpyPlbk}\n\n\\subsection{Summary}\n\\label{sec:HmtpyPlbkSumm}\n\nAs we are concerned not only with homotopy pullbacks in the\n\\emph{model categories} $\\mathbf{RelCat}$ and $\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ ($k\\ge 0$),\nbut  also in the \\emph{saturated relative categories} $\\mathbf{Rel}^{k}\\mathbf{Cat}$\n($k>1$) on which we do \\emph{not} have a model structure, we will\ndefine homotopy pullback in a more general fashion than is usually\ndone.\n\\begin{enumerate}\n\\item \\label{HmtpyPlbkSummi} In a model category we define a homotopy\n  pullback of a zigzag as \\emph{any object} which is weakly equivalent\n  to its image under a \\emph{``homotopically correct'' homotopy limit functor}.\n\\item \\label{HmtpyPlbkSummii} In a saturated relative category we then\n  define a homotopy pullback of a zigzag as \\emph{any object} weakly\n  equivalent to its image under what we will call a \\emph{weak\n    homotopy limit functor} which is a functor which has only some of\n  the properties of the above (i) homotopy limit functors.\n\\item \\label{HmtpyPlbkSummiii} Our main result then is a \\emph{global\n    equivalence lemma} which states that, if $f\\colon \\cat C \\to \\cat\n  D$ is a homotopy equivalence between saturated relative categories,\n  then $\\cat C$ \\emph{admits weak homotopy limit functors iff $\\cat D$\n    does}, and in that case $f$ \\emph{preserves homotopy pullbacks}.\n\\end{enumerate}\n\nIn view of Df.~\\ref{AbsNvFnii} and Pr.~\\ref{prop:AbsNvFncs} this\nresult not only takes care of the notion of homotopy pullback in the\ncategories $\\mathbf{Rel}^{k}\\mathbf{Cat}$ ($k>1$), but it enables us, in the proof of our\nmain result in section \\ref{sec:PrfThm}, to lift our results from the\nmodel categories $\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ ($k>1$) to the relative categories\n$\\mathbf{Rel}^{k}\\mathbf{Cat}$ ($k>1$).\n\n\\subsection{Remark}\n\\label{rem:Arbtr}\n\nThe results of this section actually hold for homotopy limit functors\n(and dually homotopy colimit functors) on \\emph{arbitrary} diagram\ncategories.\n\n\\subsection{Remark}\n\\label{rem:MdlLim}\n\nAs our definition of homotopy pullbacks is much less rigid that the\nusual ones, it might be more correct to refer to them as \\emph{models\n  for homotopy limits}.\n\n\\boldcentered{Homotopy pullbacks in model categories\\\\\n  and their left Bousfield localizations}\n\n\\subsection{Definition}\n\\label{def:HomELim}\n\nLet $\\cat E$ denote the $2$-arrow category $\\cdot \\to \\cdot \\gets\n\\cdot$.\n\nGiven a model category $\\cat M$, we then mean by a \\textbf{homotopy\n  $\\cat E$-limit functor} on $\\cat M$ a ``homotopically correct''\nhomotopy limit functor\n\\begin{displaymath}\n  \\holim^{\\cat E}\\colon \\cat M^{\\cat E} \\longrightarrow \\cat M \\rlap{\\enspace ,}\n\\end{displaymath}\ni.e.\\ a functor which, as for instance in \\cite{DHKS}*{20.1}, sends\nevery object of $\\cat M^{\\cat E}$ to a \\emph{fibrant} object of $\\cat\nM$ and every (objectwise) weak equivalence in $\\cat M^{\\cat E}$ to a\n\\emph{weak equivalence} in $\\cat M$.\n\nIt has the following property.\n\\subsection{Proposition}\n\\label{prop:HolimRtAdj}\n\n\\begin{em}\n  The functor\n  \\begin{displaymath}\n    \\Ho\\holim^{\\cat E}\\colon \\Ho(\\cat M^{\\cat E})\n    \\longrightarrow \\Ho\\cat M\n  \\end{displaymath}\n  is a right adjoint of the constant diagram functor $\\Ho\\cat M \\to\n  \\Ho(\\cat M^{\\cat E})$.\n\\end{em}\n\n\\begin{proof}\n  This is a special case of \\cite{DHKS}*{20.2}.\n\\end{proof}\n\n\\subsection{Definition}\n\\label{def:HmtpyPlbk}\n\nGiven an object $\\cat B \\in \\cat M^{\\cat E}$, we will say that an\nobject $\\cat U \\in \\cat M$ is a \\textbf{homotopy pullback} of $\\cat\nB$, if $\\cat U$ is weakly equivalent to $\\holim^{\\cat E}\\cat B$.\n\n\\boldcentered{Quasi-fibrant objects in left Bousfield localizations\\\\ of\n    left proper model categories}\n\n\\subsection{Definition}\n\\label{def:QFib}\n\nLet $\\cat M$ be a model category and let $\\mathrm{L}\\cat M$ be a left\nBousfield localization of $\\cat M$, i.e.\\ \\cite{H}*{3.3.3} a model\ncategory with the \\emph{same} cofibrations but \\emph{more} weak\nequivalences.\n\nIf $\\cat M$ is left proper, then an object $\\cat D \\in \\mathrm{L}\\cat M$ will\nbe called \\textbf{quasi-fibrant} if\n\\begin{enumerate}\n\\item \\label{QFibi} $\\cat X$ is weakly equivalent in $\\cat M$ to a\n  fibrant object in $\\mathrm{L}\\cat M$\n\\end{enumerate}\nor equivalently \\cite{H}*{3.4.6(1)}\n\\begin{resumeenumerate}{2}\n\\item \\label{Qfibii} one (and hence every) fibrant approximation of\n  $\\cat X$ in $\\cat M$ is fibrant in $\\mathrm{L}\\cat M$.\n\\end{resumeenumerate}\n\n\\subsection{Proposition}\n\\label{prop:QFpb}\n\nLet $\\mathrm{L}\\cat M$ be a left Bousfield localization of a left proper\nmodel category $\\cat M$.  Then, for every zigzag between quasi-fibrant\nobjects, its homotopy pullbacks in $\\cat M$ are quasi-fibrant in\n$\\mathrm{L}\\cat M$ and also homotopy pullbacks of this zigzag in $\\mathrm{L}\\cat M$.\n\n\\begin{proof}\n  This follows readily from \\cite{H}*{3.4.6(1)} and\n  \\cite{H}*{19.6.5}.\n\\end{proof}\n\n\n\\boldcentered{Homotopy pullbacks in saturated relative categories}\n\n\\subsection{Definition}\n\\label{def:WHLF}\n\nGive a saturated relative category $\\cat R$ (Df.~\\ref{saturi}), a\n\\textbf{weak homotopy $\\cat E$-limit functor} will be a relative functor\n\\begin{displaymath}\n  \\wholim^{\\cat E}\\colon \\cat R^{\\cat E} \\longrightarrow \\cat R\n\\end{displaymath}\nfor which the induced functor\n\\begin{displaymath}\n  \\Ho\\wholim^{\\cat E}\\colon \\Ho(\\cat R^{\\cat E}) \\longrightarrow\n  \\Ho\\cat R\n\\end{displaymath}\nis a right adjoint of the constant diagram functor $\\Ho c\\colon\n\\Ho\\cat R \\to \\Ho(\\cat R^{\\cat E})$.\n\n\\subsection{Definition}\n\\label{def:HmptPlbk}\n\nGiven an object $\\cat B \\in \\cat R^{\\cat E}$, we will say that an\nobject $\\cat U \\in \\cat R$ is a \\textbf{homotopy pullback} of $\\cat B$\nif $\\cat U$ is weakly equivalent to $\\wholim^{\\cat E}\\cat B$.\n\n\\subsection{Remark}\n\\label{rem:LocEq}\n\nWhile for a model category any two homotopy limit functors\n(Df.~\\ref{def:HomELim}) are naturally weakly equivalent, this need not\nbe the case for these \\emph{weak} homotopy limit functors.  However\nthey still have the following \\emph{local equivalence} property which,\nbecause of our \\emph{non-functorial} definition of homotopy pullbacks,\nis all we will need.\n\n\\subsection{Proposition}\n\\label{prop:HPBInd}\n\n\\begin{em}\nThe notion of homotopy pullbacks does not depend on the choice of weak\nhomotopy $\\cat E$-limit functor as\n  \\begin{enumerate}\n  \\item \\label{HPBindi}\n    for any two such functors\n    \\begin{displaymath}\n      \\wholim^{\\cat E}_{1} \\quad\\text{and}\\quad\n      \\wholim^{\\cat E}_{2}\\colon \\cat R^{\\cat E}\\longrightarrow\n      \\cat R\n    \\end{displaymath}\n    the induced functors\n    \\begin{displaymath}\n      \\Ho\\wholim^{\\cat E}_{1} \\quad\\text{and}\\quad\n      \\Ho\\wholim^{\\cat E}_{2}\\colon \\Ho(\\cat R^{\\cat E})\n      \\longrightarrow \\Ho\\cat R\n    \\end{displaymath}\n    are naturally isomorphic\n  \\end{enumerate}\n  which implies that\n  \\begin{resumeenumerate}{2}\n  \\item \\label{HPBindii} for every object $\\cat B \\in \\cat R^{\\cat\n      E}$, the objects $\\wholim^{\\cat E}_{1}\\cat B$ and $\\wholim^{\\cat\n      E}_{2}\\cat B$ are weakly equivalent.\n  \\end{resumeenumerate}\n\\end{em}\n\n\\begin{proof}\n  This follows readily from the uniqueness of adjoints and the\n  saturation of $\\cat R$.\n\\end{proof}\n\n\\intro\nWe end with another useful property.\n\\boldcentered{A global equivalence lemma}\n\n\\subsection{Lemma}\n\\label{lem:WHLEx}\n\n\\begin{em}\n  Let $f\\colon \\cat R_{1} \\to \\cat R_{2}$ be a homotopy equivalence\n  (Df.~\\ref{HomReliii}) between saturated relative categories.  Then\n  \\begin{enumerate}\n  \\item \\label{WHLExi} there exist weak homotopy $\\cat E$-limit\n    functors on $\\cat R_{1}$ iff they exist on $\\cat R_{2}$\n  \\end{enumerate}\n  in which case\n  \\begin{resumeenumerate}{2}\n  \\item \\label{WHLExii} an object $\\cat U \\in \\cat R_{1}$ is a\n    homotopy pullback of an object $B \\in \\cat R_{1}^{\\cat E}$ iff the\n    object $f\\cat U \\in \\cat R_{2}$ is a homotopy pullback of the\n    object $f\\cat B \\in \\cat R_{2}^{\\cat E}$.\n  \\end{resumeenumerate}\n\\end{em}\n\n\\begin{proof}\n  If $\\wholim^{\\cat E}_{1}\\colon \\cat R_{1}^{\\cat E} \\to \\cat R_{1}$\n  is a weak homotopy limit functor and $g\\colon \\cat R_{2} \\to \\cat\n  R_{1}$ is a homotopy inverse (Df.~\\ref{HomReliii}) of $f$, then it\n  suffices to show that the following composition is also a weak\n  homotopy limit functor\n  \\begin{displaymath}\n    \\xymatrix@C=4em{\n      {\\cat R_{2}}\n      & {\\cat R_{1}} \\ar[l]_-{f}\n      & {\\cat R_{1}^{\\cat E}} \\ar[l]_-{\\wholim_{1}^{\\cat E}}\n      & {\\cat R_{2}^{\\cat E}} \\ar[l]_-{g^{\\cat E}}\n    }\n  \\end{displaymath}\n  To do this we successively note the following.\n  \\begin{enumerate}\n  \\item The maps\n    \\begin{displaymath}\n      \\Ho f\\colon \\Ho\\cat R_{1}\\longrightarrow \\Ho\\cat R_{2}\n      \\qquad\\text{and}\\qquad\n      \\Ho g\\colon \\Ho\\cat R_{2} \\longrightarrow \\Ho\\cat R_{1}\n    \\end{displaymath}\n    are inverse equivalences of categories and hence are both left and\n    right adjoint.\n  \\item This implies the existence of the sequence of adjunctions\n    \\begin{displaymath}\n      \\xymatrix@C=4.5em{\n        {\\Ho\\cat R_{2}} \\ar@<0.75ex>[r]^-{\\Ho g}\n        \\ar@<0.25ex>[r];[]^-{\\Ho f}\n        & {\\Ho\\cat R_{1}} \\ar@<0.75ex>[r]^-{\\Ho i_{1}}\n        \\ar@<0.25ex>[r];[]^-{\\Ho \\wholim^{\\cat E}_{1}}\n        & {\\Ho(\\cat R_{1}^{\\cat E})} \\ar@<0.75ex>[r]^-{\\Ho f^{\\cat E}}\n        \\ar@<0.25ex>[r];[]^-{\\Ho g^{\\cat E}}\n        & {\\Ho(\\cat R_{2}^{\\cat E})}\n      }\n    \\end{displaymath}\n  \\item The composition of the left adjoints equals the composition\n    \\begin{displaymath}\n      \\xymatrix@C=3em{\n        {\\Ho\\cat R_{2}} \\ar[r]^-{\\Ho g}\n        & {\\Ho \\cat R_{1}} \\ar[r]^-{\\Ho f}\n        & {\\Ho\\cat R_{2}} \\ar[r]^-{\\Ho i_{2}}\n        & {\\Ho(\\cat R_{2}^{\\cat E})}\n      }\n    \\end{displaymath}\n  \\item As $(\\Ho f)(\\Ho g)$ is naturally isomorphic to the identity of\n    $\\Ho\\cat R_{2}$, the composition in (iii) is naturally isomorphic to\n    \\begin{displaymath}\n      \\Ho i_{2}\\colon \\Ho\\cat R_{2} \\longrightarrow \\Ho(\\cat\n      R_{2}^{\\cat E})\n    \\end{displaymath}\n    which implies that this map is a left adjoint of the composition\n    of the right adjoints in (iii).\\qedhere\n  \\end{enumerate}\n\\end{proof}\n\n\n\\section{Potential homotopy pullbacks}\n\\label{sec:PotHmPb}\n\n\\subsection{Summary}\n\\label{sec:PHPBsum}\n\nFor every integer $n \\ge 1$ we will in a functorial manner embed every\nzigzag $\\zigzag{f}{\\cat X}{\\cat Z}{\\cat Y}{g}$ in the categories\n$\\mathbf{Rel}^{k}\\mathbf{Cat}$ and $\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ ($k\\ge 0$) (Df.~\\ref{def:kSmpNv} and\n\\ref{def:lvlnrv}) in a commutative diagram of the form\n\\begin{displaymath}\n  \\xymatrix{\n    {\\pullback{\\cat X}{\\cat Z}{\\cat Y}} \\ar[r]^-{k} \\ar[d]\n    & {\\subcommacat{f\\cat X}{n}{g\\cat Y}} \\ar[r]^-{\\pi} \\ar[d]\n    & {\\cat Y} \\ar[d]^{g}\\\\\n    {\\cat X} \\ar[r]^-{h}\n    & {\\subcommacat{f\\cat X}{n}{\\cat Z}} \\ar[r]^-{\\pi}\n    & {\\cat Z}\n  }\n\\end{displaymath}\nin which\n\\begin{enumerate}\n\\item the squares are pullback squares,\n\\item $h$ is a weak equivalence, and\n\\item the object $\\subcommacat{f\\cat X}{n}{g\\cat Y}$ is a\n  \\emph{potential} homotopy pullback of this zigzag in the sense that,\n  under suitable restrictions on the map $f\\colon \\cat X \\to \\cat Z$\n  (which will be discussed in section \\ref{sec:PrpBnCn}), this object\n  is indeed a homotopy pullback of this zigzag.\n\\end{enumerate}\n\nWe start with an auxiliary construction.\n\\boldcentered{$n$-arrow path objects in $\\mathbf{Rel}^{k}\\mathbf{Cat}$ ($k\\ge 0$)}\n\n\\subsection{Definition}\n\\label{def:ArrPthOb}\n\nGiven an integer $n \\ge 1$ and an object $\\cat Z \\in \\mathbf{Rel}^{k}\\mathbf{Cat}$ ($k\\ge\n0$) we denote by $\\subcommacat{\\cat Z}{n}{\\cat Z} \\in \\mathbf{Rel}^{k}\\mathbf{Cat}$ the\n\\textbf{$n$-arrow path object} which has\n\\begin{enumerate}\n\\item as objects the $n$-arrow zigzags\n  \\begin{displaymath}\n    \\xymatrix{\n      {Z_{n}} \\ar@{}[r]|{{\\cdot\\;\\cdot\\;\\cdot}}\n      & {Z_{2}}\n      & {Z_{1}} \\ar[l] \\ar[r]\n      & {Z_{0}}\n    }\\quad\\text{in $w\\cat Z$,}\n  \\end{displaymath}\n\\item as maps in $w\\subcommacat{\\cat Z}{n}{\\cat Z}$ and\n  $v_{i}\\subcommacat{\\cat Z}{n}{\\cat Z}$ ($1 \\le i \\le k$) the\n  commutative diagrams of the form\n  \\begin{displaymath}\n    \\vcenter{\n      \\xymatrix{\n        {Z_{n}} \\ar@{}[r]|{{\\cdot\\;\\cdot\\;\\cdot}} \\ar[d]\n        & {Z_{2}} \\ar[d]\n        & {Z_{1}} \\ar[l] \\ar[r] \\ar[d]\n        & {Z_{0}} \\ar[d]\\\\\n        {Z'_{n}} \\ar@{}[r]|{{\\cdot\\;\\cdot\\;\\cdot}}\n        & {Z'_{2}}\n        & {Z'_{1}} \\ar[l] \\ar[r]\n        & {Z'_{0}}\n      }\n    }\n    \\qquad \\text{in $a\\cat Z$}\n  \\end{displaymath}\n  in which the vertical maps are in $w\\cat Z$ and $v_{i}\\cat Z$\n  respectively, and\n\\item as maps in $a\\subcommacat{\\cat Z}{n}{\\cat Z}$ those commutative\n  diagrams as above which are finite compositions of maps in the\n  $v_{i}\\subcommacat{\\cat Z}{n}{\\cat Z}$ ($1 \\le i \\le k$).\n\\end{enumerate}\nFurthermore\n\\begin{resumeenumerate}{4}\n\\item we denote by\n  \\begin{displaymath}\n    \\cat Z \\xrightarrow{\\enskip \\pi_{n}\\enskip}\n    \\subcommacat{\\cat Z}{n}{\\cat Z}\n    \\xrightarrow{\\enskip\\pi_{0}\\enskip} \\cat Z\n    \\qquad\\text{and}\\qquad\n    \\cat Z \\xrightarrow{\\enskip j\\enskip} \\subcommacat{\\cat Z}{n}{Z}\n  \\end{displaymath}\n  the restrictions of $\\subcommacat{\\cat Z}{n}{\\cat Z}$ to the first\n  and last entries respectively and the map which sends each object of\n  $\\cat Z$ to the alternating zigzag of its identity maps.\n\\end{resumeenumerate}\n\nThese maps have the following nice properties.\n\\subsection{Proposition}\n\\label{prop:HEWeEq}\n\\begin{em}\n  \\begin{enumerate}\n  \\item \\label{HEWeEqi} $\\pi_{n}j = \\pi_{0} j = 1$,\n  \\item \\label{HEWeEqii} $j$ is a homotopy equivalence which has\n    $\\pi_{n}$ and $\\pi_{0}$ as homotopy inverses\n  \\end{enumerate}\n  and hence (Df.~\\ref{ReeStrctiv} and (v) and\n  Pr.~\\ref{prop:AbsNvFncs})\n  \\begin{resumeenumerate}{3}\n  \\item \\label{HEWeEqiii} all three maps are weak equivalences in\n    $\\mathbf{Rel}^{k}\\mathbf{Cat}$.\n  \\end{resumeenumerate}\n\\end{em}\n\n\\begin{proof}\n  This is a straightforward computation.\n\\end{proof}\n\n\n\\boldcentered{$n$-arrow fibers and pullback objects in $\\mathbf{Rel}^{k}\\mathbf{Cat}$ ($k\n  \\ge 0$)}\n\n\\subsection{Definition}\n\\label{def:NArrFibr}\n\nGiven an integer $n \\ge 1$ and a zigzag in $\\zigzag{f}{\\cat X}{\\cat\n  Z}{\\cat Y}{g}$ in $\\mathbf{Rel}^{k}\\mathbf{Cat}$ ($k \\ge 0$)\n\\begin{enumerate}\n\\item \\label{NArrFibri} we denote by $\\subcommacat{f\\cat X}{n}{\\cat\n    Z}$ the \\textbf{$n$-arrow fibers object} which is defined by\n  \\begin{displaymath}\n    \\subcommacat{f\\cat X}{n}{\\cat Z}\n    = \\pullback{\\cat X}{\\cat Z}{\\subcommacat{\\cat Z}{n}{\\cat Z}}\n    = \\lim\\bigl(\\cat X \\xrightarrow{f}\\cat Z\n    \\xleftarrow{\\pi_{n}} \\subcommacat{\\cat Z}{n}{\\cat Z}\\bigr)\n  \\end{displaymath}\n  and note that it comes with a \\textbf{projection map}\n  \\begin{displaymath}\n    \\pi\\colon \\subcommacat{f\\cat X}{n}{\\cat Z} \\longrightarrow \\cat Z\n  \\end{displaymath}\n  induced by the map $\\pi_{0}\\colon \\subcommacat{\\cat X}{n}{\\cat Z}\n  \\to \\cat Z$, and\n\\item \\label{NArrFibrii} we denote by $\\subcommacat{f\\cat X}{n}{g\\cat\n    Y}$ the \\textbf{$n$-arrow pullback object} which is defined by\n  \\begin{displaymath}\n    \\subcommacat{f\\cat X}{n}{g\\cat Y}\n    = \\pullback{\\subcommacat{f\\cat X}{n}{\\cat Z}}{\\cat Z}{\\cat Y}\n    = \\lim\\bigl(\\subcommacat{f\\cat X}{n}{\\cat Z}\n    \\xrightarrow{\\pi} \\cat Z \\xleftarrow{g} \\cat Y\\bigr)\n  \\end{displaymath}\n  and note that it comes with a \\textbf{projection map}\n  \\begin{displaymath}\n    \\pi\\colon \\subcommacat{f\\cat X}{n}{g\\cat Y} \\longrightarrow \\cat Y\n  \\end{displaymath}\n  obtained by ``restriction to $\\cat Y$''.\n\\end{enumerate}\n\n\\boldcentered{$n$-arrow fibers and pullback objects in\n  $\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ ($k \\ge 0$)}\n\n\\subsection{Definition}\n\\label{def:NArrPB}\n\nAs (Nt.~\\ref{rem:zerorel}) $\\mathbf{Rel}^{0}\\mathbf{Cat} = \\widehat{\\mathbf{Cat}}$, the case\n$\\mathrm{s}^{0}\\widehat{\\mathbf{Cat}} = \\widehat{\\mathbf{Cat}}$ has already been taken care of in\nDf.~\\ref{def:NArrFibr}.\n\nGiven an integer $n \\ge 1$ and a zigzag $\\zigzag{f}{\\cat X}{\\cat\n  Z}{\\cat Y}{g}$ in $\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ ($k\\le 1$) we can therefore\ndefine the \\textbf{$n$-arrow fibers and pullback objects}\n$\\subcommacat{f\\cat Z}{n}{\\cat Z}$ and $\\subcommacat{f\\cat X}{n}{g\\cat\n  Y}$ and the associated \\textbf{projection maps}\n\\begin{displaymath}\n  \\pi\\colon\\subcommacat{f\\cat X}{n}{\\cat Z} \\longrightarrow \\cat Z\n  \\qquad\\text{and}\\qquad\n  \\pi\\colon\\subcommacat{f\\cat X}{n}{g\\cat Y}\\longrightarrow \\cat Y\n\\end{displaymath}\nby the requirement that, for every $k$-fold dimension $p_{*} =\n(p_{k},\\ldots,p_{1})$\n\\begin{align*}\n  \\bigl(\\subcommacat{f\\cat X}{n}{\\cat Z}\n  \\xrightarrow{\\pi}\\cat Z\\bigr)_{p_{*}}\n  &= \\bigl(\\subcommacat{f_{p_{*}}\\cat X_{p_{*}}}{n}{\\cat Z_{p_{*}}}\n  \\xrightarrow{\\pi} \\cat Z_{p_{*}}\\bigr) \\in \\widehat{\\mathbf{Cat}},\n  \\quad\\text{and}\\\\\n  \\bigl(\\subcommacat{f\\cat X}{n}{g\\cat Y}\n  \\xrightarrow{\\pi}\\cat Y\\bigr)_{p_{*}}\n  &= \\bigl(\\subcommacat{f_{p_{*}}X_{p_{*}}}{n}{g_{p_{*}}\n    \\cat Y_{p_{*}}} \\xrightarrow{\\pi} \\cat Y_{p_{*}} \\bigr) \\in\n  \\widehat{\\mathbf{Cat}}\n\\end{align*}\n\n\\intro\nThe definitions \\ref{def:NArrFibr} and \\ref{def:NArrPB} are closely\nrelated as follows.\n\\subsection{Proposition}\n\\label{prop:wpicom}\n\n\\begin{em}\n  For every integer $n \\ge 1$ and zigzag $\\zigzag{f}{\\cat X}{\\cat\n    Z}{\\cat Y}{g}$ in $\\mathbf{Rel}^{k}\\mathbf{Cat}$ ($k\\ge 1$) (Df.~\\ref{def:HighEq})\n  \\begin{align*}\n    w_{*}\\bigl(\\subcommacat{f\\cat X}{n}{\\cat Z} \\xrightarrow{\\pi}\\cat\n    Z\\bigr) &= \\bigl(\\commacat{w_{*}fw_{*}\\cat X}{w_{*}\\cat Z}\n    \\xrightarrow{\\pi} w_{*}\\cat Z\\bigr) \\in \\mathrm{s}^{k}\\widehat{\\mathbf{Cat}},\n    \\quad\\text{and}\\\\\n    w_{*}\\bigl(\\subcommacat{f\\cat X}{n}{g\\cat Y}\\xrightarrow{\\pi}\\cat\n    Y\\bigr) &= \\bigl(\\subcommacat{w_{*}fw_{*}\\cat X}{n}{w_{*}gw_{*}Y}\n    \\xrightarrow{\\pi} w_{*}Y\\bigr) \\in \\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}.\n  \\end{align*}\n\\end{em}\n\\begin{proof}\n  This follows readily from the observation that, for every $k$-fold\n  dimension $p_{*} = (p_{k},\\cdots,p_{1})$ (Df.~\\ref{def:ArrPthOb})\n  \\begin{displaymath}\n    w_{p_{*}}\\bigl(\\subcommacat{\\cat Z}{n}{\\cat Z}\n    \\xrightarrow{\\pi_{0}} \\cat Z\\bigr) =\n    \\bigl(\\subcommacat{w_{p_{*}}\\cat\n      Z}{n}{w_{p_{*}}Z}\\xrightarrow{\\pi_{0}} w_{p_{*}}Z\\bigr) \\in\n    \\widehat{\\mathbf{Cat}}. \\qedhere\n  \\end{displaymath}\n\\end{proof}\n\n\\intro\nIt remains the pull it all together as we promised in\n\\ref{sec:PHPBsum}.\n\\subsection{Proposition}\n\\label{prop:zzEmbed}\n\n\\begin{em}\n  For every integer $n \\ge 1$, every zigzag $\\zigzag{f}{\\cat X}{\\cat\n    Z}{\\cat Y}{g}$ in $\\mathbf{Rel}^{k}\\mathbf{Cat}$ or $\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ ($k\\ge 0$) can\n  in a functorial manner be embedded in a commutative diagram of the\n  form (Df.~\\ref{def:NArrFibr} and \\ref{def:NArrPB})\n  \\begin{displaymath}\n    \\xymatrix{\n      {\\pullback{\\cat X}{\\cat Z}{\\cat Y}} \\ar[r]^-{k} \\ar[d]\n      & {\\subcommacat{f\\cat X}{n}{g\\cat Y}} \\ar[r]^-{\\pi} \\ar[d]\n      & {\\cat Y} \\ar[d]^{g}\\\\\n      {\\cat X} \\ar[r]^-{h}\n      & {\\subcommacat{f\\cat X}{n}{\\cat Z}} \\ar[r]^-{\\pi}\n      & {\\cat Z}\n    }\n  \\end{displaymath}\n  in which $h$ and $k$ send each object $X \\in \\cat X$ and $(X,Y) \\in\n  \\pullback{\\cat X}{\\cat Z}{\\cat Y}$ to the alternating zigzag of\n  identity maps of $fX$, and\n  \\begin{enumerate}\n  \\item the squares are pullback squares, and\n  \\item the map $h$ is a weak equivalence.\n  \\end{enumerate}\n\\end{em}\n\n\\begin{proof}\n  That the square on the right is a pullback square follows from\n  Df.~\\ref{def:NArrFibr} and \\ref{def:NArrPB} and that the one on the\n  left is so is a simple calculation.\n\n  That $h$ is a weak equivalence follows readily from\n  Pr.~\\ref{prop:HEWeEq}.\n\\end{proof}\n\n\\section{Properties \\texorpdfstring{$B_{n}$}{Bn} and \\texorpdfstring{$C_{n}$}{Cn}}\n\\label{sec:PrpBnCn}\n\n\\subsection{Summary}\n\\label{sec:BnCnsum}\n\nIn final preparation for the formulation of our main results (in\nsection \\ref{sec:MainRlt}) we recall from \\cite{DKS}*{\\S6} the notions\nof \\emph{properties} $B_{n}$ and $C_{n}$ in $\\widehat{\\mathbf{Cat}}$ and then extend\nthese notions to the categories $\\mathbf{Rel}^{k}\\mathbf{Cat}$ and $\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ for\n$k \\ge 1$.\n\nAs these notions in $\\widehat{\\mathbf{Cat}}$ are closely related to the\n\\emph{Grothendieck construction} we start with a discussion of the\nlatter.\n\\subsection{Definition}\n\\label{def:GrConst}\n\nGiven an object $\\cat D \\in \\widehat{\\mathbf{Cat}}$ and a \\emph{not necessarily\n  relative} functor $F\\colon \\cat D \\to \\widehat{\\mathbf{Cat}}$\n(Df.~\\ref{relcatiii}), the \\textbf{Grothendieck construction} on $F$\nis the object $\\Gr F \\in \\widehat{\\mathbf{Cat}}$ which has\n\\begin{enumerate}\n\\item \\label{GrConsti} as \\emph{objects} the pairs $(D,A)$ of objects\n  $D \\in \\cat D$ and $A \\in FD$, and\n\\item \\label{GrCoonstii} as \\emph{maps} $(D_{1},A_{1}) \\to\n  (D_{2},A_{2})$ the pairs $(d,a)$ of maps\n  \\begin{displaymath}\n    d\\colon D_{1} \\longrightarrow D_{2} \\in \\cat D\n    \\qquad\\text{and}\\qquad\n    a\\colon (FD_{1})A_{1} \\longrightarrow A_{2} \\in FD_{2}\n  \\end{displaymath}\n  of which the \\emph{compositions} are given by the formula\n  \\begin{displaymath}\n    (d',a')(d,a) = \\bigl(d'd, a'\\bigl((Fd)a\\bigr)\\bigr).\n  \\end{displaymath}\n\\end{enumerate}\n\nSuch a Grothendieck construction $\\Gr F$ comes with a\n\\textbf{projection functor}\n\\begin{displaymath}\n  \\pi\\colon \\Gr F \\longrightarrow \\cat D\\in \\widehat{\\mathbf{Cat}}\n\\end{displaymath}\nwhich sends an object $(D,A)$ (resp.\\ a map $(d,a)$) to the object $D$\n(resp.\\ the map $d$) in $\\cat D$.\n\n\\intro\nThe usefulness of Grothendieck constructions is due to the following\nproperty which was noted by Bob Thomason \\cite{T1}*{1.2}.\n\\subsection{Proposition}\n\\label{prop:GrCnHmCl}\n\\begin{em}\n  \\begin{enumerate}\n  \\item The Grothendieck construction is a homotopy colimit functor on\n    $\\widehat{\\mathbf{Cat}}$\n  \\end{enumerate}\n  and hence\n  \\begin{resumeenumerate}{2}\n  \\item it is homotopy invariant in the sense that every natural weak\n    equivalence between two functors $F_{1},F_{2}\\colon \\cat D \\to\n    \\widehat{\\mathbf{Cat}}$ induces a weak equivalence $\\Gr F_{1} \\to \\Gr F_{2}$.\n  \\end{resumeenumerate}\n\\end{em}\n\n\\subsection{Example}\n\\label{ex:GrCn}\n\nGiven a map $f\\colon \\cat X \\to \\cat Z \\in \\widehat{\\mathbf{Cat}}$ and an integer $n\n\\ge 1$\n\\begin{enumerate}\n\\item \\label{GrCni} denote, for every object $Z \\in \\cat Z$ by\n  (Df.~\\ref{def:NArrFibr})\n  \\begin{displaymath}\n    \\subcommacat{f\\cat X}{n}{Z} \\subset\n    \\subcommacat{f\\cat X}{n}{\\cat Z} \\in \\widehat{\\mathbf{Cat}}\n  \\end{displaymath}\n  the category consisting of the objects and maps which end at $Z$ or\n  $1_{Z}$, and\n\\item \\label{GrCnii} denote by\n  \\begin{displaymath}\n    \\subcommacat{f\\cat X}{n}{-}\\colon \\cat Z \\longrightarrow \\widehat{\\mathbf{Cat}}\n  \\end{displaymath}\n  the \\emph{not necessarily relative} functor (Df.~\\ref{relcatiii})\n  which sends each object $Z \\in \\cat Z$ to $\\subcommacat{f\\cat\n    X}{n}{Z}$ and each map $z\\colon Z \\to Z' \\in \\cat Z$ to the\n  functor $\\subcommacat{f\\cat X}{n}{Z} \\to \\subcommacat{f\\cat\n    X}{n}{Z'}$ obtained by ``composition with $z$''.\n\\end{enumerate}\n\nThen one readily verifies that\n\\begin{resumeenumerate}{3}\n\\item \\label{GrCniii} $\\subcommacat{f\\cat X}{n}{\\cat Z} =\n  \\Gr\\subcommacat{f\\cat X}{n}{-}$, and\n\\item \\label{GrCniv} $\\bigl(\\subcommacat{f\\cat X}{n}{\\cat\n    Z}\\xrightarrow{\\pi} \\cat Z\\bigr) = \\bigl(\\Gr\\subcommacat{f\\cat\n    X}{n}{-}\\xrightarrow{\\pi} \\cat Z\\bigr)$.\n\\end{resumeenumerate}\n\n\\subsection{Example}\n\\label{ex:GrCnExAn}\n\nGiven a map $f\\colon \\cat X \\to \\cat Z \\in \\widehat{\\mathbf{Cat}}$ and an integer $n\n\\ge 1$\n\\begin{enumerate}\n\\item \\label{GrCnExAni} denote, for every pair of objects $X \\in \\cat\n  X$ and $Z \\in \\cat Z$, by (Ex.~\\ref{ex:GrCn})\n  \\begin{displaymath}\n    \\subcommacat{fX}{n}{Z} = \\subcommacat{f\\cat X}{n}{Z}\n  \\end{displaymath}\n  the category consisting of the objects and maps which start at $fX$\n  or $1_{fX}$, and\n\\item \\label{GrCnExAnii} denote by\n  \\begin{displaymath}\n    \\subcommacat{f-}{n}{Z}\\colon \\cat X\\longrightarrow \\widehat{\\mathbf{Cat}}\n    \\qquad\\text{or}\\qquad\n    \\subcommacat{f-}{n}{Z}\\colon \\cat X^{\\mathrm{op}}\\longrightarrow \\widehat{\\mathbf{Cat}}\n  \\end{displaymath}\n  the functor which sends each object $X \\in \\cat X$ to\n  $\\subcommacat{fX}{n}{Z}$ and each map $x\\colon X \\to X' \\in \\cat X$\n  to the induced functor\n  \\begin{displaymath}\n    \\subcommacat{fX}{n}{Z}\\longrightarrow \\subcommacat{fX'}{n}{Z}\n    \\qquad\\text{or}\\qquad\n    \\subcommacat{fX'}{n}{Z}\\longrightarrow \\subcommacat{fX}{n}{Z}\n  \\end{displaymath}\n  depending on whether $n$ is even or odd.\n\\end{enumerate}\n\nThen one readily verifies that\n\\begin{resumeenumerate}{3}\n\\item \\label{GrCnExAniii}\n  \\begin{displaymath}\n    \\subcommacat{f\\cat X}{n}{Z} = \\left\\{\n    \\begin{aligned}\n      \\Gr\\bigl(\\subcommacat{f-}{n}{Z}&\\colon \\cat X\n      \\longrightarrow \\widehat{\\mathbf{Cat}}\\bigr)\\\\\n      \\text{or}&\\\\\n      \\Gr\\bigl(\\subcommacat{f-}{n}{Z}&\\colon \\cat X^{\\mathrm{op}}\n      \\longrightarrow \\widehat{\\mathbf{Cat}}\\bigr)\n    \\end{aligned}\\right.\n  \\end{displaymath}\n\\end{resumeenumerate}\n\n\\boldcentered{Properties $B_{n}$ and $C_{n}$ in $\\widehat{\\mathbf{Cat}}$}\n\n\\subsection{Definition}\n\\label{def:BnCn}\n\n\\begin{enumerate}\n\\item \\label{BnCni} A map $f\\colon \\cat X \\to \\cat Z \\in \\widehat{\\mathbf{Cat}}$ is\n  said to have \\textbf{property $B_{n}$} ($n\\ge 1$) if the functor\n  $Ex.~\\ref{ex:GrCn}$\n  \\begin{displaymath}\n    \\subcommacat{f\\cat X}{n}{-}\\colon \\cat Z \\longrightarrow \\widehat{\\mathbf{Cat}}\n  \\end{displaymath}\n  is a \\emph{relative functor} (Df.~\\ref{relcativ}), and\n\\item \\label{BnCnii} an object $\\cat Z \\in \\widehat{\\mathbf{Cat}}$ is said to have\n  \\textbf{property $C_{n}$} ($n\\ge 1$) if every map\n  (Df.~\\ref{def:smpnv})\n  \\begin{displaymath}\n    \\boldsymbol 0^{w} \\longrightarrow \\cat Z \\in \\widehat{\\mathbf{Cat}}\n  \\end{displaymath}\n  has property $B_{n}$.\n\\end{enumerate}\n\n\\intro\nThe usefulness of property $C_{n}$ is due to the following result of\n\\cite{DKS}*{\\S6}.\n\\subsection{Proposition}\n\\label{prop:BnCn}\n\\begin{em}\n  If, given a map $f\\colon \\cat X \\to \\cat Z \\in \\widehat{\\mathbf{Cat}}$, the object\n  $\\cat Z$ has property $C_{n}$ ($n \\ge 1$), then the map $f\\colon\n  \\cat X \\to \\cat Z$ has property $B_{n}$.\n\\end{em}\n\n\\begin{proof}\n  In view of Ex.~\\ref{GrCnii} one has to show that every map $z\\colon\n  Z \\to Z' \\in \\cat Z$ induces a weak equivalence $\\subcommacat{f\\cat\n    Z}{n}{Zj \\to \\subcommacat{f\\cat X}{n}{Z'}}$, or equivalently\n  (Ex.~\\ref{GrCnExAniii}) a weak equivalence\n  \\begin{displaymath}\n    \\Gr\\subcommacat{f-}{n}{Z}\\longrightarrow\n    \\Gr\\subcommacat{f-}{n}{Z'}\n  \\end{displaymath}\n  But this follows readily from Ex.~\\ref{GrCnExAniii} and\n  Df.~\\ref{GrCnExAnii} and the fact that, in view of property $C_{n}$,\n  for every object $X \\in \\cat X$ the map\n  \\begin{displaymath}\n    \\subcommacat{fX}{n}{Z} \\longrightarrow \\subcommacat{fX}{n}{Z'}\n  \\end{displaymath}\n  is a weak equivalence.\n\\end{proof}\n\n\\boldcentered{Properties $B_{n}$ and $C_{n}$ in $\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ and\n$\\mathbf{Rel}^{k}\\mathbf{Cat}$ ($k\\ge 1$)}\n\n\\subsection{Definition}\n\\label{def:MpPrpB}\n\n\\begin{enumerate}\n\\item \\label{MpPrpBi} A map $f\\colon \\cat X \\to \\cat Z \\in\n  \\mathrm{s}^{k}\\mathbf{Cat}$ has \\textbf{property $B_{n}$} ($n\\ge 1$) if\n  \\begin{itemize}\n  \\item for every $k$-fold dimension $p_{*} = (p_{k},\\ldots, p_{1})$,\n    the map $f_{p_{*}}\\colon \\cat X_{p_{*}} \\to \\cat Z_{p_{*}} \\in\n    \\widehat{\\mathbf{Cat}}$ has property $B_{n}$ (Df.~\\ref{BnCni}),\n  \\end{itemize}\n\\end{enumerate}\nand\n\\begin{resumeenumerate}{2}\n\\item \\label{MpPrpBii} An object $\\cat Z \\in \\mathrm{s}^{k}\\mathbf{Cat}$ has\n  \\textbf{property $C_{n}$} ($n \\ge 1$) if\n  \\begin{itemize}\n  \\item for every $k$-fold dimension $p_{*}= (p_{k},\\ldots, p_{1})$\n    the object $Z_{p_{*}} \\in \\widehat{\\mathbf{Cat}}$ has property $C_{n}$\n    (Df.~\\ref{BnCnii}).\n  \\end{itemize}\n\\end{resumeenumerate}\n\n\\subsection{Definition}\n\\label{def:MpPrBrel}\n\n\\begin{enumerate}\n\\item \\label{MpPrBreli} A map $f\\colon \\cat X \\to \\cat Z \\in \\mathbf{Rel}^{k}\\mathbf{Cat}$\n  has \\textbf{property $B_{n}$} ($n\\ge 1$) if\n  \\begin{itemize}\n  \\item the map $w_{*}f\\colon w_{*}\\cat X \\to w_{*}\\cat Z \\in\n    \\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ has property $B_{n}$ (Df.~\\ref{MpPrpBi}\n  \\end{itemize}\n  or equivalently if\n  \\begin{itemize}\n  \\item for every $k$-fold dimension $p_{*} = (p_{k},\\ldots, p_{1})$\n    the map $w_{p_{*}}f\\colon w_{p_{*}}\\cat X \\to w_{p_{*}}\\cat Z\n    \\in \\widehat{\\mathbf{Cat}}$ has property $B_{n}$\n  \\end{itemize}\n\\end{enumerate}\nand\n\\begin{resumeenumerate}{2}\n\\item \\label{MpPrBrelii} an object $\\cat Z \\in \\mathbf{Rel}^{k}\\mathbf{Cat}$ has\n  \\textbf{property $C_{n}$} ($n\\ge 1$) if\n  \\begin{itemize}\n  \\item the object $w_{*}Z \\in \\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ has property $C_{n}$\n    (Df.~\\ref{MpPrpBii})\n  \\end{itemize}\n  or equivalently if\n  \\begin{itemize}\n  \\item for every $k$-fold dimension $p_{*} = (p_{k}, \\ldots, p_{1})$\n    the object $w_{p_{*}}\\cat Z \\in \\widehat{\\mathbf{Cat}}$ has property $C_{n}$.\n  \\end{itemize}\n\\end{resumeenumerate}\n\n\\part{The main results and their proofs}\n\n\n\\section{The main results}\n\\label{sec:MainRlt}\n\n\\subsection{Summary}\n\\label{sec:MnRsltSum}\n\nOur main results are\n\\begin{enumerate}\n\\item \\label{MpRsltSumi} \\emph{Theorem} \\ref{thm:MpPropBn} which\n  states that the presence of properties $B_{n}$ and $C_{n}$ ensure\n  that the potential homotopy pullbacks of section~\\ref{sec:PotHmPb},\n  i.e.\\ the \\emph{$n$-arrow pullback objects}, are indeed\n  \\emph{homotopy pullbacks}, and\n\\item \\label{MpRsltSumii} \\emph{Theorem} \\ref{thm:SuffCn} which states\n  that the presence of a \\emph{strict $3$-arrow calculus} implies\n  \\emph{property $C_{3}$}.\n\\end{enumerate}\n\nThese results then are applied to $(\\infty,1)$-categories and\n$(\\infty,k)$-categories for $k>1$ to prove\n\\begin{resumeenumerate}{3}\n\\item \\label{MpRsltSumiii} \\emph{Theorem} \\ref{thm:QFibQfib} which\n  combines Theorem \\ref{thm:MpPropBn} and \\ref{thm:SuffCn} with\n  results from \\cite{BK3} to show that \\emph{homotopy pullbacks in\n    $(\\infty,1)$-categories} can be described as \\emph{$3$-arrow\n    pullback objects} of zigzags between \\emph{partial model\n    categories}, i.e.\\ relative categories which have the \\emph{two\n    out of six} property and admit a \\emph{$3$-arrow calculus}, and\n\\item \\label{MpRsltSumiv} \\emph{Theorem} \\ref{thm:MpBn} which is an\n  application of Theorem \\ref{thm:MpPropBn} to $(\\infty,k)$-categories\n  for $k\\ge 1$, which is much weaker than Theorem \\ref{thm:QFibQfib},\n  because we have no model structure on $\\mathbf{Rel}^{k}\\mathbf{Cat}$ for $k\\ge 1$, nor\n  an analog for partial model categories.\n\\end{resumeenumerate}\n\nWe also give proofs of Theorems \\ref{thm:SuffCn}, \\ref{thm:QFibQfib}\nand \\ref{thm:MpBn}, but postpone the proof of Theorem\n\\ref{thm:MpPropBn} until section \\ref{sec:PrfThm}.\n\n\n\\boldcentered{The main results}\n\n\\subsection{Theorem}\n\\label{thm:MpPropBn}\n\n\\begin{em}\n  Given an integer $n \\ge 1$, let $\\zigzag{f}{\\cat X}{\\cat Z}{\\cat\n    Y}{g}$ be a zigzag in $\\mathbf{Rel}^{k}\\mathbf{Cat}$ or $\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ ($k\\ge 0$)\n  (Df.~\\ref{def:kSmpNv} and \\ref{def:lvlnrv}) with the property that\n  \\begin{enumerate}\n  \\item \\label{MpPropBni} the map $f\\colon \\cat X \\to \\cat Z$ has\n    property $B_{n}$ (Df.~\\ref{BnCni}, \\ref{MpPrpBi} and\n    \\ref{MpPrBreli})\n  \\end{enumerate}\n  which (Pr.~\\ref{prop:BnCn}) is in particular the case if\n  \\begin{resumeenumerate}{2}\n  \\item \\label{MpPropBnii} the object $\\cat Z$ has property $C_{n}$\n    (Df.~\\ref{BnCnii}, \\ref{MpPrpBii} and \\ref{MpPrBrelii}).\n  \\end{resumeenumerate}\n  Then\n  \\begin{resumeenumerate}{3}\n  \\item \\label{MpPropBniii} the $n$-arrow pullback object\n    $\\subcommacat{f\\cat X}{n}{g\\cat Y}$ (Df.~\\ref{def:NArrFibr} and\n    \\ref{def:NArrPB}) is a homotopy pullback (Df.~\\ref{def:HmtpyPlbk}\n    and \\ref{def:HmptPlbk}) of this zigzag.\n  \\end{resumeenumerate}\n\\end{em}\n\nThe main tools for proving this are the \\emph{fibrillations} of\nHopkins and Rezk which we will discuss in section \\ref{sec:HopRzF} and\nwe therefor postpone the proof of Theorem \\ref{thm:MpPropBn} until\nsection \\ref{sec:PrfThm}.\n\n\\subsection{Corollary}\n\\label{cor:SuffHPB}\n\n\\begin{em}\n  A sufficient condition in order that the pullback $\\pullback{\\cat\n    X}{\\cat Z}{\\cat Y}$ of the above zigzag is also a homotopy\n  pullback is that the obvious map\n  \\begin{displaymath}\n    \\pullback{\\cat X}{\\cat Z}{\\cat Y} \\xrightarrow{\\enspace k\\enspace}\n    \\subcommacat{f\\cat X}{n}{g\\cat Y}\n  \\end{displaymath}\n  of Pr.~\\ref{prop:zzEmbed} is a weak equivalence.\n\\end{em}\n\n\\subsection{Theorem}\n\\label{thm:SuffCn}\n\n\\begin{em}\n  A sufficient condition in order that an object $\\cat Z \\in \\mathbf{Rel}^{k}\\mathbf{Cat}$\n  ($k\\ge 0$) has property $C_{3}$ (Df.~\\ref{MpPrBrelii}) is that $\\cat\n  Z$ admits a strict $3$-arrow calculus (Df.~\\ref{def:strcalculus} and\n  \\ref{def:Str3ar}).\n\\end{em}\n\n\\begin{proof}\n  \\leavevmode\n  \\begin{enumerate}\n  \\item The case $k=0$.  This follows from Rk.~\\ref{rem:whystrict},\n    Df.~\\ref{def:strcalculus} and \\cite{DK}*{3.3, 6.1, 6.2 and 8.2}.\n  \\item The case $k\\ge 1$.  It follows readily from\n    Df.~\\ref{def:Str3ar} and \\ref{def:HighEq} that, for every $k$-fold\n    dimension $p_{*} = (p_{k},\\ldots, p_{1})$, the object\n    $w_{p_{*}}\\cat Z \\in \\widehat{\\mathbf{Cat}}$ admits a strict $3$-arrow calculus\n    and hence (i) has property $C_{3}$.  The desired result then\n    follows from Df.~\\ref{MpPrBrelii}.\\qedhere\n  \\end{enumerate}\n\\end{proof}\n\n\n\\boldcentered{Applications to $(\\infty,1)$-categories and\n  $(\\infty,k)$-categories for $k>1$}\n\nBefore formulating the application mentioned in \\ref{MpRsltSumiii} we\nrecall some results from \\cite{BK3}.\n\n\\subsection{Remark}\n\\label{rem:CSSfib}\n\nRecall from \\cite{BK3} that a \\textbf{partial model category} is an\nobject $\\cat X \\in  \\mathbf{RelCat}$ which admits a $3$-arrow calculus\n(Df.~\\ref{def:calculus}) and has the \\textbf{two out of six property}\nthat, for every three maps $r$, $s$ and $t \\in \\cat X$ for which the\n\\emph{two} compositions $sr$ and $ts$ exist and are weak equivalences,\nthe other $four$ maps $r$, $s$, $t$ and $tsr$ are also weak\nequivalences.\n\nIt then was shown in \\cite{BK3} that\n\\begin{enumerate}\n\\item \\label{CSSfibi} \\emph{for every partial model category $\\cat X\n    \\in \\mathbf{RelCat}$, one (and hence every) Reedy fibrant approximation to\n  $\\mathrm{s}\\N\\cat X \\in \\mathrm{s}\\cat S$ (Df.~\\ref{def:smpnv}) is a complete\n  Segal space, i.e.\\ a fibrant object in $\\mathrm{L}\\mathrm{s}\\cat S$\n  (Df.~\\ref{RzkStrcti})} \n\\end{enumerate}\nwhich in view of \\cite{BK1}*{6.1} implies that\n\\begin{resumeenumerate}{2}\n\\item \\label{CSSfibii} \\emph{$\\cat X$ is a quasi-fibrant object\n    (Df.~\\ref{def:QFib}) of $\\mathrm{L}\\mathbf{RelCat}$ (Df.~\\ref{RzkStrctiii})}.\n\\end{resumeenumerate}\nMoreover\n\\begin{resumeenumerate}{3}\n\\item \\label{CSSfibiii}\\emph{every complete Segal space is Reedy\n    equivalent to the simplicial nerve of a partial model category.}\n\\end{resumeenumerate}\n\n\\subsection{Theorem}\n\\label{thm:QFibQfib}\n\nIf $\\zigzag{f}{\\cat X}{\\cat Z}{\\cat Y}{g}$ is a zigzag in $\\mathbf{RelCat}$ in\nwhich\n\\begin{enumerate}\n\\item $\\cat X$ and $\\cat Y$ are quasi-fibrant in $\\mathrm{L}\\mathbf{RelCat}$ and\n  $\\cat Z$ is a partial model category,\n\\end{enumerate}\nwhich in particular is the case if\n\\begin{resumeenumerate}{2}\n\\item $\\cat X$, $\\cat Y$ and $\\cat Z$ are all three partial model\n  categories,\n\\end{resumeenumerate}\nthen\n\\begin{resumeenumerate}{3}\n\\item \\emph{$\\subcommacat{f\\cat X}{3}{g\\cat Y}$ is a quasi-fibrant\n    object of $\\mathrm{L}\\mathbf{RelCat}$, which is a homotopy pullback of this\n    zigzag not only in $\\mathbf{RelCat}$, but also in $\\mathrm{L}\\mathbf{RelCat}$.}\n\\end{resumeenumerate}\n\n\\begin{proof}\n  This follows readily from Rk.\\ref{rem:whystrict},\n  Pr.~\\ref{prop:QFpb} and Th.~\\ref{thm:MpPropBn} and \\ref{thm:SuffCn}.\n\\end{proof}\n\n\\intro It remains to deal with the result that was mentioned in\n\\ref{MpRsltSumiv}.\n\\subsection{Theorem}\n\\label{thm:MpBn}\n\n\\begin{em}\n  Let $\\zigzag{f}{\\cat X}{\\cat Z}{\\cat Y}{g}$ be a zigzag in\n  $\\mathbf{Rel}^{k}\\mathbf{Cat}$ ($k\\ge 1$) for which\n  \\begin{enumerate}\n  \\item $\\mathrm{s}^{k}\\N\\cat X$, $\\mathrm{s}^{k}\\N\\cat Z$ and $\\mathrm{s}^{k}\\N\\cat\n    Y$ are quasi-fibrant objects of $\\mathrm{L}\\mathrm{s}^{k}\\cat S$\n    (Df.~\\ref{RzkStrcti})\n  \\end{enumerate}\n  or equivalently (Pr.~\\ref{prop:AbsNvFncs})\n  \\begin{resumeenumerate}{2}\n  \\item $w_{*}\\cat X$, $w_{*}\\cat Z$ and $w_{*}\\cat Y$ are\n    quasi-fibrant objects of $\\mathrm{L}\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$\n  \\end{resumeenumerate}\n  and assume that for some integer $n \\ge 1$\n  \\begin{resumeenumerate}{3}\n  \\item the map $f\\colon \\cat X \\to \\cat Z$ has property $B_{n}$\n  \\end{resumeenumerate}\n  which (Pr.~\\ref{prop:BnCn}) in particular is the case if\n  \\begin{resumeenumerate}{4}\n  \\item the object $\\cat Z$ has property $C_{n}$,\n  \\end{resumeenumerate}\n  then\n  \\begin{resumeenumerate}{5}\n  \\item $\\subcommacat{f\\cat X}{n}{g\\cat Y}$ is a homotopy pullback of\n    this zigzag not only in $\\mathbf{Rel}^{k}\\mathbf{Cat}$ but also in $\\mathrm{L}\\mathbf{Rel}^{k}\\mathbf{Cat}$\n    (Df.~\\ref{RzkStrctiii}).\n  \\end{resumeenumerate}\n\\end{em}\n\n\\begin{proof}\n  In view of Df.~\\ref{def:NArrPB} and \\ref{def:MpPrBrel},\n  Pr.~\\ref{prop:wpicom} and Th.~\\ref{thm:MpPropBn}\n  $w_{*}\\subcommacat{f\\cat X}{n}{g\\cat Y}$ is a homotopy pullback if\n  the zigzag\n  \\begin{displaymath}\n    \\zigzag{w_{*}f}{w_{*}\\cat X}{w_{*}\\cat Z}{w_{*}\\cat Y}{w_{*}g}\n  \\end{displaymath}\n  in $\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ as well as, in view of Pr.~\\ref{prop:QFpb}, in\n  $\\mathrm{L}\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$.\n\n  The desired result now follows readily from the \\emph{Global\n    equivalence lemma} \\ref{lem:WHLEx} and the fact that\n  (Df.~\\ref{def:AbsNvFn} and Pr.~\\ref{prop:AbsNvFncs}) $w_{*}$ is a\n  \\emph{homotopy equivalence}.\n\\end{proof}\n\n\n\\section{Hopkins-Rezk fibrillations}\n\\label{sec:HopRzF}\n\n\\subsection{Summary}\n\\label{sec:HopRzFSum}\n\nOur proof of Theorem~\\ref{thm:MpPropBn} (in section \\ref{sec:PrfThm})\nwill consist of two parts.  The first consists of a proof for the\n\\emph{model} categories $\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ and $\\mathbf{RelCat}$.  In the\nsecond we lift these results for the model categories\n$\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ ($k>1$) to the \\emph{relative} categories\n$\\mathbf{Rel}^{k}\\mathbf{Cat}$ by means of the \\emph{Global equivalence lemma}\n\\ref{lem:WHLEx}.\n\nOur aim in this section is to describe three lemmas which we will need\nfor the first part.  They involve, each in a different way, the\n\\emph{fibrillations} of Hopkins and Rezk as follows.\n\\begin{enumerate}\n\\item \\label{HopRzFSumi} The first lemma is a \\emph{Quillen\n    fibrillation lemma} which will produce the fibrillations to get us\n  started.\n\n  It is essentially a reformulation in terms of relative functors and\n  fibrillations as well as a slight strengthening of the lemma that\n  Quillen used to prove his Theorem~B and states that\n  \\begin{itemize}\n  \\item given an object $\\cat D \\in \\widehat{\\mathbf{Cat}}$, a functor $f\\colon \\cat\n    D \\to \\widehat{\\mathbf{Cat}}$ is a \\emph{relative} functor iff the projection\n    functor from its Grothendieck construction to $\\cat D$\n    (Df.~\\ref{def:GrConst}) is a \\emph{fibrillation} in $\\widehat{\\mathbf{Cat}}$.\n  \\end{itemize}\n\\item \\label{HopRzFSumii} The second lemma is a \\emph{Fibrillation\n    lifting lemma} which enables us to obtain more fibrillation as it\n  provides\n  \\begin{itemize}\n  \\item a sufficient condition on a relative functor in order that it\n    \\emph{reflects fibrillations}.\n  \\end{itemize}\n\\item \\label{HopRzFSumiii} The third lemma is a \\emph{Hopkins-Rezk\n    fibrillation lemma} which shows how, in a right proper model\n  category, some of the fibrillations obtained in (i) and (ii) can be\n  used to construct homotopy pullbacks.\n\\end{enumerate}\n\n\n\\boldcentered{Fibrillations}\n\nWe start with recalling from \\cite{R2}*{\\S2} the notion of what\nCharles Rezk called \\emph{sharp maps} but which, because of their\nfibration-like properties (see \\ref{prop:PBfibril} below), we prefer\nto call \\emph{fibrillations}.\n\n\\subsection{Definition}\n\\label{def:fibril}\n\nGiven a relative category $\\cat R$ \\textbf{with pullbacks} (i.e.\\\nwhich is closed under pullbacks) a map $p \\in \\cat R$ is called a\n\\textbf{fibrillation} if every diagram in $\\cat R$ of the form\n\\begin{displaymath}\n  \\xymatrix{\n    {\\cdot} \\ar[r]_{j} \\ar[d]\n    & {\\cdot} \\ar[r] \\ar[d]\n    & {\\cdot} \\ar[d]^{p}\\\\\n    {\\cdot} \\ar[r]_{i}\n    & {\\cdot} \\ar[r]\n    & {\\cdot}\n  }\n\\end{displaymath}\nin which\n\\begin{itemize}\n\\item the squares are pullback squares, and\n\\item $i$ is a weak equivalence\n\\end{itemize}\nhas the property that\n\\begin{itemize}\n\\item $j$ is also a weak equivalence.\n\\end{itemize}\n\nThis definition readily implies that, just like the \\emph{fibrations}\nin a \\emph{right proper} model category, fibrillations have the\nfollowing properties.\n\n\\subsection{Proposition}\n\\label{prop:PBfibril}\n\n\\begin{em}\n  \\begin{enumerate}\n  \\item \\label{PBfibrili} Every pullback of a fibrillation is again a\n    fibrillation, and\n  \\item \\label{PBfibrilii} every pullback of a weak equivalence along\n    a fibrillation is again a weak equivalence.\n  \\end{enumerate}\n\\end{em}\n\n\\begin{proof}\n  This is straightforward.\n\\end{proof}\n\n\\boldcentered{The fibrillation lifting lemma}\n\n\n\\subsection{Lemma}\n\\label{lem:FibLftLm}\n\n\\begin{em}\n  Let $f\\colon \\cat R_{1} \\to \\cat R_{2}$ be a relative functor\n  between relative categories with pullbacks which\n  \\begin{itemize}\n  \\item preserves pullbacks (e.g.\\ is a right adjoint), and\n  \\item reflects weak equivalences.\n  \\end{itemize}\n  Then\n  \\begin{itemize}\n  \\item it also reflects fibrillations.\n  \\end{itemize}\n\\end{em}\n\n\\begin{proof}\n  Given a pullback diagram in $\\cat R_{1}$ of the form\n  \\begin{displaymath}\n    \\xymatrix{\n      {\\cdot} \\ar[r]_{j} \\ar[d]\n      & {\\cdot} \\ar[r] \\ar[d]\n      & {\\cdot} \\ar[d]^{f}\\\\\n      {\\cdot} \\ar[r]_{i}\n      & {\\cdot} \\ar[r]\n      & {\\cdot}\n    }\n  \\end{displaymath}\n  in which $i$ is a weak equivalence and of which the image in $\\cat\n  R_{2}$ is a similar diagram in which $fj$ is a fibrillation.  Then\n  $fj$ is a weak equivalence and hence so is $j$.\n\\end{proof}\n\n\\subsection{Example}\n\\label{ex:FibLm}\n\nExamples of functors which satisfy the conditions of the fibrillation\nlifting lemma \\ref{lem:FibLftLm} are the abstract nerve functors of\nDf.~\\ref{def:AbsNvFn} and Pr.~\\ref{prop:AbsNvFncs}.\n\n\n\\boldcentered{The Hopkins-Rezk fibrillation lemma}\n\nWe recall from \\cite{R2}*{\\S2} the following\n\\subsection{Lemma}\n\\label{lem:HPBzz}\n\n\\begin{em}\n  Let\n  \\begin{displaymath}\n    \\xymatrix{\n      & {\\cat A} \\ar[r] \\ar[d]\n      & {\\cat Y} \\ar[d]^{g}\\\\\n      {\\cat X} \\ar[r]^{h}\n      & {\\cat B} \\ar[r]^{\\pi}\n      & {\\cat Z}\n    }\n  \\end{displaymath}\n  be a diagram in a right proper model category in which\n  \\begin{itemize}\n  \\item the square is a pullback square\n  \\item $h$ is a weak equivalence, and\n  \\item $\\pi$ is a fibrillation.\n  \\end{itemize}\n  Then $\\cat A$ is a homotopy pullback of the zigzag\n  \\begin{displaymath}\n    \\zigzag{\\pi}{\\cat B}{\\cat Z}{\\cat Y}{g}\n  \\end{displaymath}\n  and hence (Df.~\\ref{def:HmtpyPlbk}) also of the zigzag\n  \\begin{displaymath}\n    \\zigzag{\\pi h}{\\cat X}{\\cat Z}{\\cat Y}{g}\n  \\end{displaymath}\n\\end{em}\n\n\\boldcentered{The Quillen fibrillation lemma}\n\n\\subsection{Lemma}\n\\label{lem:QFLm}\n\n\\begin{em}\n  Given an object $\\cat D \\in \\widehat{\\mathbf{Cat}}$ (Df.~\\ref{kzeroi}) and a\n  functor $F\\colon \\cat D \\to \\widehat{\\mathbf{Cat}}$ (Df.~\\ref{relcatiii}), the\n  following three statements are equivalent.\n  \\begin{enumerate}\n  \\item \\label{QFLmi} $F$ is a relative functor (Df.~\\ref{relcativ}),\n  \\item \\label{QFLmii} the map $\\N\\pi\\colon \\N\\Gr F \\to \\N\\cat D \\in\n    \\cat S$ (Nt.~\\ref{kzeroiii} and Df.~\\ref{def:GrConst}) is a\n    fibrillation, and\n  \\item \\label{QFLmiii} the map $\\pi\\colon \\Gr F \\to \\cat D \\in\n    \\widehat{\\mathbf{Cat}}$ is a fibrillation.\n  \\end{enumerate}\n\\end{em}\n\n\\begin{proof}\n  To prove (i)$\\Rightarrow$(ii) we note that\n  \\begin{itemize}\n  \\item Quillen's proof of this lemma \\cite{Q}*{\\S1} implies that, for\n    every integer $p \\ge 0$ and map $\\Delta[p]= \\N\\cat p \\to \\N\\cat\n    D$, the pullback\n    \\begin{displaymath}\n      \\pullback{\\N\\cat p}{\\N\\cat D}{\\N\\Gr F}\n    \\end{displaymath}\n    of the zigzag $\\N\\cat p \\to \\N\\cat D\\gets \\N\\Gr F$ is a homotopy\n    pullback\n  \\end{itemize}\n  and that\n  \\begin{itemize}\n  \\item in view of \\cite{R2}*{4.1(i and (ii))} this implies that the\n    map $\\N\\pi \\to \\N\\Gr F \\to \\N D \\in \\cat S$ is a fibrillation.\n  \\end{itemize}\n\n  That (ii)$\\Rightarrow$(iii) then follows from the \\emph{Fibrillation\n    lifting lemma} \\ref{lem:HPBzz} and Ex.~\\ref{ex:FibLm}.\n\n  Finally (iii)$\\Rightarrow$(i) follows by a simple calculation from\n  the fact that, in view of the \\emph{Hopkins-Rezk fibrillation lemma}\n  \\ref{lem:HPBzz} the pullbacks of the form\n  (Df.~\\ref{def:strcalculus})\n  \\begin{displaymath}\n    \\pullback{\\boldsymbol 0^{w}}{\\cat D}{\\Gr F}\n    \\qquad\\text{and}\\qquad\n    \\pullback{\\boldsymbol 1^{w}}{\\cat D}{\\Gr F}\n  \\end{displaymath}\n  are both homotopy pullbacks.\n\\end{proof}\n\n\n\\section{A proof of Theorem \\ref{thm:MpPropBn}}\n\\label{sec:PrfThm}\n\n\\subsection{Preliminaries}\n\\label{sec:Prelim}\n\nWe start with recalling from Pr.~\\ref{prop:zzEmbed} that\n\\begin{enumerate}\n\\item \\label{Prelimi} every zigzag $\\zigzag{f}{\\cat X}{\\cat Z}{\\cat\n    Y}{g}$ in $\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ ($k \\ge 0$) or $\\mathbf{RelCat}$ can, for\n  every integer $n \\ge 1$, in a functorial manner be embedded in a\n  commutative diagram of the form (Df.~\\ref{def:NArrFibr} and\n  \\ref{def:NArrPB})\n  \\begin{displaymath}\n    \\xymatrix{\n      & {\\subcommacat{f\\cat X}{n}{g\\cat Y}} \\ar[r]^{\\qquad\\pi} \\ar[d]\n      & {Y} \\ar[d]^{g}\\\\\n      {X} \\ar[r]^{h\\qquad}\n      & {\\subcommacat{f\\cat X}{n}{\\cat Z}} \\ar[r]^{\\qquad\\pi}\n      & {Z}\n    }\n  \\end{displaymath}\n  in which\n  \\begin{itemize}\n  \\item the square is a pullback square, and\n  \\item $h$ is a weak equivalence\n  \\end{itemize}\n\\end{enumerate}\nand note that, in view of the \\emph{Hopkins-Rezk fibrillation lemma}\n\\ref{lem:HPBzz}\n\\begin{resumeenumerate}{2}\n\\item \\label{Prelimii} if the map $\\pi\\colon \\commacat{f\\cat X}{\\cat\n    Z} \\to \\cat Z$ is a fibrillation, then the object\n  $\\subcommacat{f\\cat X}{n}{g\\cat Y}$ is a homotopy pullback of that\n  zigzag.\n\\end{resumeenumerate}\n\n\n\\subsection{A proof for the category $\\widehat{\\mathbf{Cat}}$}\n\\label{sec:PrfCtht}\n\n\\begin{enumerate}\n\\item \\label{PrfCthti} It follows from Df.~\\ref{def:GrConst} and\n  \\ref{BnCni}, Ex.~\\ref{GrCniii} and the \\emph{Quillen fibrillation\n    lemma} \\ref{lem:QFLm} that the map $\\pi\\colon \\subcommacat{f\\cat\n    X}{n}{\\cat Z} \\to \\cat Z$ is a fibrillation, and\n\\item \\label{PrfCthtii} the desired result now follows from\n  \\ref{sec:Prelim}.\n\\end{enumerate}\n\n\\subsection{A proof for the categories $\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ ($k\\ge 1$)}\n\\label{sec:PrfSmpk}\n\n\\begin{enumerate}\n\\item In view of \\ref{PrfCthti} and Df.~\\ref{def:MpPrpB}, for every\n  $k$-fold dimension $p_{*}$, the map\n  \\begin{displaymath}\n    \\pi_{p_{*}}\\colon\n    \\subcommacat{f_{p_{*}}\\cat X_{p_{*}}}{n}{\\cat Z_{p_{*}}}\n    \\longrightarrow \\cat Z_{p_{*}} \\in \\widehat{\\mathbf{Cat}}\n  \\end{displaymath}\n  is a fibrillation, and\n\\item if $\\prod_{p_{*}} p_{*}$ denotes the product of these maps for\n  all $k$-fold dimensions and $\\prod_{p_{*}} \\widehat{\\mathbf{Cat}}$ denotes the\n  corresponding product of copies of $\\widehat{\\mathbf{Cat}}$, then clearly the same\n  holds for the map\n  \\begin{displaymath}\n    \\prod_{p_{*}}\\bigl(\\pi_{p_{*}}\\colon\n        \\subcommacat{f_{p_{*}}\\cat X_{p_{*}}}{n}{\\cat Z_{p_{*}}}\n        \\to \\cat Z_{p_{*}}\\bigr) \\in \\prod_{p_{*}} \\widehat{\\mathbf{Cat}}\n  \\end{displaymath}\n\\item Moreover one readily verifies that, in view of\n  Df.~\\ref{def:NArrPB}, the obvious map $\\mathrm{s}^{k}\\widehat{\\mathbf{Cat}} \\to\n  \\prod_{p_{*}}\\widehat{\\mathbf{Cat}}$ satisfies the conditions of the\n  \\emph{Fibrillation lifting lemma} \\ref{lem:FibLftLm}, which implies\n  that the map $\\pi\\colon \\subcommacat{f\\cat X}{n}{\\cat Z} \\to \\cat Z\n  \\in \\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ is also a fibrillation.\n\\item The desired result now follows from \\ref{sec:Prelim}.\n\\end{enumerate}\n\n\\subsection{Two proofs for the category $\\mathbf{RelCat}$}\n\\label{sec:TwoPrf}\n\n\\begin{enumerate}\n\\item \\label{TwoPrfi} As the map $f\\colon \\cat X \\to \\cat Z \\in\n  \\mathbf{RelCat}$ has property $B_{n}$ (by assumption), so does, in view of\n  Df.~\\ref{def:MpPrBrel}, the map $w_{*}f\\colon w_{*}\\cat X \\to\n  w_{*}\\cat Z \\in \\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$.\n\\item \\label{TwoPrfii} It follows from \\ref{sec:PrfSmpk} that the map\n  $\\pi\\colon \\commacat{w_{*}fw_{*}\\cat X}{w_{*}\\cat Z} \\to w_{*}\\cat Z\n  \\in \\mathrm{s}^{k}\\widehat{\\mathbf{Cat}}$ is a fibrillation.\n\\item \\label{TwoPrfiii} Moreover as (Ex.~\\ref{ex:FibLm}) $w_{*}$\n  satisfies the conditions of the \\emph{Fibrillation lifting lemma}\n  \\ref{lem:FibLftLm} it follows from \\ref{prop:wpicom} that the map\n  $\\pi\\colon \\subcommacat{f\\cat X}{n}{\\cat Z} \\to \\cat Z \\in \\mathbf{RelCat}$\n  is also a fibrillation and the desired result then follows from\n  \\ref{sec:Prelim}.\n\\end{enumerate}\n\nHowever if one does not want to use the model structure on $\\mathbf{RelCat}$\none can also, instead of using (ii) and (iii) proceed as follows:\n\\begin{itemize}\n\\item [(ii)$'$] It follows from \\ref{sec:PrfSmpk} that the object\n  $\\commacat{w_{*}fw_{*}\\cat X}{w_{*}gw_{*}\\cat Y}$ is a homotopy\n  pullback of the zigzag $\\zigzag{w_{*}f}{w_{*}\\cat X}{w_{*}\\cat\n    Z}{w_{*}\\cat Y}{w_{*}g}$, and\n\\item [(iii)$'$] as (Pr.~\\ref{prop:AbsNvFncs}) $w_{*}$ is a homotopy\n  equivalence the desired result now follows from Pr.~\\ref{prop:wpicom}\n  and the \\emph{Global equivalence Lemma} \\ref{lem:WHLEx}.\n\\end{itemize}\n\n\\subsection{A proof for the categories $\\mathbf{Rel}^{k}\\mathbf{Cat}$ ($k>1$)}\n\\label{sec:PrfRlkCt}\n\nThis is essentially the same as the second proof for $\\mathbf{RelCat}$\n(\\ref{TwoPrfi}, (ii)$'$ and (iii)$'$).\n\n\n\n\n\n\n\n\n\\begin{bibdiv} \n  \\begin{biblist}\n\n\n    \\bib{B}{thesis}{\n      label={B},\n      author={Barwick, Clark},\n      title={$(\\infty,n)$-$\\mathbf{Cat}$ as a closed model category},\n      organization={University of Pennsylvania},\n      date={2005}\n    }\n\n    \\bib{BK1}{article}{\n      label={BK1},\n      author={Barwick, Clark},\n      author={Kan, Daniel M.},\n      title={Relative categories: another model for the homotopy theory of\n        homotopy theories},\n      journal={Indag. Math. (N.S.)},\n      volume={23},\n      date={2012},\n      number={1-2},\n      pages={42--68},\n      issn={0019-3577},\n    }\n\t\t\n    \\bib{BK2}{misc}{\n      label={BK2},\n      author={Barwick, Clark},\n      author={Kan, Daniel M.},\n      title={$n$-relative categories: a model for the homotopy theory\n        of $n$-fold homotopy theories},\n      status={To appear}\n    }\n\n    \\bib{BK3}{misc}{\n      label={BK3},\n      author={Barwick, Clark},\n      author={Kan, Daniel M.},\n      title={Partial model categories and their simplicial nerves},\n      status={To appear}\n    }\n\n    \\bib{DHKS}{book}{\n      label={DHKS},\n      author={Dwyer, William G.},\n      author={Hirschhorn, Philip S.},\n      author={Kan, Daniel M.},\n      author={Smith, Jeffrey H.},\n      title={Homotopy limit functors on model categories and homotopical\n        categories},\n      series={Mathematical Surveys and Monographs},\n      volume={113},\n      publisher={American Mathematical Society},\n      place={Providence, RI},\n      date={2004},\n    }\n\n    \\bib{DK}{article}{\n      label={DK},\n      author={Dwyer, W. G.},\n      author={Kan, D. M.},\n      title={Calculating simplicial localizations},\n      journal={J. Pure Appl. Algebra},\n      volume={18},\n      date={1980},\n      number={1},\n      pages={17--35},\n    }\n\n    \\bib{DKS}{article}{\n      label={DKS},\n      author={Dwyer, W. G.},\n      author={Kan, D. M.},\n      author={Smith, J. H.},\n      title={Homotopy commutative diagrams and their realizations},\n      journal={J. Pure Appl. Algebra},\n      volume={57},\n      date={1989},\n      number={1},\n      pages={5--24},\n    }\n\n    \\bib{GZ}{book}{\n      label={GZ},\n      author={Gabriel, P.},\n      author={Zisman, M.},\n      title={Calculus of fractions and homotopy theory},\n      series={Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35},\n      publisher={Springer-Verlag New York, Inc., New York},\n      date={1967},\n    }\n\n    \\bib{H}{book}{\n      label={H},\n      author={Hirschhorn, Philip S.},\n      title={Model categories and their localizations},\n      series={Mathematical Surveys and Monographs},\n      volume={99},\n      publisher={American Mathematical Society},\n      place={Providence, RI},\n      date={2003},\n    }\n\n    \\bib{La}{article}{\n      label={La},\n      author={Latch, Dana May},\n      title={The uniqueness of homology for the category of small categories},\n      journal={J. Pure Appl. Algebra},\n      volume={9},\n      date={1976\/77},\n      number={2},\n      pages={221--237},\n    }\n\n    \\bib{Le}{article}{\n      label={Le},\n      author={Lee, Ming Jung},\n      title={Homotopy for functors},\n      journal={Proc. Amer. Math. Soc.},\n      volume={36},\n      date={1972},\n      pages={571--577; erratum, ibid. 42 (1973), 648--650},\n    }\n\n    \\bib{Lu}{article}{\n      label={Lu},\n      author={Lurie, Jacob},\n      title={$(\\infty,2)$-categories and the Goodwillie calculus I},\n      note={Available at \\url{http:\/\/arxiv.org\/abs\/0905.0462}},\n      status={To appear}\n    }\n    \n    \\bib{Q}{article}{\n      label={Q},\n      author={Quillen, Daniel},\n      title={Higher algebraic $K$-theory. I},\n      conference={\n        title={Algebraic $K$-theory, I: Higher $K$-theories (Proc. Conf.,\n          Battelle Memorial Inst., Seattle, Wash., 1972)},\n      },\n      book={\n        publisher={Springer},\n        place={Berlin},\n      },\n      date={1973},\n      pages={85--147. Lecture Notes in Math., Vol. 341},\n    }\n\n    \\bib{R1}{article}{\n      label={R1},\n      author={Rezk, Charles},\n      title={A model for the homotopy theory of homotopy theory},\n      journal={Trans. Amer. Math. Soc.},\n      volume={353},\n      date={2001},\n      number={3},\n      pages={973--1007 (electronic)},\n    }\n\n    \\bib{R2}{misc}{\n      label={R2},\n      author={Rezk, Charles},\n      title={Fibrations and homotopy colimits of simplicial sheaves},\n    }\n\n    \\bib{T1}{article}{\n      label={T1},\n      author={Thomason, R. W.},\n      title={Homotopy colimits in the category of small categories},\n      journal={Math. Proc. Cambridge Philos. Soc.},\n      volume={85},\n      date={1979},\n      number={1},\n      pages={91--109},\n      issn={0305-0041},\n    }\n\n    \\bib{T2}{article}{\n      label={T2},\n      author={Thomason, R. W.},\n      title={Cat as a closed model category},\n      journal={Cahiers Topologie G\\'eom. Diff\\'erentielle},\n      volume={21},\n      date={1980},\n      number={3},\n      pages={305--324},\n      issn={0008-0004}\n    }\n\n\n  \\end{biblist}\n\\end{bibdiv}\n\n\n\n\n\\end{document}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThanks to extremely sensitive near-infrared (NIR) imaging obtained\nusing Wide Field Camera 3 (WFC3) on the {\\em Hubble Space Telescope}\nit is now possible to routinely identify galaxies \\ at very high\nredshift ($z>6$: e.g., Oesch et al.\\ 2010; Bunker et al.\\ 2010;\nWilkins et al.\\ 2010; Wilkins et al.\\ 2011a; Bouwens et al.\\ 2011b;\nFinkelstein et al.\\ 2010, 2012; Oesch et al.\\ 2012; McLure et\nal.\\ 2013; Schmidt et al.\\ 2014) with the first samples now being\nidentified at $z\\sim 10$ and beyond, less than 500 Myr after the Big\nBang (Bouwens et al.\\ 2011a; Zheng et al.\\ 2012; Ellis et al.\\ 2013;\nOesch et al.\\ 2013, 2014, 2015b; Zitrin et al.\\ 2014; Ishigaki et\nal.\\ 2015; Bouwens et al.\\ 2015b).\n\nOne area of significant interest in the study of distant galaxies\nregards their spectral characteristics.  At very early times, we might\nexpect galaxies to potentially have very different SEDs than at latter\ntimes.  In particular, one would expect galaxies to be bluer in terms\nof their rest-frame UV colours and UV-optical breaks, both due to a\nyounger stellar population (Wilkins et al.\\ 2013a) and a lower dust\ncontent (e.g., Wilkins et al.\\ 2011b; Bouwens et al.\\ 2009, 2012;\nFinkelstein et al.\\ 2012).  In fact, there has been some evidence for\nbluer UV colours at high redshift (Lehnert \\& Bremer 2003; Papovich et\nal.\\ 2004; Stanway et al.\\ 2005; Bouwens et al.\\ 2006, 2009, 2012,\n2014a; Wilkins et al.\\ 2011b; Finkelstein et al.\\ 2012; Kurczynski et\nal.\\ 2014) though the strength of the evolution at the highest\nredshifts has been subject to some debate (Dunlop et al.\\ 2012;\nRobertson et al.\\ 2013).  Evolution in the size of the Balmer break is\nless easily inferred (Stark et al.\\ 2009; Gonzalez et al.\\ 2010;\nMcLure et al.\\ 2011) largely due to the presence of strong nebular\nemission lines (Shim et al.\\ 2011; Schaerer \\& de Barros 2010; Wilkins\net al. 2013b), but now seems clear from $z\\sim6$ to $z\\sim2$ (Stark et\nal.\\ 2013; Gonzalez et al.\\ 2014; Schaerer \\& de Barros 2013;\nLabb{\\'e} et al.\\ 2013; Smit et al.\\ 2014; Salmon et al.\\ 2015).\n\nThe spectral characteristics of galaxies at $z\\sim8$ and $z\\sim8.5$\nhave also been explored (Bouwens et al.\\ 2010, 2013; Finkelstein et\nal.\\ 2010, 2012; Dunlop et al.\\ 2013), but are more difficult to\nrobustly quantify due to the limited leverage in wavelength available\nfor constraining these slopes and corrections required to remove the\nimpact of the IGM absoprtion (Bouwens et al.\\ 2014a).  This is\nparticularly true for $UV$-continuum slope determinations at\n$z\\sim8.5$-9.5.  Given the challenges in deriving the spectral\ncharacteristics of galaxies at $z\\sim8.5$-9.5, it may seem that\nfurther advances may need to wait until the {\\em James Webb Space\n  Telescope} (JWST).\n\nFortunately, as we will show, we can make immediate progress on this\nissue by taking advantage of the deep IRAC observations available for\ngalaxy samples at $z\\sim10$ from the {\\em Spitzer Space Telescope}.\nOver the redshift interval $z\\sim9.5$-10.5, the {\\em Hubble} $H$-band\nband and the IRAC $3.6\\mu$m band fall in the $UV$-continuum.  The\nwavelength leverage is sufficient between these bands that one can\nplausibly quantify the UV-continuum slopes of galaxies more accurately\nat $z\\sim10$ than is possible at $z\\sim8$ and especially at\n$z\\sim8.5$-9, as we demonstrate in \\S\\ref{sec:observations.slope} of\nthis paper.\n\nOpportunities to use $z\\sim10$ samples to perform such studies now\nexist thanks to the deep near-IR data available from {\\em Hubble} in\nthe Cosmic Assembly Near-Infrared Deep Extragalactic Survey (CANDELS,\nGrogin et al. 2011; Koekemoer et al. 2011), {\\em Hubble} Ultra-Deep\nField 2009\/2012 programs (Bouwens et al.\\ 2011; Koekemoer et al. 2013;\nEllis et al. 2013)\\footnote{The HUDF12 programme \\ obtained\n  $Y_{f105w}$ and $J_{f160w}$. Both the HUDF09 and HUDF12\n  observations, along with imaging from various other sources have\n  been released as part of the eXtreme Deep Field (XDF) project\n  (Illingworth et al. 2013).}, and Cluster Lensing and Supernovae\nsurvey with {\\em Hubble} (CLASH: Postman et al.\\ 2012) programs.  The\nsamples range from particularly faint sources over the HUDF\/XDF (Ellis\net al. 2013; Oesch et al. 2013) and the Frontier Fields (Zitrin et\nal.\\ 2014) to brighter sources located over CANDELS (Oesch et\nal. 2014) and CLASH (e.g. Zheng et al. 2012, Coe et al. 2013, and\n\\ Zitrin et al. 2014).\\footnote{It is important to note that not all\n  such candidates necessarily lie at $z\\sim 10$; indeed UDFj-39546284\n  (Bouwens et al. 2011a) for instance, which was initially reported to\n  be a $z\\sim 10$ candidate, no longer has a favoured high-redshift\n  identification (Ellis et al.\\ 2013; Brammer et al. 2013, Bouwens et\n  al. 2013a).}\n\nParticularly valuable for probing the spectral characteristics of\ngalaxies in the early universe are those $z\\sim10$ candidates that are\nintrinsically bright or lensed, since those sources have sufficient\nS\/N with IRAC that one can use them to quantify the mean\n$UV$-continuum slope of galaxies at very early times.  To date, five\nsuch galaxies have been identified in the magnitude range 26-27 mag\nfrom the CANDELS fields (Oesch et al.\\ 2014) and behind lensing\nclusters (Zheng et al.\\ 2012).\n\nIn this paper, we make use of these 5 particularly bright $z\\sim10$\ncandidates to derive, for the first time, the mean $UV$-continuum\nslope $\\beta$ at $z\\sim10$ for a multi-object sample.  We compare the\nobserved properties with predictions from recent cosmological galaxy\nformation simulations to provide some context.  Such simulations are\nvaluable given that the $UV$-continuum slopes and $UV$-optical colours\ncan be affected by a complex mixture of different factors including\nthe joint distribution of stellar masses, ages, metallicities, and the\nescape fraction -- which makes it difficult to interpret the\nobservations in terms of a single variable (Bouwens et al.\\ 2010;\nWilkins et al.\\ 2011).\n\nThis paper is organised as follows: in Section \\ref{sec:observations}\nwe describe recent observations of candidate $z\\sim 10$ star forming\ngalaxies. In Section \\ref{sec:dust_interp} we interpret observations\nof these systems in the context of dust emission.  Finally, in Section\n\\ref{sec:conclusions} we present our conclusions. Throughout this work\nmagnitudes are calculated using the $AB$ system (Oke \\& Gunn 1983). In\ncalculating absolute magnitudes we assume a $\\Omega_{M}=0.3$,\n$\\Omega_{\\Lambda}=0.7$, $h=0.7$ cosmology.\n\n\\section{Observations of the UV-continuum slope}\\label{sec:observations}\n\n\\subsection{Data and Sample Selection}\\label{sec:data}\n\nThis paper is based on the bright $z\\sim10$ galaxy sample selected\nover the GOODS fields from Oesch et al. (2014) as well the $z\\sim9.6$\nCLASH source behind MACS1149 identified in Zheng et al. (2012).  For\ndetails on the datasets, we refer the reader to the discovery\npapers. We summarise the photometry of these sources in Table\n\\ref{tab:photometry}.\n\nIn brief, the candidates from Oesch et al. (2014) were identified\nusing the complete {\\em Hubble} dataset from the CANDELS survey\n(Grogin et al. 2011, Koekemoer et al. 2011) in addition to ancillary\nAdvanced Camera for Surveys (ACS) data mostly from the GOODS\nsurvey. The central area of the GOODS-South and North fields represent\nthe CANDELS Deep survey, which reach to H$_{f160w}=27.8$ mag\n($5\\sigma$), while the outer regions are covered with slightly\nshallower data H$_{f160w}=27.1$ mag.  Galaxy candidates at $z\\sim10$\nwere identified based on the spectral break shortward of\nLyman-$\\alpha$ resulting in a red color of\n$($J$_{f125w}-$H$_{f160w})>1.2$ and complete non-detection in all\nshorter wavelength filters. While Oesch et al. (2014) also selected\n$z\\sim9$ galaxies with a weaker continuum break, here we restrict our\nanalysis to the redder color selection resulting in galaxies with\n$z_{phot}\\gtrsim9.5$.\n\nHere we make use of a redetermination of the IRAC photometry for the\nfour sources in the Oesch et al.\\ (2014) sample.  For these\nmeasurements, we take advantage of new reductions of the Spitzer\/IRAC\n3.6\\,$\\mu$m and 4.5\\,$\\mu$m imaging data over the GOODS fields from\nLabbe et al.\\ (2015).  These reductions include all data from the\noriginal GOODS, the Spitzer Extended Deep Survey (SEDS: Ashby et\nal.\\ 2013), the IRAC Ultra Deep Field (IUDF: Labb\\'e et al.\\ 2015),\nand S-CANDELS (Ashby et al.\\ 2015) programs.  The average 5$\\sigma$\ndepths of the IRAC data within 1\\arcsec\\ radius apertures are 27.0 and\n26.7 mag in the two channels, respectively.\n\nOur measurements again make use of the \\textsc{mophongo} software\n(Labb{\\'e} et al. 2006, 2010, 2013) to model and subtract the flux\nfrom neighboring sources, but include significantly improved PSF\nmodeling.  Our derived PSFs account for the orientation of all\nexposures that contribute to our IRAC reductions.  Our rederived IRAC\nfluxes are completely consistent within the errors with those given in\nOesch et al. (2014).  Three of the four $z\\sim10$ galaxy candidates\nfrom Oesch et al. (2014) are significantly detected ($>4.5\\sigma$) in\nthese data in at least one filter. In particular, the brightest source\nGN-z10-1 with a photometric redshift of $z_{phot} = 10.2\\pm0.4$ is\nrobustly detected in the 3.6\\,$\\mu$m channel at $\\sim7\\sigma$,\nallowing for an individual estimate a UV-continuum slope for this\nsource.\n\nEqually important for robust measurements of the $H_{160}-[3.6]$\ncolors are accurate measurements of the total $H_{160}$-band flux.\nOur determination of the total $H_{160}$-band flux is as described in\nOesch et al.\\ (2014) and includes all the light inside an elliptical\naperture extending to 2.5 Kron (1980) radii.  The measured flux inside\nthe utilized Kron aperture is corrected to total (typically a\n$\\sim$0.2 mag correction) based on the expected light outside this\naperture (Dressel et al.\\ 2012).\n\nAs a correction to total is performed for both the HST $H_{160}$-band\nphotometry and for the Spitzer\/IRAC photometry, colors derived from\nHST to Spitzer\/IRAC should not suffer from significant biases.\nNevertheless, it was worthwhile to verify that this was the case by\napplying the HST-to-Spitzer PSF-correction kernel to the HST data\n(derived from \\textsc{mophongo}) and then measuring total magnitudes\nfrom the HST data in the same way as the Spitzer\/IRAC data.  Given the\nrelatively small number of $z\\sim10$ candidates in our sample and\ntheir limited S\/N after convolving with the IRAC PSF correction\nkernel, we perform this test on a sample of bright ($H_{160,AB}<25$)\n$z\\sim4$ galaxies from the Bouwens et al.\\ (2015a) catalogs.  We found\nthat the total magnitudes we recover by applying this procedure to the\nHST observations of these $z\\sim4$ sources were consistent to $<0.1$\nmag in the median with that derived using our primary method.\n\nTo probe intrinsically fainter galaxies while maintaining a sufficiently high signal-to-noise, we can take advantage of sources which are lensed by foreground galaxies (or clusters of galaxies). Several $z\\sim 10$ candidates have now been discovered in cluster searches (Zheng et al. 2012; Coe et al. 2013; Zitrin et al. 2014), with more likely to be identified in the near future as a result of the ongoing {\\em Hubble} Frontier Fields observations. \n\nOf the three currently known $z\\sim 10$ candidates we concentrate on MACS1149-JD (Zheng et al. 2012). The object presented by Zitrin et al. (2014) does not currently have sufficiently deep IRAC imaging to provide anything other than a weak upper limit on the rest-frame UV-continuum slope. The object presented by Coe et al. (2013) has a photometric redshift of $z\\sim 10.7$, at which redshift the $H_{160}$-band flux could be affected by the position of the Lyman-$\\alpha$ break within the $H_{160}$ band and\/or Lyman-$\\alpha$ line emission. \n\nMACS1149-JD was identified in a search of 12 CLASH clusters and has strong detections in both the JH$_{f140w}$ and H$_{f160w}$ bands with weaker detections in both Y$_{f105w}$ and J$_{f125w}$ and non-detections in several optical bands. Photometric redshift fitting of the sources photometry suggests it is $z\\sim 9.6$.\\footnote{Bouwens et al.\\ 2014b estimate a photometric redshift of 9.7$\\pm$0.1.}  Zheng et al. (2012) presented {\\em Spitzer}\/IRAC photometry of MACS1149-JD ([3.6]$<160\\,{\\rm nJy},\\,1\\sigma$) based on observations taken under Program ID 60034 (PI: Egami). We augment these observations with Frontier Fields observations and archival data taken as part of the Spitzer UltRa Faint SUrvey Program (Surfs'Up, Brada{\\v c} et al. 2014) and measure a [3.6] flux of $175\\pm44$ nJy (see Bouwens et al.\\ 2014b). This is consistent with that reported by both Zheng et al. (2012) ($<160\\,{\\rm nJy},\\,1\\sigma$) and Brada{\\v c} et al. (2014) ($190\\pm 87\\,{\\rm nJy}$).\n\n\n\\begin{table*}\n\\caption{Photometry and derived properties for the $z\\sim10$ sources, and the bright stack, considered in this work.\\label{tab:photometry}}\n\\begin{tabular}{lccccccccc}\n\\hline\n & $z$ & $M_{1500}$ & H$_{f160w}$\/nJy & [3.6]\/nJy & H$_{f160w}-$[3.6] & $\\beta_{\\rm obs}$ & $A_{1500,{\\rm C00}}$ & $A_{1500,{\\rm SMC}}$ &\\\\\n& & & & & & & \\multicolumn{2}{c}{(assumes $\\beta_{\\rm int} = -2.54$)$^7$} \\\\\n\\hline\nGN-z10-1$^{1,2}$ & $10.2$ & $-21.6$ & $152\\pm10$ & $136\\pm29$ & $-0.1\\pm0.2$ & $-2.1\\pm0.3$ & $0.9\\pm0.6$ & $0.5\\pm0.3$ \\\\\nGN-z10-2$^{1,2}$ & $9.8$ & $-20.7$ & $68\\pm9$ & $45\\pm26$ & $-0.5\\pm0.6$ & $-2.5\\pm0.7$ & $0.1\\pm1.6$ & $0.1\\pm0.9$ \\\\\nGN-z10-3$^{1,2}$ & $9.5$ & $-20.6$ & $73\\pm8$ & $59\\pm24$ & $-0.2\\pm0.5$ & $-2.3\\pm0.5$ & $0.6\\pm1.1$ & $0.3\\pm0.6$ \\\\\nGS-z10-1$^{1,2}$ & $9.9$ & $-20.6$ & $66\\pm9$ & $71\\pm27$ & $0.1\\pm0.4$ & $-1.9\\pm0.5$ & $1.4\\pm1.1$ & $0.8\\pm0.6$ \\\\\nMACS1149-JD1$^{4}$ & $9.6$ & $-19.4$$^4$ & $190\\pm13.3$$^5$ & $177\\pm44$ & $-0.1\\pm0.3$ & $-2.1\\pm0.3$ & $1.0\\pm0.7$ & $0.6\\pm0.4$ \\\\\n\\hline\n\\bf STACK & - & - &  - &  - & $-0.1\\pm0.2$$^6$ & $-2.1\\pm0.3$$^6$ & $0.9\\pm0.6$ & $0.5\\pm0.3$  \\\\\n\\hline\n\\end{tabular}\n\n\\medskip\n$^1$ Oesch et al. (2014). $^{2}$ included in stack. $^3$ Zheng et al. (2012), Bouwens et al.\\ (2014b). $^{4}$ MACS1149-JD is gravitationally lensed by a foreground cluster, we determine the un-lensed absolute magnitude using the best-fit magnification of $14.5$. $^{5}$ We independently measure the [3.6] flux for MACS1149-JD using the Frontier Fields {\\em Spitzer}\/IRAC observations combined with observations taken as part of the Spitzer UltRa Faint SUrvey Program (SURFSUP, Brada{\\v c} et al. 2014) and Program ID 60034 (PI: Egami). Our flux measurements are consistent with those reported by both Zheng et al. (2012) ($<160\\,{\\rm nJy},\\,1\\sigma$) and Brada{\\v c} et al. (2014) ($190\\pm 87\\,{\\rm nJy}$).  $^6$ The uncertainty on the mean increases to $\\pm0.3$, if we assume there is significant intrinsic scatter in the $\\beta$ distribution as observed at $z\\sim4$-5 (Bouwens et al.\\ 2009, 2012; Castellano et al.\\ 2012; Rogers et al.\\ 2014: see \\S2.2).  $^7$ The intrinsic $UV$-continuum slope predicted in dynamical simulation at $z\\sim 10$ assuming $f_{\\rm esc}=0$ (\\S3).\n\\end{table*}\n\n\n\n\\subsubsection{Bright Stack}\\label{sec:bright_stack}\n\nThe uncertainties on the $UV$-continuum slopes of individual sources\nis large enough that it is useful to consider constraints for the\naverage $z\\sim10$ source from the Oesch et al.\\ (2014) bright sample.\nWe therefore stack the photometry of the 4 $z\\sim 10$ candidates\npresented in Oesch et al. (2014).  The H$_{f160w}-$[3.6] colour of the\nstack is $-0.1\\pm0.2$. If we stack only GN-z10-1 and GN-z10-2 (i.e. excluding sources without JH$_{f140w}$ detections) we find H$_{f160w}-$[3.6]$=-0.2\\pm0.2$.\n\n\\subsection{Measuring the UV continuum slope}\\label{sec:observations.slope}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=20pc]{figures\/measuring_beta.eps}\n\\caption{Relative observed near-IR photometry of a model star-forming galaxy at $z\\in\\{7,8,9,10\\}$ highlighting the bands available to measure the rest-frame UV-continuum slope. At $z>9.6$, the {\\em Spitzer}\/IRAC [3.6] band can be combined with the H$_{f160w}$ band to measure the UV continuum slope over a large wavelength baseline, minimising its uncertainty. At $z\\sim 8$ only the JH$_{f140w}$ and H$_{f160w}$ bands are uncontaminated by the Lyman-$\\gamma$ break providing only a small wavelength baseline and leaving the uncertainty on the observed UV continuum slope very large.}\n\\label{fig:measuring_beta}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=20pc]{figures\/wavelength_baseline.eps}\n\\caption{{\\em Top panel} - The number of {\\em Hubble}\/WFC3 (Y$_{f098m}$, J$_{f125w}$, JH$_{f140w}$, and H$_{f160w}$) and {\\em Spitzer}\/IRAC ([3.6]) bands probing the rest frame $1216<\\lambda_{\\rm rest}\/{\\rm\\AA}<3700$ UV continuum as a function of redshift. {\\em Second panel} - The rest-frame UV wavelength baseline accessible by {\\em Hubble}\/WFC3 and {\\em Spitzer}\/IRAC observations as a function of redshift. {\\em Third panel } - The ratio of the mean wavelength of the bluest and reddest filters to probe the rest-frame UV continuum. The arrows at the top denote the redshift range over which an individual band probes the rest-frame $1216<\\lambda_{\\rm rest}\/{\\rm\\AA}<3700$ UV continuum. {\\em Bottom panel } - The expected uncertainty on the measurement of $\\beta$ as a function of redshift. This assumes a $\\beta=-2$, $m=26$ source, for which $\\beta$ is measured from the colour providing the longest usable wavelength baseline.}\n\\label{fig:wavelength_baseline}\n\\end{figure}\n\nFor $z>7$ galaxies, observations of the rest-frame UV continuum are\ntypically limited to just 3 WFC3\/IR bands uncontaminated by the\nLyman-$\\alpha$ break (i.e., $J_{f125w}$, $JH_{f140w}$, and\n$H_{f160w}$), and therefore suitable to measure the UV continuum slope\n$\\beta$. This is demonstrated, in Figure \\ref{fig:measuring_beta}, for\nsources at $z\\in\\{7,8,9,10\\}$.\n\nWhen only two bands are utilised for $UV$-continuum slope estimates, the relationship\\footnote{Assuming the underlying spectral flux density is described by a power-law, i.e. $f_{\\nu}\\propto\\lambda^{\\beta+2}$} between the observed colour and UV continuum slope can simply be written as:\n\n\\begin{equation}\n\\beta = p\\times (m_{1}-m_{2})_{\\rm AB}-2,\n\\end{equation}\nwhere $(m_{1}-m_{2})_{\\rm AB}$ is the observed colour assuming the AB magnitude system and $p$ is a value sensitive to the choice of bands. The value $p$ is approximately related to the ratio of the effective wavelengths ($\\lambda_1$, $\\lambda_2$) of the filters used to probe the slope\\footnote{The value of $p$ is also sensitive to the shape of the filter and when calculating the final value appropriate for our observations we take this into account.}, i.e.,\n\n\\begin{equation}\np^{-1} = 2.5\\cdot\\log_{10}(\\lambda_1\/\\lambda_2).\n\\end{equation}\n\nThe value of $\\log_{10}(\\lambda_1\/\\lambda_2)$ and the rest-frame UV wavelength baseline ($\\lambda_2-\\lambda_1$) accessible by {\\em Hubble}\/WFC3 and {\\em Spitzer}\/IRAC observations as a function of redshift are shown in Figure \\ref{fig:wavelength_baseline}.\n\nThis accessible wavelength baseline is important as the uncertainty on $\\beta$ will also scale with $p$. For example, at $z\\sim 8$ where we only have observations using the JH$_{f140w}$ and H$_{f160w}$ bands, the value of $p$ using these bands is approximately $9.0$ (Bouwens et al.\\ 2014a). A typical error on $m_{1}$ and $m_{2}$ of $0.1$ then translates into a large uncertainty on $\\beta$ ($\\approx 9.0\\cdot\\sqrt{0.1^{2}+0.1^{2}}\\approx 1.3$). At $z>9.5$ the {\\em Spitzer}\/IRAC [3.6]-band probes the UV continuum, which, when combined with H$_{f160w}$-band observations, provides an especially extended wavelength baseline.  The end result is a particularly small value for $p$ ($\\approx 1.1$) and thus small uncertainty on $\\beta$. This long wavelength baseline compensates for the lower sensitivity of observations with {\\em Spitzer}\/IRAC [3.6], allowing the the UV continuum to be estimated much more robustly than at $z\\sim 8$ and on a par with $z\\sim 6-8$ for the same observed apparent magnitude. This is demonstrated in the bottom panel of Figure \\ref{fig:wavelength_baseline}. \n\nThe observed values of the UV-continuum slope $\\beta$ for the $z\\sim\n10$ candidates (and the stack) are listed in Table\n\\ref{tab:photometry} and shown in Figure \\ref{fig:M_beta}. For the\nbrightest candidate (GN-z10-1) we find $\\beta_{\\rm obs}=-2.1\\pm 0.2$\nwhile for the bright stack (see \\S\\ref{sec:bright_stack}) we find\n$\\beta_{\\rm obs}=-2.1\\pm 0.3$.  If we stack only those sources which\nhave $JH_{140}$-band observations (providing a second WFC3\/IR filter\nwhere the $z\\sim10$ candidates are detected), i.e., GN-z10-1 and\nGN-z10-2, we find $\\beta_{\\rm obs}=-2.2\\pm 0.3$.\n\nWhile the formal random error on the mean $\\beta$ is 0.2, at lower\nredshifts the $\\beta$ distribution for luminous galaxies appears to\nshow a significant intrinsic scatter of $\\sigma_{\\beta}\\sim0.35$\n(Bouwens et al.\\ 2009, 2012; Castellano et al.\\ 2012; Rogers et\nal.\\ 2014).  Assuming a similar scatter at $z\\sim10$ translates to a\nslightly larger random error on the mean $\\beta$ of 0.3.\n\n\n\\subsubsection{Selection Biases}\n\nIt is useful to consider briefly whether our mean $\\beta$ results\ncould be biased because of the selection criteria that were applied in\nsearching for $z\\sim10$ galaxies.  Such issues became an important\naspect of the debate regarding $UV$-continuum slopes at $z\\sim7$\n(e.g., Wilkins et al.\\ 2012; Dunlop et al.\\ 2012; Bouwens et\nal.\\ 2012, 2014a), and it is important that we ensure that such issues\ndo not become important again.\n\n\nIn identifying bright $z\\sim9$-10 candidates over CANDELS GOODS-S and\nGOODS-N, Oesch et al. (2014) did not consider sources with\nparticularly red H$_{f160w}-$[4.5]$>2$ colors redward of the Lyman\nbreak.  Although the H$_{f160w}-$[4.5] colour, unlike the\nH$_{f160w}-$[3.6] colour, does not directly probe the UV continuum\nslope, they are correlated and noise in the two colors is not\nindependent.\\footnote{This is particularly true at\n  H$_{f160w}-$[4.5]$>0$ where the colour is dominated by the effect of\n  dust reddening.}. A colour of H$_{f160w}-[4.5]\\sim2$ corresponds to\n$\\beta\\approx -0.1$.  Given that all of our candidates have measured\ncolors of $\\beta<-1.8$ with small uncertainties, it is unlikely that a\n$\\beta<-0.1$ selection will have an important impact on the mean\n$\\beta$ measured for the sample, since the limit is $>3\\sigma$ away\nfrom the mean $\\beta$ observed and $3\\sigma$ away from the mean\n$\\beta$ assuming no evolution from $z\\sim5$-7.\n\nIt is encouraging that Bouwens et al.\\ (2015a) selected exactly the\nsame $z\\sim10$ candidates as Oesch et al.\\ (2014), utilizing a\nslightly modified criteria (H$_{f160w}-$[3.6]$<1.6$: corresponding to\n$\\beta<-0.2$).  This further confirms that the Oesch et al. (2014)\nsample of bright sources is not affected by strong $\\beta$-dependent\nselection biases.\n\n\\subsection{Possible Evolution in the observed UV-continuum slope}\n\nIn the previous section, we presented the first determination of the\nmean $UV$-continuum slope $\\beta$ for a multi-object sample at\n$z\\sim10$.  Previously, UV continuum slopes for similarly luminous\ngalaxies could only be determined up to $z\\sim8$ (Finkelstein et\nal.\\ 2012).\n\nWith these new measurements in hand, it is interesting to look for\nevidence of evolution in the mean $\\beta$ of galaxies versus redshift.\nFigure~\\ref{fig:beta_z_M} presents the current measurements and\ncompares it against previous measurements at $z\\sim4$-7 from Bouwens\net al.\\ (2014a) and Rogers et al.\\ (2014).\n\nAs is evident from Figure~\\ref{fig:beta_z_M}, the observed UV\ncontinuum slopes of individual galaxies at $z\\sim 10$ are found to be\nmuch bluer ($\\Delta\\beta\\approx 0.4$) than those at $z\\sim 4$-7.\nThese new results are interesting, as they suggest either a gradual\nevolution in $\\beta$ towards bluer colors or no evolution over a wide\nredshift baseline.\n\nThe comparison of measurements at $z=4$-7 with $z\\sim 10$ is not\nnecessarily straightforward as the $UV$-continuum slope is measured\nover very different wavelength baselines which may introduce a\nsystematic bias.  This is explored in more detail in Appendix A where\nwe conclude that this is unlikely to be an important factor in this\ncase, but could be as large as $\\Delta\\beta\\sim 0.2$.  If we allow for\npotential $\\sim$10\\% systematics in our HST - IRAC color measurements\n(possible if the total magnitudes we estimate from the HST or\nSpitzer\/IRAC data are not quite identical), the total systematic error\nrelevant to the inferred evolution in $\\beta$ could be as large as\n$\\sim0.22$.\n\nAccounting for both random and systematic errors, we find the observed\n$\\beta$ for luminous galaxies at $z\\sim10$ is only $1\\sigma$ bluer\nthan at $z\\sim7$.  Therefore, our results are consistent with either a\nmild reddening of the $UV$-continuum slopes with cosmic time or no\nevolution at all.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=20pc]{figures\/M_beta.eps}\n\\caption{Mean value of the UV-continuum slope as a function of absolute rest-frame UV magnitude for candidate high-redshift ($z=4-10$) star forming galaxies (Bouwens et al.\\ 2014; Rogers et al.\\ 2014). The thin horizontal line denotes the intrinsic slope implicit in the empirical Meurer et al. (1999) relation and the two horizontal bands show the range of intrinsic slopes expected from the {\\sc MassiveBlack-II} hydrodynamical simulation at $z\\sim 10$ assuming both $f_{\\rm esc}=1$ (pure stellar) and $f_{\\rm esc}=0$. It is important to note that at lower redshift the {\\sc MassiveBlack-II}  simulation predicts significantly bluer intrinsic slopes (see Wilkins et al. 2013a).}\n\\label{fig:M_beta}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=20pc]{figures\/beta_z_M.eps}\n\\caption{The observed UV continuum slope as a function of redshift for candidate high-redshift ($z=4-10$) star forming galaxies. Results at $z=4-7$ are based on the bi-weight mean of galaxies with $M_{1500}\\approx -20.75$ from Bouwens et al. (2014a).}\n\\label{fig:beta_z_M}\n\\end{figure}\n\n\n\n\n\n\n\n\n\\section{Inferred Dust Attenuation}\\label{sec:dust_interp}\n\n\n\\subsection{Relating the observed UV continuum slope to dust attenuation}\n\nThe presence of dust causes the observed UV continuum slope to redden relative to the intrinsic slope. The relationship between between the observed slope $\\beta_{\\,\\rm obs}$ and the attenuation $A_{\\lambda}$ can be written as (e.g. Meurer et al. 1999, Wilkins et al. 2013a),\n\\begin{equation}\\label{eq:abeta}\nA_{\\lambda}=D_{\\lambda}\\times[\\beta_{\\,\\rm obs}-\\beta_{\\,\\rm int}],\n\\end{equation} \nwhere $\\beta_{\\rm int}$ is the intrinsic UV continuum slope, $\\beta_{\\rm obs}$ is the observed slope, and $D_{\\lambda}$ ($={\\rm d}A_{\\lambda}\/{\\rm d}\\beta$) describes the change in the attenuation as a function of the change in $\\beta$, and is sensitive to the choice of attenuation\/extinction curve. The intrinsic UV continuum slope is sensitive to a range of properties, including the star formation and metal enrichment histories, and the ionising photon escape fraction (see \\S\\ref{sec:dust_interp.intbeta}, and Wilkins et al. 2012, Wilkins et al. 2013a). \n\nBoth $\\beta_{\\rm int}$ and $D_{\\lambda}$ can be constrained empirically (e.g. Meurer et al. 1999, Heinis et al. 2013) using a combination of rest-frame UV and far-IR observations of a sample of galaxies. $D_{\\lambda}$ can be determined for any attenuation curve which extends over the rest-frame UV, and thus doesn't necessarily require far-IR observations to be constrained. \n\n\\subsection{The intrinsic UV continuum slope}\\label{sec:dust_interp.intbeta}\n\nEmpirical constraints on $\\beta_{\\rm int}$ are, at present, due to the lack of sufficiently deep far-IR\/sub-mm observations, limited to low-intermediate redshift (e.g. Heinis et al. 2013). The value of $\\beta_{\\rm int}$ at low\/intermediate redshift is unlikely to reflect that at very-high redshift as $\\beta_{\\rm int}$ is sensitive to both the age and metallicity of the stellar population, both of which are expected to decrease in typical star forming galaxies to high-redshift (e.g. Wilkins et al. 2013a).\n\nThe sensitivity of the intrinsic UV-continuum slope $\\beta_{\\rm int}$ to various properties, including the joint distribution of stellar masses, ages, and metallicities (themselves determined the recent star formation and metal enrichment histories, and initial mass function) and the presence of nebular continuum, and to a lesser extent, line emission is discussed in Wilkins et al. (2012) and Wilkins et al. (2013a). We demonstrate this sensitivity in Figure \\ref{fig:colours_ageZ_constant_z10} utilising the {\\sc Pegase.2} stellar population synthesis code (Fioc \\& Rocca-Volmerange 1997, 1999). We determine the H$_{f160w}-$[3.6] colour (our proxy for the UV continuum slope at $z\\sim 10$) as a function of the duration of previous constant star formation, for two stellar metallicities, and also assuming $f_{\\rm esc}=0$ and $f_{\\rm esc}=1$ (i.e. a pure stellar continuum). Increasing the metallicity, the strength of nebular emission (i.e. decreasing the escape fraction $f_{\\rm esc}$), and the duration of previous star formation all result in the intrinsic UV continuum slope becoming redder. \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=20pc]{figures\/colours_ageZ_constant_z10.eps}\n\\caption{The sensitivity of the H$_{f160w}$-[3.6] colour (our proxy for the UV continuum slope) to the duration of previous (constant) star formation. The blue and purple lines show the result for $Z=0.02$ and $Z=0.0004$ respectively, while the solid and dashed lines show the result assuming $f_{\\rm esc}=0$ (i.e. including nebular continuum and line emission) and $f_{\\rm esc}=1$ (i.e. pure stellar emission) respectively. The solid and hatched horizontal bands show the predictions from the {\\sc MassiveBlack-II} simulations assuming $f_{\\rm esc}=0$ and $f_{\\rm esc}=1$ respectively.}\n\\label{fig:colours_ageZ_constant_z10}\n\\end{figure}\n\n\\subsubsection{Sensitivity to the candidate redshift}\n\nIn addition to the physical properties outlined above the UV continuum slope inferred from observations will also be sensitive to the redshift of the source. This is demonstrated in Figure \\ref{fig:colours_z} where the predicted intrinsic H$_{f160w}-$[3.6] colour (our proxy for the observed UV-continuum slope) of a stellar population that has been forming stars constantly for 50 Myr is shown as a function of redshift. Three different nebular emission scenarios are shown: (i) $f_{\\rm esc}=1$ (i.e. pure stellar), (ii) $f_{\\rm esc}=0$, and (iii) $f_{\\rm esc}=1$ but with Lyman-$\\alpha$ suppressed. In all three cases, in the interval $z=9.6-10.4$, the observed colour exhibits virtually no variation. At $z<9.6$ various strong emission lines, beginning with [OII]$\\lambda 3727{\\rm\\AA}$, enter the {\\em Spitzer}\/IRAC [3.6]-band resulting in generally redder colours\\footnote{The {\\em Spitzer}\/IRAC [3.6]-band also no longer probes the rest-frame UV continuum}.\n\nAt $z>10.4$ the H$_{f160w}$-band filter overlaps with the wavelength\nof rest-frame Lyman-$\\alpha$. Assuming no Lyman-$\\alpha$ emerges\n(scenario {\\em iii}) this results in the observed H$_{f160w}$-band\nencompassing the Lyman-$\\alpha$ break resulting in a decreased flux,\nand consequently redder H$_{f160w}-$[3.6] colours. If strong\nLyman-$\\alpha$ emerges the H$_{f160w}-$[3.6] colour would rapidly\nbecome very blue before gradually becoming redder.  Spectroscopic\nfollow-up of $z\\gtrsim6.5$ galaxies strongly points towards very\nlittle Ly$\\alpha$ emission in $z\\gtrsim10.4$ galaxies (e.g., Stark et\nal.\\ 2010; Ono et al.\\ 2012; Schenker et al.\\ 2012; Pentericci et\nal.\\ 2011; Caruana et al.\\ 2012; Treu et al.\\ 2013; Finkelstein et\nal.\\ 2013; Oesch et al.\\ 2015a: but see also Zitrin et al.\\ 2015).\n\n\\subsubsection{Simulation Predictions for the Intrinsic Slope}\n\nWhile empirical constraints on the intrinsic slope $\\beta_{\\rm int}$ exist they are only available at low-redshift, and are therefore unlikely to be representative of the very-high redshift Universe. We then also employ predictions from galaxy formation models for the intrinsic slope and spectral energy distribution. Specifically, we utilise predictions from the {\\sc MassiveBlack} and {\\sc MassiveBlack-II} hydro-dynamical simulations (see Khandai et al. 2015 for a general description of the simulations, and Wilkins et al. 2013a for predictions of the intrinsic UV continuum slope). At $z=10$ these simulations predict a median intrinsic slope of $\\beta_{\\rm int}\\approx -2.54$ (assuming $f_{\\rm esc}=0$) and $\\beta_{\\rm int}\\approx -2.78$ for a pure stellar SED. In both cases the {\\sc Pegase.2} SPS model is assumed, however other commonly used models models produce similar results (Wilkins et al. 2013ab).\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=20pc]{figures\/beta_z.eps}\n\\caption{{\\em lower-panel} The sensitivity of the H$_{f160w}$-[3.6] colour (our proxy for the UV continuum slope) to the redshift for a stellar population which has been constantly forming stars for 50 Myr (with $Z_{}=0.004$, i.e. $\\approx 1\/5\\,Z_{\\odot}$). The three lines show the result of different nebular emission scenarios: ({\\em dashed}) $f_{\\rm esc}=1$ (i.e. pure stellar), ({\\em solid}) $f_{\\rm esc}=0$, and ({\\em dotted}) $f_{\\rm esc}=1$ but with Lyman-$\\alpha$ suppressed. The predictions from {\\sc MassiveBlack-II} are shown by the shaded horizontal bands. {\\em upper-panel} The redshift probability distributions of the 4 bright $z\\sim 10$ candidates.}\n\\label{fig:colours_z}\n\\end{figure}\n\n\n\\subsection{Inferred Dust Attenuation}\n\nBy combining the observed UV continuum slope $\\beta_{\\rm obs}$ with a choice of intrinsic slope $\\beta_{\\rm int}$ and dust curve we can infer the level of dust attenuation using Equation \\ref{eq:abeta}. We do this both utilising the empirical Meurer relation and by assuming the intrinsic slope predicted by the {\\sc MassiveBlack-II} simulation combined with various attenuation\/extinction curves. \n\nIn Figure \\ref{fig:M_A1500_M99} we first show the inferred UV\nattenuation assuming the empirical Meurer et al. (1999) relation. The\ncombination of very blue observed colours and the $\\beta\\approx -2.23$\nintrinsic slope implicit in the Meurer relation result in individual objects\nhaving best fit attenuations formally consistent with $A_{1500}=0$. The attenuation inferred from the bright stack is $A_{1500}=0.3\\pm 0.5$.\n\nFigure \\ref{fig:M_A1500} is similar but instead shows the attenuation when we assume the intrinsic slope predicted by the {\\sc MassiveBlack-II} simulation (assuming $f_{\\rm esc}=0.0$) along with both the Calzetti et al. (2000) and SMC (from Pei et al. 1992) dust curves. In this case all of the individual observations as well of the stack yield positive values of $A_{1500}$. If instead a higher escape fraction is assumed (which would make the intrinsic slope bluer) the inferred attenuation increases by between $0.25-0.5$ mags, depending on the choice of attenuation\/extinction curve.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=20pc]{figures\/M_A1500_M99.eps}\n\\caption{The inferred rest-frame UV attenuation $A_{1500}$ (right-hand axes) assuming the Meurer et al. (1999) relation.}\n\\label{fig:M_A1500_M99}\n\\end{figure}\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=20pc]{figures\/M_A1500.eps}\n\\caption{The inferred rest-frame UV attenuation $A_{1500}$ (right-hand axes) assuming the intrinsic slope predicted by {\\sc MassiveBlack} along with the Calzetti et al. (2000) {\\em starburst} and Pei et al. (1992) SMC dust curves.}\n\\label{fig:M_A1500}\n\\end{figure}\n\n\\subsubsection{Evolution of dust attenuation}\n\nWe now focus on the evolution of the rest-frame UV attenuation from $z\\sim 10$ to $z=4-7$. In Figure~\\ref{fig:A1500_zevo} we compare our predictions for the UV attenuation at $z\\sim 10$ to those at $z=4-7$ applying the same methodology - using both the Meurer et al. 1999 relation and by combining predictions of the intrinsic slope with the Calzetti et al. (2000) dust curve. Assuming a constant intrinsic slopes hints at a significant increase in the UV attenuation from $z\\sim 10$ to $z\\sim 7$ followed by little evolution between $z=7$ and $z=4$. Utilising the intrinsic slopes predicted by {\\sc MassiveBlack} (which vary with redshift) weakens this evolution.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=20pc]{figures\/A1500_zevo.eps}\n\\caption{The evolution of the rest-frame UV attenuation $A_{1500}$ inferred from observations of the UV continuum slope in bright $z\\in\\{4,5,6,7,10\\}$ star forming galaxies. Both the attenuation calculated using the Meurer et al. (1999) relation and combining the Calzetti et al. (2000) attenuation curve with the {\\sc MassiveBlack} predictions are shown.}\n\\label{fig:A1500_zevo}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\\section{Conclusions}\\label{sec:conclusions}\n\nHere we make use of the deep {\\em Hubble} and Spitzer observations available\nover 5 particularly bright $z\\sim10$ candidates (Oesch et al.\\ 2014;\nZheng et al.\\ 2012) to provide a first characterization of the mean\nUV-continuum slope for a multi-object sample of galaxies at\n$z\\sim10$.\n\nWe find:\\\\\n\n\\begin{itemize}\n\n\\item Combining {\\em Hubble} and {\\em Spitzer} we have measured the\n  mean UV continuum slope of star formation galaxy candidates at\n  $z\\sim 10$.  We find a mean $\\beta$ of $-2.1\\pm0.3\\pm0.2$.  We allow\n  for up to a $0.2$ error in this measurement due to systematic errors\n  that may derive from the wavelength baseline used to derive $\\beta$\n  (Appendix A) or from small systematics in the photometry.  The\n  average observed UV continuum slope of a stack of bright $z\\sim 10$\n  sources is only bluer than those at $z<8$ ($\\beta_{\\rm obs}\\approx\n  -1.7$) by $1\\sigma$.  These measurements are more robust than those\n  at $z\\sim 8$ due to the wide wavelength baseline provided by the\n  combination of the {\\em Hubble}\/WFC3 H$_{f160w}$-band and {\\em\n    Spitzer}\/IRAC [3.6]-band.  The only previous measurement of\n  $\\beta$ at $z\\sim10$ was by Oesch et al.\\ (2014) for the most\n  luminous $z\\sim10$ galaxy in their selection.\\\\\n\n\\item These slopes are redder than the intrinsic slope predicted by galaxy formation models, suggesting the existence of some dust attenuation, even at $z\\approx 10$. For the brightest candidate, GN-z10-1 (Oesch et al. 2014), we infer a rest-frame UV attenuation of $\\approx 0.9$ assuming the Calzetti et al. (2000) attenuation curve ($\\approx 0.5$ assuming an SMC extinction law), while for the stack of bright candidates we find $0.9\\pm 0.4$ assuming the Calzetti et al. (2000) law ($0.6\\pm 0.2$ assuming an SMC extinction law). \\\\\n\n\\end{itemize}\n\n\n\n\\subsection{Acknowledgements}\n\nWe acknowledge useful conversations with Dan Coe on this topic, who\nhas been attempting similar measurements on his triply imaged\n$z\\sim11$ candidate.  SMW acknowledges support from the Science and\nTechnology Facilities Council.  IL acknowledges support from the\nEuropean Research Council grant HIGHZ no. 227749 and the Netherlands\nOrganisation for Scientific Research Spinoza grant. The Dark Cosmology\nCentre is funded by the Danish National Research Foundation.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}