# The Volumetric Star Formation Law in the Almost Edge-on Galaxy NGC 4302
Revealed by ALMA

Kijeong Yim Department of Astronomy and Space Science, Chungnam National
University, Daejeon 34134, Republic of Korea  Korea Astronomy and Space
Science Institute, 776 Daedeok-daero, Yuseong-gu, Daejeon 34055, Republic of
Korea  Tony Wong Department of Astronomy, University of Illinois, 1002 West
Green Street, Urbana, IL 61801, USA  Richard J. Rand Department of Physics and
Astronomy, University of New Mexico, 1919 Lomas Blvd NE, Albuquerque, NM
87131-1156, USA

(Received 2022 June 27 ; Accepted 2022 October 7)

###### Abstract

We observe the almost edge-on ($i\sim 90\arcdeg$) galaxy NGC 4302 using ALMA
(CO) and VLA (H I) to measure the gas disk thickness for investigating the
volumetric star formation law (SFL). The recent star formation rate (SFR) is
estimated based on a linear combination of IR 24 $\micron$ and H$\alpha$
emissions. The measured scale heights of CO and H I increase significantly
with radius. Using the scale heights along with the vertically integrated
surface densities, we derive the mid-plane volume densities of the gas
($\rho_{\rm HI}$, $\rho_{\rm H_{2}}$, and $\rho_{\rm gas}$= $\rho_{\rm
HI}$+$\rho_{\rm H_{2}}$) and the SFR ($\rho_{\rm SFR}$) and compare the
volumetric SFL ($\rho_{\rm SFR}$$\propto$ $\rho_{\rm gas}$n) with the
vertically integrated SFL ($\Sigma_{\rm SFR}$$\propto$ $\Sigma_{\rm gas}$N).
We find tight power-law correlations between the SFR and the gas (H I, H2, and
the total gas) in both volume and surface densities. The power-law indices of
the total gas and H I for the volumetric SFL are noticeably smaller than the
indices for the vertically integrated SFL while the H2 indices for both cases
are similar to each other. In terms of the star formation efficiency (SFE), we
find that the molecular and total gas SFEs are roughly constant, while the
atomic SFE is clearly decreasing with radius in both cases.

Interstellar medium (847); Barred spiral galaxies (136); Star formation
(1569); Galaxy structure (622); Galaxy kinematics (602); Galaxy dynamics
(591); Radio astronomy (1338)

##  1 Introduction

A power-law correlation between the SFR ($\Sigma_{\rm SFR}$) and gas (e.g.,
$\Sigma_{\rm H_{2}}$, $\Sigma_{\rm HI}$, $\Sigma_{\rm gas}$) based on surface
densities in nearby galaxies is reported by a number of observational studies
(e.g., Wong & Blitz 2002; Bigiel et al. 2008; Schruba et al. 2011; Yim et al.
2014, 2016) following the work by Kennicutt (1998), who demonstrated the
correlation between $\Sigma_{\rm SFR}$ and $\Sigma_{\rm gas}$ (=$\Sigma_{\rm
H_{2}}$+$\Sigma_{\rm HI}$) averaged over the disks: $\Sigma_{\rm SFR}$
$\propto\Sigma_{\rm gas}^{1.4}$. However, the volume (not surface) density is
the relevant quantity for the Jeans instability and the free-fall time scale,
and indeed the original Schmidt (1959) star formation law was based on volume
density. Krumholz et al. (2012) showed that the volumetric star formation law
is well matched to observational data including Milky Way molecular clouds,
Local Group galaxies, and high-redshift galaxies. Recently, Bacchini et al.
(2019) investigated the relationship between gas and the SFR volume densities
estimated from hydrostatic equilibrium and found a strong correlation between
them with a smaller scatter compared to the vertically integrated star
formation law (Kennicutt-Schmidt law) in nearby disk galaxies. Bacchini et al.
(2020) suggested that the volumetric SFL is valid even in dwarf galaxies and
outer regions of spiral galaxies where the vertically integrated SFL appears
to break down due to H I predominance and low metallicity in the regions. In
our previous study (Yim et al., 2020), we investigated the volumetric SFL by
measuring the gas disk thickness directly from the edge-on galaxies observed
by the Combined Array for Research in Millimeter-wave Astronomy (CARMA) in
contrast to volumetric SFL studies using the inferred disk thickness from
hydrostatic equilibrium (e.g., Abramova & Zasov 2008; Bacchini et al. 2019,
2020). However, the CO disk thickness is not well-resolved by CARMA and the
angular resolution is about 3$\arcsec$, corresponding to a physical scale of
$\sim$200 pc in the edge-on galaxy sample. For this reason, we use the Atacama
Large Millimeter/submillimeter Array (ALMA), providing unprecedented
sensitivity and resolution, to resolve the CO disk structure for studying the
volumetric SFL.

NGC 4302 is an excellent target, allowing us to measure the disk thickness in
the most straightforward way since its inclination is almost $90\arcdeg$
(Heald et al. 2007; Zschaechner et al. 2015). It is located at a distance of
14 Mpc based on a local flow model by Mould et al. (2000) with $H_{0}$ = 73 km
s-1 Mpc-1. NGC 4302 has a companion galaxy NGC 4298, but the disk of NGC 4302
is not obviously distorted by the interaction (Rand 1996; Howk & Savage 1999).
On the other hand, Zschaechner et al. (2015) detected a faint H I bridge
between NGC 4302 and NGC 4298 in a few channel maps, probably due to ram
pressure stripping by the intracluster medium in the Virgo Cluster.

Figure 1: CO and H I integrated intensity maps. The northern disk is placed in
negative $x$. Contour levels are $10.0\times 2.5^{n}$ K km s-1, with n=0, 1,
2, 3, 4 for CO and $750.0\times 1.6^{n}$ K km s-1, with n=0, 1, 2, 3 for H I.
The lowest contour levels are $\sim$10$\sigma$ for CO and $\sim$5$\sigma$ for
H I. The synthesized beam is shown in the lower right corner of each panel.
The $Spitzer$ 24 $\micron$ contour levels are 0.2 $\times$ 3.4n MJy sr-1, with
n = 0, 1, 2, and 3 and the H$\alpha$ contour levels are 0.0003 $\times$ 2.5n
MJy sr-1, with n = 0, 1, and 2. The lowest contours are $\sim$10$\sigma$ for
24 $\micron$ and $\sim$3$\sigma$ for H$\alpha$. The point spread function
(PSF) FWHM is shown in the lower right corner of each panel: 5.9$\arcsec$ for
24 $\micron$ and 1.5$\arcsec$ for H$\alpha$.

##  2 Observations and Data Reduction

We observed 12CO ($J=1\rightarrow 0$) emission toward NGC 4302 using ALMA
(Cycle 7) 12m in array configurations C43-4 (6.7 hours) and C43-1 (1.9 hours)
with a twenty-five point mosaic in 2019 October 20th – 2020 March 4th. Using a
pipeline script of the Common Astronomy Software Applications (CASA) package
(McMullin et al., 2007) provided by ALMA, we calibrated separately the
visibility data in different configurations. The calibrated data were combined
using the CASA task CONCAT and imaged using the CASA task TCLEAN with Briggs
weighting (ROBUST = 0.5). A primary beam correction was applied to the cleaned
image using IMPBCOR. The combined image has an angular resolution of
1$\farcs$42 $\times$ 1$\farcs$16 (P.A. = $-39.2\arcdeg$) and a velocity
resolution of 5 km s-1. The rms noise per channel is $\sim$1.3 mJy Beam-1
($\sim$72.6 mK). Figure 1 (top) shows the integrated intensity map of the CO
cube. In order to place the redshifted side (southern disk) on the right, the
map is rotated by 90.93$\arcdeg$ based on a position angle of -0.93$\arcdeg$
that we measured from $Spitzer$ 3.6 $\micron$ image using the MIRIAD task
IMFIT. The galactic center ($\alpha$ = 12h21m42$\fs$32 and $\delta$ =
14$\arcdeg$35$\arcmin$52.23$\arcsec$) is located at $x=0$ and $z=0$ in the
map. We masked the CO cube to reduce noise in the integrated intensity map by
blanking regions below a 3$\sigma$ threshold in a cube smoothed to FWHM =
3$\arcsec$. The total flux of the integrated intensity map is $\sim$650 Jy km
s-1. The sensitivity falls to 50% at $x$-offset of $\sim$120$\arcsec$. In
order to examine the flux recovery, we compare the ALMA observations with 45 m
single-dish NRO (Nobeyama Radio Observatory) observations (Komugi et al.,
2008) in the same region over the velocity range of 990–1170 km s-1, chosen by
Komugi et al. (2008) to calculate the integrated intensity. The ALMA data are
convolved to the circular beam of 16$\arcsec$, which is the angular resolution
of the NRO observations. The integrated intensities over the same region from
990 km s-1 to 1170 km s-1are 25 K km s-1 for NRO and 16 K km s-1 for ALMA.
This indicates that the ALMA observations recover about 64% of the flux.

The H I observations were carried out with VLA in B configuration in 2009
(Zschaechner et al., 2015) and C configuration in 2005 (Chung et al., 2009).
We used the CASA package tasks FLAGCMD (mode=‘manual’) for flagging bad data,
SETJY for setting the flux density of the amplitude calibrator, BANDPASS for
the bandpass correction, GAINCAL for the gain calibration, FLUXSCALE for
bootstrapping the flux density of the gain calibrator, APPLYCAL for applying
the calibration solutions to the target, and CONCAT for combining the
separately calibrated B and C data. Finally, we cleaned the combined data
using the task CLEAN and subtracted continuum emission from the image using
the task IMCONTSUB. In the cleaning process, we used Briggs weighting with a
robustness of 0.5, which resulted in an angular resolution of 6$\farcs$56
$\times$ 5$\farcs$59 (P.A. = $-87.4\arcdeg$). The masked integrated intensity
map of the H I cube, consisting from 940 km s-1 to 1340 km s-1 with a channel
width of 20 km s-1, is shown in Figure 1 (middle). The mask was created from
the 3$\sigma$ threshold of a cube smoothed to 15$\arcsec$. The rms noise per
channel is $\sim$2.3 K and the total flux of the integrated intensity map is
28 Jy km s-1. For comparison, the integrated flux density from the Arecibo
Legacy Fast ALFA (ALFALFA) survey is 26.81 Jy km s-1 (Giovanelli et al.,
2007). The single dish flux is in good agreement with our measurement within
uncertainties.

In order to estimate the SFR, we used both 24 $\micron$ and H$\alpha$
emissions from dust and H II regions surrounding young massive stars,
respectively. For the $Spitzer$ MIPS 24 $\micron$ (Program ID 30945; PI:
Jeffrey Kenney), we downloaded the basic calibrated data (BCD) from the
Spitzer Heritage Archive and removed the residual artifacts such as latents
and gradients (MIPS Instrument
Handbook111https://irsa.ipac.caltech.edu/data/SPITZER/docs/mips/) in the BCD
via Image Reduction and Analysis Facility (IRAF) tasks. After mosaicking the
BCD images using MOPEX (Mosaicking and Point Source Extraction), we employed
the Groningen Image Processing System (GIPSY; van der Hulst et al. 1992) task
BLOT to mask out bright stars of the 24 $\micron$ image in Figure 1 (third
panel). The H$\alpha$ image in the bottom panel was observed by Rand (1996)
with the 2.1 m Kitt Peak National Observatory (KPNO). The continuum
subtraction was carried out in the image and there is no contamination by the
[N II] lines (Rand, 1996).

Figure 2: Position-velocity diagrams integrated vertically for CO (top) and H
I (middle). CO contours are $1.0\times 1.8^{n}$ K arcsec, with n=0, 1, 2, 3,
4. H I contour levels are $175.0\times 1.5^{n}$ K arcsec, with n=0, 1, 2, 3,
4. The lowest contour level is 5$\sigma$. The horizontal dotted lines indicate
the heliocentric systemic velocity of 1140 km s-1. Bottom: Rotation curves of
CO (red squares) and H I (blue circles) obtained from p-v diagrams along the
mid-plane. The rotation velocities are overlaid on the p-v diagrams as red
filled circles. The uncertainties from the correction term in Equation 1 are
shown in the upper left and right corners.

##  3 Kinematics

###  3.1 Position-Velocity Diagrams

The position-velocity (p-v) diagram allows us to obtain the rotation curve
using the envelope tracing method which uses the terminal velocity corrected
for the instrumental resolution and gas velocity dispersion (Sofue & Rubin
2001; see Section 3.3) and the radial density distribution based on the PVD
method which assumes axisymmetry (Yim et al. 2011; see Section 4.1). We
obtained the p-v diagrams (Figure 2) by integrating the data cubes over
$\pm$10$\arcsec$ for CO and $\pm$30$\arcsec$ for H I from the mid-plane. The
horizontal dotted line indicates the systemic velocity $V_{\rm sys}$ of 1140
km s-1 that we adopted based on both the consistency of the redshifted and
blueshifted rotation curves and HyperLEDA222http://leda.univ-lyon1.fr (Makarov
et al., 2014).

The p-v diagram of CO (top panel of Figure 2) shows a central elongated
feature followed by a gap, possibly caused by a bar such as in NGC 891 (Yim et
al., 2011) and NGC 4013 (Yim et al., 2020). The maximum radial velocity of the
central feature ($\sim$1280 km s-1) is lower than the almost flat velocity
($\sim$1320 km s-1) beyond $\sim$30$\arcsec$ in the redshifted side. The lower
velocity compared to the flat velocity suggests an end-on bar (e.g. NGC 4013)
while a higher velocity in the central feature suggests a side-on bar as in
NGC 891 when we are seeing the $x_{2}$ periodic orbits in the center
(Athanassoula & Bureau, 1999). Therefore, we suggest that a bar of NGC 4302 is
located close to the line of sight (i.e. end-on bar). Unlike the CO p-v
diagram, the elongated feature in the central region does not appear in the H
I p-v diagram (middle panel of Figure 2); this is perhaps because the central
region has little H I emission. The p-v diagram clearly shows that the
emission in the blueshifted side is much further extended than the emission in
the redshifted side. Chung et al. (2007) reported the extended emission as a
long one-sided H I tail and they suggest that the H I tail could be caused by
ram pressure stripping.

###  3.2 Rotation Curve

We derived the CO and H I rotation curves using the p-v diagrams along the
mid-plane through the envelope tracing method. This method calculates the
rotation velocities ($V_{\rm rot}$) based on the terminal velocities ($V_{\rm
ter}$) of the p-v diagram, the observational velocity resolution ($\sigma_{\rm
obs}$) and the velocity dispersion of gas ($\sigma_{\rm g}$):

$V_{\rm rot}=V_{\rm ter}-\sqrt{\sigma^{2}_{\rm obs}+\sigma^{2}_{\rm g}},$ (1)

where $\sigma^{2}_{\rm obs}$ for CO and H I is 2.2 km s-1 and 8.5 km s-1 based
on the channel resolutions, respectively. The adopted velocity dispersions of
CO and H I are 4 km s-1 (Wilson et al. 2011; Mogotsi et al. 2016; Marasco et
al. 2017) and 8 km s-1 (Shostak & van der Kruit 1984; Blitz & Rosolowsky
2006), respectively. The adopted values do not noticeably affect the results.
The terminal velocity is defined as the highest velocity at the 3$\sigma$
contour level on the mid-plane p-v diagram. The obtained rotation velocities
are shown as red circles overlaid on the integrated p-v diagrams in Figure 2.
The rotation curve is very uncertain near the center due to the existence of a
bar. The blueshifted side is in good agreement with the redshifted side for
both CO and H I. The rotation curves in the bottom panel of Figure 2 show
average values of the blueshifted and redshifted velocities as red open
circles for CO and blue crosses for H I. The differences between the
blueshifted and redshfited velocities are used as the error bars on the
averages. In addition, the uncertainties from the correction term
($\sqrt{\sigma^{2}_{\rm obs}+\sigma^{2}_{\rm g}}$) in Equation 1 are shown in
the upper left and right corners in the figure.

##  4 Radial Distribution

The density distributions as a function of radius are necessary to compare the
gas with the SFR for investigating the star formation law. We obtained the
radial profiles using the PVD method for the gas and the GIPSY task RADPROF
for the SFR. The CO, H I, 24 $\micron$, and H$\alpha$ maps are convolved to a
circular beam of 6.56$\arcsec$ $\times$ 6.56$\arcsec$ based on the beam size
of H I before deriving the radial profiles in order to compare and/or combine
them. When we convolve the 24 $\micron$ and H$\alpha$ maps, we use the kernels
provided by Aniano et al. (2011). The assumed original point spread functions
for 24 $\micron$ and H$\alpha$ are 5.9$\arcsec$ and 1.5$\arcsec$ (Rand, 1996),
respectively.

###  4.1 Molecular and Atomic Gas

Figure 3:  Top: radial surface density profiles of H2 (red squares), H I (blue
circles), SFR (magenta diamonds), and the total gas (black triangles). The
vertical error bars represent the standard deviation of data in a radial bin.
Bottom: SFR radial profiles from the linear combination of H$\alpha$ and 24
$\micron$ (magenta diamonds) and the radio continuum (blue circles). The
vertical error bars are the standard deviation of the average in each bin of
25$\arcsec$.
---
Figure 4: Scale heights of H I (blue circles) and CO (red squares) as
functions of radius. The open (blueshifted side) and filled (redshifted side)
circles are average values in bins of 3$\arcsec$ (CO) and 7$\arcsec$ (H I).
The lines are the linear least-squares fit to the average data points. The
shaded regions around the best fit lines show uncertainties of the best-fits
and the vertical error bars show uncertainties of the Gaussian fitting for the
scale height. The right three panels show Gaussian fits (line) to the vertical
profiles (filled circles) of CO at $x$-offsets of $-$34$\arcsec$, 56$\arcsec$,
and 78$\arcsec$.

We use the vertically integrated p-v diagrams in Figure 2 to obtain the radial
profiles of $\Sigma_{\rm HI}$ and $\Sigma_{\rm H_{2}}$ using the PVD method,
which is based on an assumption of circular rotation. Each pixel position in
the p-v diagram allows us to derive a galactic radius and the face-on surface
brightness at the radius:

$R=V_{\rm c}\left<\frac{x}{V_{\rm los}-V_{\rm sys}}\right>\quad\quad\rm{with}\
\it|V_{\rm los}-V_{\rm sys}|<V_{\rm c},$ (2)

where $V_{\rm c}$ is the assumed circular speed obtained from the rotation
curve (bottom panel of Figure 2) and $V_{\rm los}$ is the observed line-of-
sight velocity at each position $x$. The angle brackets denote the mean value
within a pixel. When we use the p-v diagram, we exclude the central regions
($|x|<40$$\arcsec$ and $|V_{\rm los}-V_{\rm sys}|<40$ km s-1) to avoid non-
circular motions and confusion caused by the line of sight blending of
emission from many different radii in the regions. Using the rotation curve,
we find a radius corresponding to a position of $x$ and $V_{\rm los}$ in the
p-v diagram. Finally, we convert the flux in the pixel into the face-on
surface brightness and obtain the surface mass densities from the surface
brightnesses using a standard Galactic value for the CO-to-H2 conversion
factor (Strong & Mattox 1996; Dame et al. 2001) and assuming optically thin H
I emission:

$\mbox{$\Sigma_{\rm H_{2}}$}\,[\mbox{M${}_{\odot}$ pc${}^{-2}$}]=3.2I_{\rm
CO}\,[\rm K\,\mbox{km~{}s${}^{-1}$}],$ (3) $\mbox{$\Sigma_{\rm
HI}$}\,[\mbox{M${}_{\odot}$ pc${}^{-2}$}]=0.0146I_{\rm HI}\,[\rm
K\,\mbox{km~{}s${}^{-1}$}].$ (4)

We average the surface mass densities within each 10$\arcsec$ bin and plot the
average values of H2 (red squares) and H I (blue circles) as a function of
radius in Figure 3 (top). The molecular and atomic gas densities include a
factor of 1.36 for the helium correction. This figure shows that the molecular
gas ($\Sigma_{\rm H_{2}}$) is centrally concentrated while the atomic gas
($\Sigma_{\rm HI}$) is almost uniformly distributed over the disk even beyond
the CO disk. The surface density of the total gas ($\Sigma_{\rm gas}$) is the
combination of the molecular and atomic gas densities. The vertical error bars
show the standard deviation of the average data within each bin. We also
estimate the uncertainty from error propagation using the rms noise in the p-v
diagram; this uncertainty is significantly smaller than the standard
deviation. The uncertainties from the center outward are 0.14–0.01 M⊙ pc-2 for
H2 and 2.15–0.04 M⊙ pc-2 for H I. The horizontal error bars indicate the
maximum and minimum radius obtained by using the angular and velocity
resolutions in the calculation of radius (Equation 2).

Since the circular speed could be affected by a bar, we do not exactly know
the rotation curve near the center. Therefore, we also use a circular speed of
180 km s-1 for $V_{\rm c}$ instead of the rotation curve to examine how the
radial profiles vary in the central regions within $\sim$40$\arcsec$, where
the rotation curve does not match to the assumed flat velocity of 180 km s-1.
The maximum difference between the surface densities using the adopted
rotation curve and the assumed flat rotation curve (180 km s-1) in the central
regions is a factor of $\sim$2 for CO and $\sim$5 for H I. However, the
surface densities using the different $V_{\rm c}$ values are consistent with
each other beyond $R\sim$40$\arcsec$.

###  4.2 Star Formation Rate

We use a linear combination of H$\alpha$ and 24 $\micron$ (Calzetti et al.
2007; Schruba et al. 2011) to estimate the recent SFR density:

$\mbox{$\Sigma_{\rm SFR}$}[\mbox{M${}_{\odot}$}\,{\rm
kpc^{-2}\,yr^{-1}}]=0.029\,I_{\rm H\alpha}+0.0025\,I_{24\,\mu m},$ (5)

where the intensities of H$\alpha$ and 24 $\micron$ in units of MJy sr-1 are
obtained from RADPROF, which provides the radial profile in a face-on disk.
The linear combination is meant to capture both obscured and unobscured star
formation. Calzetti et al. (2007) derived the calibration from a sample of 33
nearby galaxies obtained by the SINGS ($Spitzer$ Infrared Nearby Galaxies
Survey; Kennicutt et al. 2003) project.

Since the data do not include the velocity information, the PVD method is not
usable to obtain the SFR radial profile. Instead, we use the GIPSY task
RADPROF, requiring integrated intensity strips and some input parameters
(position angle, inclination, beam size, sigma, and initial estimate for the
density distribution) and providing the radial surface density distribution
based on the Lucy iterative scheme assuming axisymmetry (Lucy, 1974) developed
by Warmels (1988). The SFR surface density ($\Sigma_{\rm SFR}$) from RADPROF
is shown as magenta diamonds in Figure 3 (top). The SFR density profile is
dominated by the 24 $\micron$ term in Equation 5. The vertical error bars show
the standard deviation of data in a bin of 10$\arcsec$. The SFR density is
also centrally concentrated like the CO density, but the emission is extended
beyond the CO disk, up to the H I disk. We also derive another SFR estimate
using the VLA 1.4 GHz radio continuum data (Vollmer et al., 2013) based on the
method given by Murgia et al. (2002). We have compared the SFR radial profiles
from the linear combination of H$\alpha$ and 24 $\micron$ and the radio
continuum at the same resolution of 22$\arcsec$ in Figure 3 (bottom). Overall,
they are very consistent with each other except for the central regions within
40$\arcsec$, where the largest difference is a factor of 1.9. Vargas et al.
(2018) found that edge-on galaxies have a measurable optical depth at 24
$\micron$ and have lower flux ratios of 25/100 $\micron$ flux ratios compared
to less inclined galaxies by a factor of 1.36.

##  5 Gas Disk Thickness

NGC 4302 is the almost edge-on galaxy that allows us to measure the disk
thickness directly. First, we integrate the velocity channels over 950–980 km
s-1 for the redshifted side and 1300–1330 km s-1 for the blueshifted side to
obtain the terminal-velocity integrated intensity maps using the CO and H I
masked cubes. The radial ranges where the measurements are available are about
25$\arcsec$–80$\arcsec$ for CO and about 25$\arcsec$–140$\arcsec$ for H I.
Second, we fit a Gaussian function to vertical density profiles of the
integrated intensity map at each $x$-offset (which approaches a galactic
radius at the terminal velocity). Finally, we use the averaged Gaussian widths
in radial bins of 3$\arcsec$ for CO and 7$\arcsec$ for H I as the scale
heights, where the beam size is deconvolved. Figure 4 (left) shows the scale
heights of H I (blue circles) and CO (red squares) as open (blueshifted side)
and filled (redshifted side) circles with vertical error bars, representing
the uncertainties of the Gaussian fits. The right three panels in the figure
show the Gaussian fits to the vertical density profiles at different
$x$-offsets. The uncertainty of the CO scale height is clearly smaller than
that of the edge-on galaxy NGC 4013 ($i\sim 90\arcdeg$) using CARMA (Yim et
al., 2020) thanks to the unprecedented sensitivity of ALMA. The blue and red
lines are the linear least-squares fits to the average data points and the
best fits for the scale heights ($h$),

$h_{\rm{HI}}=(39.6\pm 0.7)\times R~{}[\rm kpc]+(78.5\pm 3.7)\quad[\rm{pc}],$
(6)

$h_{\rm H_{2}}=(18.2\pm 0.3)\times R~{}[\rm kpc]+(2.6\pm 0.9)\quad[\rm{pc}],$
(7)

will be used to derive the gas volume density. The uncertainties of the best
fits are shown as the shaded (cyan) region around the lines. Both the CO and H
I scale heights increase as a function of radius, but the H I gradient is much
steeper than the CO gradient. NGC 4302 has a significantly steeper slope in
the gas scale height compared to the other edge-on galaxies presented in the
previous studies (Yim et al. 2014, 2020); steeper by a factor of $\sim$4 for
CO and a factor of $\sim$2 for H I compared to the average CO and H I
gradients of the other galaxies.

##  6 The Volumetric Star Formation Law

We derive the mid-plane volume densities using the surface densities and the
scale heights under assumptions of Gaussian (gas) and exponential (SFR)
distributions along the vertical direction $z$. The surface densities are

$\mbox{$\Sigma_{\rm gas}$}=\int_{-\infty}^{+\infty}\rho_{\rm
gas}e^{\frac{-z^{2}}{2h^{2}}}dz,\mbox{$\Sigma_{\rm
SFR}$}=\int_{-\infty}^{+\infty}\rho_{\rm SFR}e^{\frac{-|z|}{h}}dz.$ (8)

Therefore, the volume densities are

$\rho_{\rm gas}=\frac{\mbox{$\Sigma_{\rm gas}$}}{h_{\rm
gas}\sqrt{2\pi}},\qquad\rho_{\rm SFR}=\frac{\mbox{$\Sigma_{\rm SFR}$}}{2h_{\rm
SFR}},$ (9)

where we use the CO scale height for $h_{\rm SFR}$ under the assumption that
the molecular gas should have about the same scale height as the SF. The 24
$\micron$ scale height is likely not the SF scale height because of high-
latitude dust being heated by a thinner SF layer.

---
Figure 5: Top panels: SFR volume density $\rho_{\rm SFR}$ as a function of
$\rho_{\rm HI}$ (blue circles), $\rho_{\rm H_{2}}$ (red squares), and
$\rho_{\rm gas}$ (black triangles). SFR surface density $\Sigma_{\rm SFR}$ as
a function $\Sigma_{\rm HI}$, $\Sigma_{\rm H_{2}}$, and $\Sigma_{\rm gas}$.
The gas depletion time (SFE-1) is labeled on the corresponding dashed lines.
Bottom panels: H2, H I, and the total gas SFEs as a function of radius for
volume (left) and surface (right) densities. The vertical error bars represent
the standard deviation of data in each bin.

We plot $\rho_{\rm SFR}$ against $\rho_{\rm HI}$, $\rho_{\rm H_{2}}$, and
$\rho_{\rm gas}$ and $\Sigma_{\rm SFR}$ against $\Sigma_{\rm HI}$,
$\Sigma_{\rm H_{2}}$, and $\Sigma_{\rm gas}$ in the top panels of Figure 5 to
investigate and compare the star formation laws based on volume and surface
densities. We find tight power-law correlations between the SFR and H I, H2,
and the total gas by fitting the ordinary least-squares bisector (Isobe et
al., 1990) on a log-log scale:

$\mbox{$\rho_{\rm SFR}$}\propto\rho_{\rm HI}^{2.15\pm 0.17},\rho_{\rm
SFR}\propto\rho_{\rm H_{2}}^{0.86\pm 0.07},\rho_{\rm SFR}\propto\rho_{\rm
gas}^{0.96\pm 0.07},$ (10)

and

$\mbox{$\Sigma_{\rm SFR}$}\propto\Sigma_{\rm HI}^{3.25\pm
0.22},\mbox{$\Sigma_{\rm SFR}$}\propto\Sigma_{\rm H_{2}}^{0.83\pm
0.09},\mbox{$\Sigma_{\rm SFR}$}\propto\Sigma_{\rm gas}^{1.34\pm 0.08}.$ (11)

The molecular power-law index of the volumetric SFL is comparable to that of
the vertically integrated SFL while the atomic and total gas indices of the
volume density are noticeably flatter than the indices of the surface density.
When we correct the SFR for the extinction in the central region ($R\leq$
30$\arcsec$) by a factor of 1.9 that we obtained from the comparison between
the radio continuum and the linear combination of H$\alpha$ and 24 $\micron$,
the both SFLs have a bit steeper slopes: 1.00 for the both molecular slopes,
1.09 (volume) and 1.57 (surface) for the total gas, and 2.44 (volume) and 3.82
(surface) for the atomic gas. The main reason for quite different indices
between surface and volume densities for the atomic and total gas is that the
H I scale height increases with radius rapidly, causing a big difference in
the gas surface and volume densities. In both cases, the H I power-law index
is obviously larger compared to the indices for the molecular and total gas
since the H I density is roughly constant until it decreases slowly in the
outer regions, whereas the SFR density decreases rapidly with radius. On the
other hand, a tight correlation between SFR and H I in volume and surface
densities is not shown in many previous studies (e.g., Bigiel et al. 2008;
Schruba et al. 2011; Yim et al. 2020) though in a recent study, Bacchini et
al. (2019) present a strong correlation between $\rho_{\rm SFR}$ and
$\rho_{\rm HI}$. The correlation shown in H I, unlikely from the previous
study of Yim et al. (2020), is caused by the different radial density
distributions: this study using the derived rotation curve and the previous
study assuming a flat rotation curve.

In terms of the star formation efficiency (SFE = $\rho_{\rm SFR}/\rho_{\rm
gas}$ or $\Sigma_{\rm SFR}$/$\Sigma_{\rm gas}$) in the bottom panels of Figure
5, the SFE${}_{\rm H_{2}}$ and SFEgas appear to be roughly constant over the
disk, while the atomic gas SFEHI decreases as a function of radius in both
volume and surface densities. The molecular gas depletion time (SFE${}_{\rm
H_{2}}^{-1}$) is in ranges of 1.8–6.6 Gyr (volume) and 2.2–8.3 Gyr (surface)
and the total gas depletion time is in ranges of 2.4–7.4 Gyr (volume) and
3.8–14.1 Gyr (surface).

##  7 Summary and Conclusion

Using ALMA (12CO $J=1\rightarrow 0$) and VLA (H I), we derive the molecular
and atomic gas surface densities of the edge-on galaxy NGC 4302. We also
estimate the SFR surface density from the linear combination of 24 $\micron$
and H$\alpha$ emissions. In order to obtain the mid-plane volume density, we
measure the CO and H I scale heights by fitting the Gaussian function to the
vertical profiles and find that the scale heights increase significantly as a
function of radius. Uncertainty of the CO scale height using ALMA is smaller
than the uncertainty using CARMA given by Yim et al. (2020): the gradient
uncertainty of the ALMA scale height is smaller by a factor of $\sim$7.

Using the measured scale heights and surface densities, we infer the volume
densities of CO, H I, and SFR (24 $\micron$ $\pm$ H$\alpha$) to investigate
the volumetric star formation law and compare the volumetric SFL with the
vertically integrated SFL. we find strong power-law correlations between
$\rho_{\rm HI}$, $\rho_{\rm H_{2}}$, $\rho_{\rm gas}$ and $\rho_{\rm SFR}$
with the indices of 2.15, 0.86, and 0.96, respectively. These volumetric
power-law indices for H I and the total gas (H I+H2) are quite smaller than
the indices based on the surface densities while the H2 indices for both
volume and surface densities are similar to each other. We also find that the
SFE${}_{\rm H_{2}}$ and SFEgas are roughly constant, while the SFEHI decreases
as a function of radius in both volume and surface densities.

We thank the anonymous referee for useful suggestions that improved this
paper. K.Y. was supported by Basic Science Research Program through the
National Research Foundation of Korea (NRF) funded by the Ministry of
Education (grant number NRF-2021R1I1A1A01049891). T.W. acknowledges financial
support from the University of Illinois Vermilion River Fund for Astronomical
Research. This paper makes use of the following ALMA data:
ADS/JAO.ALMA#2019.1.00562.S. ALMA is a partnership of ESO (representing its
member states), NSF (USA) and NINS (Japan), together with NRC (Canada), MOST
and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the
Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and
NAOJ. The National Radio Astronomy Observatory is a facility of the National
Science Foundation operated under cooperative agreement by Associated
Universities, Inc.

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# Mapping spaces in homotopy coherent nerves

Fabian Hebestreit  Mathematisches Institut, RFWU Bonn, Germany
<EMAIL_ADDRESS>and  Achim Krause  Mathematisches Institut, WWU
Münster, Germany<EMAIL_ADDRESS>
###### Abstract.

We give a direct proof that middle mapping spaces in coherent nerves of Kan
enriched categories have the same homotopy type as the original mapping
spaces.

###### Contents

  1. 1 Introduction
  2. 2 Recollections
  3. 3 Directed two-object simplicial sets and categories
  4. 4 Proof of the main result

##  1\. Introduction

For a quasi-category $\mathcal{C}$, there are three common definitions of
morphism complexes, often termed left, right and middle. For morphisms from
$x$ to $y$ they can be defined as the fibres of the canonical maps

$\mathcal{C}_{/y}\rightarrow\mathcal{C},\quad\mathcal{C}_{x/}\rightarrow\mathcal{C}\quad\text{and}\quad\mathrm{Ar}(\mathcal{C})\rightarrow\mathcal{C}\times\mathcal{C}$

over $x,y$ and $(x,y)$, respectively, and we shall denote them by
$\operatorname{Hom}^{\mathrm{L}}_{\mathcal{C}}$,
$\operatorname{Hom}^{\mathrm{R}}_{\mathcal{C}}$ and
$\operatorname{Hom}_{\mathcal{C}}$.

There are canonical inclusions

$\operatorname{Hom}^{\mathrm{L}}_{\mathcal{C}}(x,y)\longrightarrow\operatorname{Hom}_{\mathcal{C}}(x,y)\longleftarrow\operatorname{Hom}^{\mathrm{R}}_{\mathcal{C}}(x,y)$

that are homotopy equivalences of Kan complexes (though not usually
isomorphisms unless $\mathcal{C}$ is an ordinary category) by non-trivial
results of Joyal. Though all of these have their uses, arguably it is the
middle one which is the most natural: For example, its definition is symmetric
in source and target, it readily compares to morphism spaces of complete Segal
spaces, has the most clearly defined composition law and features in the
standard proof that fully faithful and essentially surjective functors are
equivalences.

Now, one of the most prominent constructions of quasi-categories is the
coherent nerve $\mathfrak{N}$ of Cordier and Porter, which takes as input a
category enriched in Kan complexes. There arises the problem of relating, for
such a category $\mathrm{C}$ the given morphism complexes $\mathrm{C}(x,y)$
with those of $\mathfrak{N}(\mathrm{C})$. This was done by Lurie by producing
an explicit equivalence
$\mathrm{C}(x,y)\rightarrow\operatorname{Hom}^{\mathrm{L}}_{\mathfrak{N}(\mathrm{C})}(x,y)$,
which can be combined with Joyal’s result above to obtain an equivalence
$\sigma\colon\mathrm{C}(x,y)\rightarrow\operatorname{Hom}_{\mathfrak{N}(\mathrm{C})}(x,y)$.
The main goal of the present note is to present a direct proof of this latter
result avoiding the rather less elementary theorem of Joyal.

######  Theorem.

The map

$\sigma\colon\mathrm{C}(x,y)\longrightarrow\operatorname{Hom}_{\mathfrak{N}(\mathrm{C})}(x,y)$

is a homotopy equivalence of Kan complexes for every Kan enriched category
$\mathrm{C}$ with objects $x$ and $y$.

By formal considerations there is in fact a tautological isomorphism

$\operatorname{Hom}_{\mathfrak{N}(\mathrm{C})}(x,y)\cong\operatorname{Sing}_{\mathrm{W}}(\mathrm{C}(x,y))$

for some particular
$\mathrm{W}\colon{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}}\rightarrow\mathrm{sSet}$; here and in general we denote for a
functor $T\colon\mathrm{A}\rightarrow\mathrm{B}$ by

$|\cdot|_{T}\colon\mathcal{P}(\mathrm{A})\longleftrightarrow\mathrm{B}\nobreak\mskip
6.0mu plus 1.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip
2.0mu\operatorname{Sing}_{T}$

the adjunction arising by colimit extension of $T$ (assuming of course that
$A$ is small and $B$ cocomplete).

The map $\sigma$ of the theorem is then induced by an explicit transformation
$\sigma\colon\mathrm{W}\rightarrow\Delta$, where $\Delta$ denotes the
cosimplicial object of $\mathrm{sSet}$ formed by the simplices (which
satisfies
$|\cdot|_{\Delta}=\mathrm{id}_{\mathrm{sSet}}=\operatorname{Sing}_{\Delta}$).
Our proof of the theorem boils down to the fact that the cosimplicial object
$\mathrm{W}$ is both Reedy cofibrant and termwise contractible. This is the
exact same strategy employed in the analogous result for left mapping spaces,
except that the relevant cosimplicial object is a lot less complicated in that
case. We, in fact, verify both cases simultaneously with comparatively little
explicit combinatorial manipulation.

We also provide a somewhat more detailed analysis of mapping spaces in
coherent nerves, in particular producing (to us) surprising isomorphisms

$\operatorname{Hom}^{\mathrm{L}}_{\mathfrak{N}(\mathrm{C})}(x,y)\cong\operatorname{Hom}^{\mathrm{R}}_{\mathfrak{N}(\mathrm{C})}(x,y)^{\mathrm{op}}\quad\text{and}\quad\operatorname{Hom}_{\mathfrak{N}(\mathrm{C})}(x,y)\cong\operatorname{Hom}_{\mathfrak{N}(\mathrm{C})}(x,y)^{\mathrm{op}}.$

######  Remark.

Our hope is that the proof we present here can be used in a first course on
quasi-categories, which in our experience is better served by considering
middle mapping spaces, than either the left or right variants. It was
precisely this context which originally provided the motivation for the
present note.

Our proofs are therefore designed to work with minimal input (though we are
not aware that they could be simplified much by investing results other than
the comparison of left and middle mapping spaces). We will, however, either
have to rely on the non-trivial fact that cofibrations of simplicial sets that
are also weak homotopy equivalences are in fact anodyne extensions or on the
consequence of Joyal’s lifting theorem that middle mapping spaces in quasi-
categories are Kan; this dependence could be avoided by directly checking that
$|\cdot|_{\mathrm{W}}$ preserves anodyne extensions, but we refrain from doing
so here, since we would expect at least one of these results to be available
by the time the statements we prove here become relevant in any context.

### Acknowledgements

We want to heartily thank Thomas Nikolaus and Christoph Winges for several
very useful discussions, and the participants of the lecture course
‘Introduction to higher categories’ in the winter term 19/20 at the University
of Bonn for enduring the first exposition of the material we present here.

FH is a member of the Hausdorff Center for Mathematics at the University of
Bonn, and AK of the Cluster ‘Mathematics Münster: Dynamics-Geometry-
Structure’, both funded by the German Research Foundation (DFG) under grant
nos. EXC 2047, project-ID 390685813 and EXC 2044, project-ID 390685587,
respectively. AK was further funded by the DFG through the collaborative
research centre ‘Geometry: Deformations an Rigidity’ under grant no. SFB 1442,
project-ID 427320536.

##  2\. Recollections

Following directly from the definition of the three kinds of mapping spaces we
can describe their simplices explicitly as follows: An $n$-simplex in
$\operatorname{Hom}_{\mathcal{C}}(x,y)$ consists of a map
$\Delta^{n}\times\Delta^{1}\to\mathcal{C}$, which on $\Delta^{n}\times\\{0\\}$
is the constant map to $x$, and on $\Delta^{n}\times\\{1\\}$ the constant map
to $y$. An $n$-simplex in the left mapping space
$\operatorname{Hom}^{\mathrm{L}}_{\mathcal{C}}(x,y)$ consists of an
$(n+1)$-simplex $\Delta^{n+1}\to\mathcal{C}$ which sends the initial vertex
$\\{0\\}$ to $x$, and the face $d_{0}\Delta^{n+1}$ opposite it via the
constant map to $y$. Analogously, an $n$-simplex in
$\operatorname{Hom}^{\mathrm{R}}_{\mathcal{C}}(x,y)$ consists of an
$(n+1)$-simplex $\Delta^{n+1}\to\mathcal{C}$ taking the terminal vertex
$\\{n+1\\}$ to $y$ and the face $d_{n+1}\Delta^{n+1}$ opposite it to $x$ via
the constant map.

The comparison maps among the three types of mapping spaces are then induced
by certain maps

$\Delta^{n+1}/d_{0}\Delta^{n+1}\longleftarrow\Delta^{n}\times\Delta^{1}\amalg_{\Delta^{n}\times\\{0,1\\}}\\{0,1\\}\longrightarrow\Delta^{n+1}/d_{n+1}\Delta^{n+1}.$

The other ingredient into the statements, the coherent nerve $\mathfrak{N}$,
is defined as
$\operatorname{Sing}_{\mathfrak{C}}\colon\operatorname{Cat}^{\mathrm{sSet}}\rightarrow\mathrm{sSet}$,
via a certain cosimplicial simplicially enriched category
$\mathfrak{C}\colon{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}}\rightarrow\operatorname{Cat}^{\mathrm{sSet}}$.
$\mathfrak{C}(\Delta^{n})$ has objects $\\{0,\dots,n\\}$ and morphisms from
$i$ to $j$ are given as the nerve of the poset $\\{T\subseteq[i,j]\mid i,j\in
T\\}$ ordered by inclusion, and composition is induced by union of subsets.
Note that this nerve is isomorphic to $(\Delta^{1})^{\times j-i-1}$. It is a
theorem of Cordier and Porter [CP86] that $\mathfrak{N}(\mathrm{C})$ is a
quasi-category if all mapping complexes in $\mathrm{C}$ are Kan. As usual we
will also denote the colimit extension
$|\cdot|_{\mathfrak{C}}\colon\mathrm{sSet}\rightarrow\operatorname{Cat}^{\mathrm{sSet}}$
simply by $\mathfrak{C}$. The mapping complexes in the categories
$\mathfrak{C}(X)$ are most easily described by necklaces in $X$, as expounded
in [DS11]. While one can also use this description to obtain a proof of our
main result, this route is combinatorially involved and we will rather exploit
a simplified description (valid only for very particular $X$), that we explain
in the next section.

Now to start the analysis of
$\operatorname{Hom}_{\mathfrak{N}(\mathrm{C})}(x,y)$ for Kan enriched
$\mathrm{C}$, note that an $n$-simplex in it corresponds to a map
$\Delta^{n}\times\Delta^{1}\amalg_{\Delta^{n}\times\\{0,1\\}}\\{0,1\\}\to\mathfrak{N}(\mathrm{C})$
which sends the two $0$-simplices to $x$ and $y$, respectively. Let us write
$S\Delta^{n}$ for the domain of this map, since it is precisely the usual
(unreduced) suspension; more generally we shall write $SK$ for
$K\times\Delta^{1}\amalg_{K\times\\{0,1\\}}\\{0,1\\}$. By adjunction such a
map is the same as a functor of simplicially enriched categories,
$\mathfrak{C}[S\Delta^{n}]\to\mathrm{C}$ sending the two objects $0,1$ to
$x,y$.

As we will soon see, in the source category, the endomorphisms of $0$ and $1$
are trivial (i.e. consist of just the identity regarded as a discrete
simplicial set), and the simplicial set of maps $1\to 0$ is empty. The only
interesting part of $\mathfrak{C}[S\Delta^{n}]$ is therefore the simplicial
set $\big{(}\mathfrak{C}[S\Delta^{n}]\big{)}(0,1)$, which one readily checks
to depend functorially on $n\in\Delta$.

######  2.1 Definition.

Denote the cosimplicial simplicial set
$n\mapsto\big{(}\mathfrak{C}[S\Delta^{n}]\big{)}(0,1)$ by
$\mathrm{W}\colon{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}}\rightarrow\mathrm{sSet}$.111The abbreviation $\mathrm{W}$ is
meant to indicate the word ‘Wurst’, since in particular $\mathrm{W}_{3}$ looks
like one.

$d_{1}$$d_{2}$$d_{0}$$d_{0}$$d_{1}$$d_{2}$$d_{3}$$Q_{0,3}$$Q_{1,2}$$Q_{2,1}$$Q_{3,0}$
Figure 1. $W_{2}$ and $W_{3}$ together with their faces, red areas are
collapsed to points. For the labeling of the cubes, see 3.10 and the
discussion immediately preceding it.

The data of a simplicially enriched functor
$\mathfrak{C}[S\Delta^{n}]\rightarrow\mathrm{C}$ sending $0,1$ to $x,y$ is
thus simply a map $\mathrm{W}_{n}\to\mathrm{C}(x,y)$. Since this
correspondence is natural in $n$, we in total obtain an isomorphism

$\operatorname{Hom}_{\mathfrak{N}(\mathrm{C})}(x,y)\cong\operatorname{Sing}_{\mathrm{W}}(\mathrm{C}(x,y))$

as desired. Similar considerations yield cosimplicial objects
$\mathrm{Q}^{\mathrm{L}},\mathrm{Q}^{\mathrm{R}}\colon{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}} \end{picture}}}\rightarrow\mathrm{sSet}$
such that

$\operatorname{Hom}^{\mathrm{L}}_{\mathfrak{N}(\mathrm{C})}(x,y)\cong\operatorname{Sing}_{\mathrm{Q}^{\mathrm{L}}}(\mathrm{C}(x,y))\quad\text{and}\quad\operatorname{Hom}^{\mathrm{R}}_{\mathfrak{N}(\mathrm{C})}(x,y)\cong\operatorname{Sing}_{\mathrm{Q}^{\mathrm{R}}}(\mathrm{C}(x,y)),$

where $SK$ is replaced $S^{L}(K)=(\Delta^{0}*K)\amalg_{K}\Delta^{0}$ and
$S^{R}(K)=(K*\Delta^{0})\amalg_{K}\Delta^{0}$, respectively.

The common feature of these simplicial sets, is that they are equipped with a
canonical map to $\Delta^{1}$, whose fibres are trivial. It is this feature
which allows for a fairly simple discussion of the cosimplicial objects
$\mathrm{W},\mathrm{Q}^{\mathrm{L}}$ and $\mathrm{Q}^{\mathrm{R}}$, as we will
show in the next section.

##  3\. Directed two-object simplicial sets and categories

In this section, we first explore the behavior of the adjoint pair
$\mathfrak{C}$ and $N$ on the category of simplicially enriched categories
with just two objects, and morphisms only “in one direction”, as well as a
counterpart in the world of simplicial sets. The upshot is that such
simplicial sets $X$ have an associated bisimplicial set
$\operatorname{dec}(X)$, and there is a certain bicosimplicial object
$\mathrm{Q}\colon{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}}\times{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}}\rightarrow\mathrm{sSet}$, such that

$\big{(}\mathfrak{C}(X)\big{)}(0,1)\cong|\operatorname{dec}(X)|_{\mathrm{Q}}.$

Since we will find $\mathrm{Q}_{i,0}=\mathrm{Q}^{\mathrm{L}}_{i}$ and
$\mathrm{Q}_{0,i}=\mathrm{Q}^{\mathrm{R}}_{i}$ this observation offers no
simplification concerning the analysis of left and right morphism complexes in
coherent nerves, but with $X=S\Delta^{n}$ the right hand side allows for a
simple analysis of $\mathrm{W}_{n}$, and thus middle morphism complexes, as
well.

######  3.1 Definition.

We call a simplicially enriched category $\mathrm{C}$ with two objects
$0,1\in\mathrm{C}$ a _directed two-object simplicially enriched category_ if
the following holds:

  1. (1)

It has no other objects.

  2. (2)

We have $\mathrm{C}(0,0)=\Delta^{0}$, $\mathrm{C}(1,1)=\Delta^{0}$.

  3. (3)

We have $\mathrm{C}(1,0)=\emptyset$.

Note that a directed two-object simplicially enriched category is determined
by its mapping space $\mathrm{C}(0,1)$. In particular, there is a functor
$F:\mathrm{sSet}\to\\{0,1\\}/\operatorname{Cat}^{\mathrm{sSet}}$, taking $K$
to the unique directed two-object category $\mathrm{C}(0,1)=K$. One
immediately finds:

######  3.2 Lemma.

The functor $F$ is left adjoint to the functor
$\\{0,1\\}/\operatorname{Cat}^{\mathrm{sSet}}\rightarrow\mathrm{sSet},\mathrm{C}\mapsto\mathrm{C}(0,1)$.
Furthermore, $F$ is an equivalence onto the full subcategory
$\operatorname{Cat}^{\mathrm{sSet}}_{\mathrm{dir}}\subseteq\\{0,1\\}/\operatorname{Cat}^{\mathrm{sSet}}$
spanned by the directed two-object simplicially enriched categories.

In particular, directed two-object simplicially enriched categories are closed
under colimits in $\\{0,1\\}/\operatorname{Cat}^{\mathrm{sSet}}$ and the
functor $\mathrm{sSet}\rightarrow\operatorname{Cat}^{\mathrm{sSet}}$ obtained
by composing $F$ with forgetting the marked objects preserves all colimits
over connected diagrams.

For the final claim recall that the forgetful functor of any comma category
preserves connected colimits.

######  3.3 Corollary.

The functor $\\{0,1\\}/\operatorname{Cat}^{\mathrm{sSet}}\to\mathrm{sSet}$,
$\mathrm{C}\mapsto\mathrm{C}(0,1)$, preserves colimits of diagrams of directed
two-object simplicially enriched categories.

We can make a definition similar to 3.1 on the side of simplicial sets:

######  3.4 Definition.

We call an object $K\in\partial\Delta^{1}/\mathrm{sSet}$ a _directed two-
object simplicial set_ if it can be endowed with a map $K\to\Delta^{1}$ such
that the diagram

${\partial\Delta^{1}}$${K}$${\partial\Delta^{1}}$${\Delta^{1}}$$\scriptstyle{\mathrm{id}}$

commutes and is a pullback diagram.

Note immediately, that such a map $K\rightarrow\Delta^{1}$ is unique if one
exists, and that this is the case if and only if $K$ only has two
$0$-simplices $0,1$ and every simplex $\Delta^{n}\rightarrow K$ can be
decomposed (necessarily uniquely) as
$\Delta^{n}=\Delta^{i}*\Delta^{j}\rightarrow K$, constant with value $0$ on
$\Delta^{i}$ and constant with value $1$ on $\Delta^{j}$.

######  3.5 Example.

All three of $S(K),S^{L}(K)$ and $S^{R}(K)$ are naturally directed two-object
simplicial sets.

Again the following is immediate:

######  3.6 Lemma.

The directed two-object simplicial sets form a full subcategory
$\mathrm{sSet}_{\mathrm{dir}}\subseteq\partial\Delta^{1}/\mathrm{sSet}$ closed
under all colimits and the forgetful functor
$\mathrm{sSet}_{\mathrm{dir}}\to\mathrm{sSet}$ commutes with all connected
colimits.

######  3.7 Proposition.

The functor
$\mathrm{J}:{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}}\times{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}}\to\mathrm{sSet}_{\mathrm{dir}}$,

$(i,j)\longmapsto(\Delta^{i}*\Delta^{j})\amalg_{(\Delta^{i}\amalg\Delta^{j})}(\Delta^{0}\amalg\Delta^{0})$

gives rise to an equivalence

$|\cdot|_{\mathrm{J}}\colon\mathrm{ssSet}\longrightarrow\mathrm{sSet}_{\mathrm{dir}}\nobreak\mskip
6.0mu plus 1.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip
2.0mu\operatorname{Sing}_{\mathrm{J}}.$

The functor $\operatorname{Sing}_{\mathrm{J}}$ is the functor
$\operatorname{dec}$, for decomposition, from the introduction to this
section, and we shall use that name once the proposition is established;
decoding definitions $\operatorname{dec}(X)$ is explicitly given by

$(i,j)\longmapsto\operatorname{Hom}_{\mathrm{sSet}/\Delta^{1}}(\Delta^{i}*\Delta^{j},X)$

where the structure map on the left is
$\Delta^{i}*\Delta^{j}\rightarrow\Delta^{0}*\Delta^{0}=\Delta^{1}$.

###### Proof.

We will employ the following standard criterion: Suppose given a small
category $A$, a cocomplete category $B$, and a functor $F:A\to B$ such that

  1. (1)

$F$ is fully faithful and takes values in the small objects of $B$ (i.e. those
$x\in B$ for which $\operatorname{Hom}_{B}(x,-)$ commutes with colimits), and

  2. (2)

the associated functor $\operatorname{Sing}_{F}$ is conservative.

Then the adjunction

$|\cdot|_{F}\colon\mathcal{P}(A)\longleftrightarrow B\nobreak\mskip 6.0mu plus
1.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 2.0mu\operatorname{Sing}_{F}$

consists of inverse equivalences, since by (1) the unit
$\mathrm{id}_{\mathcal{P}(A)}\Rightarrow\operatorname{Sing}_{F}\circ|\cdot|_{F}$
is a transformation between colimit preserving functors, that is an
equivalence of representables, thus an equivalence itself, and by (2) the
triangle equality then gives the same for the counit.

These assumptions are satisfied for the functor
$J:{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}}\times{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}}\to\mathrm{sSet}_{\mathrm{dir}}$: To see that it is fully
faithful observe first that

$\operatorname{Hom}_{\mathrm{sSet}_{\mathrm{dir}}}(\mathrm{J}_{i,j},\mathrm{J}_{k,l})\cong\operatorname{Hom}_{\Delta^{i}\amalg\Delta^{j}/\mathrm{sSet}}(\Delta^{i}*\Delta^{j},\mathrm{J}_{i,j})$

But then since $\Delta^{i}*\Delta^{j}$ is small in $\mathrm{sSet}$, this
receives a surjection from the subset of
$\operatorname{Hom}_{\mathrm{sSet}}(\Delta^{i}*\Delta^{j},\Delta^{k}*\Delta^{l})$
taking $\Delta^{i}$ to $\Delta^{k}$ and $\Delta^{j}$ to $\Delta^{l}$. But by
definition of $\mathrm{J}_{k,l}$ this projection identifies two maps if their
entire images lie in either $\Delta^{k}$ or $\Delta^{l}$, which is impossible,
so it is in fact an isomorphism. Since maps between simplices are determined
by their effect on vertices we find the described subset isomorphic to
$\operatorname{Hom}_{{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}} \end{picture}}}^{\times
2}}((i,j),(k,l))$ and the composite isomorphism is obviously a retract to the
effect of $\mathrm{J}$.

To see that $\mathrm{J}$ takes values in the small objects note that the
$n$-simplices of an object $K\in\mathrm{sSet}_{\mathrm{dir}}$ come with a
natural partition

($\ast$)
$K_{n}=\\{0\\}\amalg\\{1\\}\amalg\coprod_{i+j=n-1}\operatorname{Hom}_{\mathrm{sSet}_{\mathrm{dir}}}(\mathrm{J}_{i,j},K)$

witnessing the fact that every $n$-simplex $\sigma:\Delta^{n}\to K_{n}$ comes
with a partition of its vertices into
$\sigma^{-1}(0),\sigma^{-1}(1)\subseteq[n]$ and factors canonically over the
respective quotient of $\Delta^{n}$. The formula for colimits in
$\mathrm{sSet}_{\mathrm{dir}}\subseteq\\{0,1\\}/\mathrm{sSet}$ established
above then immediately implies that $\mathrm{J}_{i,j}$ is indeed small, and it
also clear from ($\ast$ ‣ 3) that $\operatorname{Sing}_{\mathrm{J}}$ is
conservative. ∎

Let us denote the exterior product
$\mathrm{sSet}\times\mathrm{sSet}\rightarrow\mathrm{ssSet}$ by $\boxtimes$.

######  3.8 Example.

It is easy to understand the images of the relevant simplicial sets under this
equivalence:

  1. (1)

There are canonical natural isomorphisms

$\operatorname{dec}(S^{L}(\Delta^{n}))\cong\Delta^{n}\boxtimes\Delta^{0}\quad\text{and}\quad\operatorname{dec}(S^{R}(\Delta^{n}))\cong\Delta^{0}\boxtimes\Delta^{n}$

since a map $\Delta^{i}*\Delta^{j}\rightarrow S^{L}(\Delta^{n})$ over
$\Delta^{1}$ is the same as a map
$\Delta^{i}*\Delta^{j}\rightarrow\Delta^{n+1}$ taking $\Delta^{i}$ to the
front face and $\Delta^{j}$ to the vertex $n+1$.

Since the functor
$S^{L}\colon\mathrm{sSet}\rightarrow\mathrm{sSet}_{\mathrm{dir}}$ is easily
checked to preserve colimits, we deduce that generally
$\operatorname{dec}(S^{L}X)\simeq X\boxtimes\Delta^{0}$ and similarly for
$S^{R}$.

  2. (2)

Similarly, if we denote by $\operatorname{Cut}^{n}$ denote the bisimplicial
set whose $i,j$-simplices are given by

$\operatorname{Cut}^{n}_{i,j}=\operatorname{Hom}_{\mathrm{sSet}}(\Delta^{i}*\Delta^{j},\Delta^{n})$

we find

$\operatorname{dec}(S\Delta^{n})\cong\operatorname{Cut}^{n}$

naturally in
$n\in{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}} \end{picture}}}$ and thus generally

$\operatorname{dec}(SX)\cong|X|_{\operatorname{Cut}}$

by regarding $\operatorname{Cut}$ as a functor
${\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}}\rightarrow\mathrm{ssSet}$; since simplices are small in
$\mathrm{sSet}$ we find $|X|_{\operatorname{Cut}}$ given in bidegree $(i,j)$
by $\operatorname{Hom}_{\mathrm{sSet}}(\Delta^{i}*\Delta^{j},X)$.

  3. (3)

Under these identifications the natural maps

$S^{L}(K)\longleftarrow SK\longrightarrow S^{R}(K)$

giving rise to the comparisons of mapping spaces correspond to the maps

$\Delta^{n}\boxtimes\Delta^{0}\longleftarrow\operatorname{Cut}^{n}\longrightarrow\Delta^{0}\boxtimes\Delta^{n}$

that simply restrict a map $\Delta^{i}*\Delta^{j}\rightarrow\Delta^{n}$ to
$\Delta^{i}$ and $\Delta^{j}$, respectively.

Next, let us observe that the coherent nerve makes these two notions of two-
object objects correspond:

######  3.9 Lemma.

The adjunction
$\mathfrak{C}\colon\mathrm{sSet}\longleftrightarrow\operatorname{Cat}^{\mathrm{sSet}}\nobreak\mskip
6.0mu plus 1.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 2.0mu\mathfrak{N}$
induces an adjunction

$\mathfrak{C}\colon\mathrm{sSet}_{\mathrm{dir}}\longleftrightarrow\operatorname{Cat}^{\mathrm{sSet}}_{\mathrm{dir}}\nobreak\mskip
6.0mu plus 1.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip
2.0mu\mathfrak{N}.$

###### Proof.

$\mathfrak{C}$ sends the discrete simplicial set $\\{0,1\\}$ to the discrete
simplicially-enriched category $\\{0,1\\}$, and $\mathfrak{N}$ does the
opposite. Thus, we obtain an adjunction between slice categories
${\\{0,1\\}/}\mathrm{sSet}\leftrightarrow{\\{0,1\\}/}\operatorname{Cat}^{\mathrm{sSet}}$,
that we claim restricts to the one from the statement. Given on the one hand
an object $K\in\mathrm{sSet}_{\mathrm{dir}}$, it is immediate for example from
the necklace description of mapping complexes in [DS11, 4.4 Corollary] that
$\mathfrak{C}(K)\in\operatorname{Cat}^{\mathrm{sSet}}_{\mathrm{dir}}$; this
also follows from 3.10 below (which is independent of the present discussion):
By 3.7 $K$ is the colimit of the objects
$\mathrm{J}_{i,j}\in\mathrm{sSet}_{\mathrm{dir}}$ over the tautological
diagram indexed by the category of (bi)simplices of $\operatorname{dec}(K)$.
By adding the initial object $\\{0,1\\}$ of $\mathrm{sSet}_{\mathrm{dir}}$ to
that diagram, we get a colimit diagram which is connected and hence preserved
by the forgetful functor to $\mathrm{sSet}$ by 3.6. By 3.10 $\mathfrak{C}$
takes all the terms of this diagram to directed two-object simplicially
enriched categories, so since
$\mathfrak{C}:\mathrm{sSet}\to\operatorname{Cat}^{\mathrm{sSet}}$ preserves
colimits, and the forgetful functor
$\\{0,1\\}/\operatorname{Cat}^{\mathrm{sSet}}\to\operatorname{Cat}^{\mathrm{sSet}}$
detects connected colimits, we see that $\mathfrak{C}$ takes
$\mathrm{sSet}_{\mathrm{dir}}$ to
$\operatorname{Cat}^{\mathrm{sSet}}_{\mathrm{dir}}$.

The analogous statement for $\mathfrak{N}$ follows by direct inspection: if
$\mathrm{C}$ is a directed two-object simplicially enriched category, then
$\mathfrak{N}(\mathrm{C})$ has two $0$-simplices $0,1$. A functor from
$\mathfrak{C}[\Delta^{n}]\to\mathrm{C}$ necessarily is an increasing map
$[n]\to[1]$ on objects, and thus induces a map
$\mathfrak{N}(\mathrm{C})\to\Delta^{1}$. Functors
$\mathfrak{C}[\Delta^{n}]\to\mathrm{C}$ sending all objects to $0$ or all
objects to $1$ are constant, so
$\partial\Delta^{1}\times_{\Delta^{1}}\mathfrak{N}(\mathrm{C})=\partial\Delta^{1}$
as required. ∎

Under the equivalences $\mathrm{sSet}_{\mathrm{dir}}\simeq\mathrm{ssSet}$ and
$\operatorname{Cat}^{\mathrm{sSet}}_{\mathrm{dir}}\simeq\mathrm{sSet}$ from
3.2 and 3.7 the functor
$\mathfrak{C}\colon\mathrm{sSet}_{\mathrm{dir}}\rightarrow\operatorname{Cat}^{\mathrm{sSet}}_{\mathrm{dir}}$
therefore corresponds to a colimit preserving functor
$\mathrm{ssSet}\rightarrow\mathrm{sSet}$. Unwinding definitions its governing
bicosimplicial object is

$(i,j)\longmapsto\big{(}\mathfrak{C}(\mathrm{J}_{i,j})\big{)}(0,1),$

which is the object $\mathrm{Q}$ from the introduction to this section.

We immediately note that by 3.8 we have
$\mathrm{W}_{n}\cong|\operatorname{Cut}^{n}|_{\mathrm{Q}}$, whereas

$\mathrm{Q}^{\mathrm{L}}_{n}\cong|\Delta^{n}\boxtimes\Delta^{0}|_{\mathrm{Q}}=\mathrm{Q}_{n,0}\quad\text{and}\quad\mathrm{Q}^{\mathrm{R}}_{n}\cong|\Delta^{0}\boxtimes\Delta^{n}|_{\mathrm{Q}}=\mathrm{Q}_{0,n},$

naturally in
$n\in{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}} \end{picture}}}$. Now the computation of
the left morphism complexes in $\mathfrak{N}(\mathrm{C})$ is achieved by
understanding $\mathrm{Q}^{\mathrm{L}}$, see [Lu09, Section 2.2.2], and the
analysis immediately generalises to give:

######  3.10 Lemma.

For $i,j\geq 0$, we have that $\mathfrak{C}[\mathrm{J}_{i,j}]$ is the two-
object directed simplicially enriched category with mapping space

$\mathrm{Q}_{i,j}=\mathrm{N}(\\{S\subseteq[i]*[j]\mid[i]\cap S,[j]\cap
S\neq\emptyset\\})/\sim$

where two chains of subsets $S_{0}\subseteq\ldots\subseteq S_{n}$ and
$S^{\prime}_{0}\subseteq\ldots\subseteq S^{\prime}_{n}$ are identified if
there exist $i_{0}\in[i]$ and $j_{0}\in[j]$ with $\\{i_{0},j_{0}\\}\subseteq
S_{0},S_{0}^{\prime}$ and
$S_{k}\cap[i_{0},j_{0}]=S_{k}^{\prime}\cap[i_{0},j_{0}]$ for all $k$.

The identification is natural for
$(i,j)\in{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}} \end{picture}}}^{\times 2}$ for the
obvious bicosimplicial structure on the right hand side.

Decoding, one finds $\mathrm{Q}_{i,j}$ a quotient of an $i+j$-dimensional cube
with a bunch of faces collapsed to lower-dimensional cubes.

###### Proof.

Let us abbreviate

$\mathrm{N}(\\{S\subseteq[i]*[j]\mid[i]\cap S,[j]\cap S\neq\emptyset\\})/\sim\
\cong\mathrm{N}(\\{S\subseteq[i+1+j]\mid 0,i+1+j\in S\\})/\sim$

to $\mathrm{N}_{i,j}$. We thus have to produce a natural bijection, for any
simplicially enriched category $\mathrm{C}$, between the set of maps
$\mathrm{J}_{i,j}\rightarrow\mathfrak{N}(\mathrm{C})$ and the set of
simplicially enriched functors $F(\mathrm{N}_{i,j})\rightarrow\mathrm{C}$.
Both sets of data decompose as disjoint unions over those maps taking the
$0$-simplices/objects to $x,y\in\mathrm{C}$. It is thus enough to produce, for
any fixed $x,y\in\mathrm{C}$, a natural bijection between the set of maps
$\mathrm{J}_{i,j}\rightarrow\mathfrak{N}(\mathrm{C})$ and the set of functors
$F(\mathrm{N}_{i,j})\rightarrow\mathrm{C}$ each sending $0,1$ to $x,y$.

Now a map $\mathrm{J}_{i,j}\rightarrow\mathfrak{N}(\mathrm{C})$ corresponds to
a map $\Delta^{i}*\Delta^{j}\to\mathfrak{N}(\mathrm{C})$, which is constant on
$\Delta^{i}$ and $\Delta^{j}$ with values $x$ and $y$, respectively. By
adjunction, this corresponds to a functor
$\mathfrak{C}[\Delta^{i+1+j}]=\mathfrak{C}[\Delta^{i}*\Delta^{j}]\to\mathrm{C}$
which takes the full simplicial subcategory spanned by the objects
$0,\ldots,i$ to the object $x$ (and its identity morphism), and that spanned
by $i+1,\ldots,i+j+1$ to $y$. We will abbreviate
$\mathfrak{C}[\Delta^{i+1+j}]$ to $\mathfrak{C}$ for the remainder of this
proof.

Note that a functor $\mathfrak{C}\rightarrow\mathrm{C}$ as just described is
completely characterised by its restriction to
$\mathfrak{C}(0,i+j+1)\to\mathrm{C}(x,y)$: On $\mathfrak{C}(k,l)$ it is
required to be constant if $k,l\leq i$ or $k,l\geq i+1$, and compatibility
with composition translates to the commutativity of the diagrams

${\mathfrak{C}(l,i+j+1)\times\mathfrak{C}(k,l)\times\mathfrak{C}(0,k)}$${\mathfrak{C}(0,i+j+1)}$${\mathfrak{C}(k,l)}$${\mathrm{C}(x,y)}$

for each $k\leq i<l$, which determine the value of the functor on
$\mathfrak{C}(k,l)$ in terms of the lower horizontal map, since the left
vertical map is surjective. What’s more, any collection of maps
$\mathfrak{C}(k,l)\to\mathrm{C}(x,y)$ fitting into such commutative diagrams
is easily checked to determine a functor $\mathfrak{C}\rightarrow\mathrm{C}$
as desired. But essentially by definition we have a pushout diagram

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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i<l}\mathfrak{C}(l,i+1+j)\times\mathfrak{C}(k,l)\times\mathfrak{C}(0,k)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
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3.0pt}}}}}}\ignorespaces{\hbox{\kern 107.66461pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
0.0pt\raise-20.16035pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.8264pt\hbox{$\scriptstyle{\mathrm{pr}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-29.82071pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
107.66461pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathfrak{C}(0,i+1+j)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
141.67679pt\raise-20.16035pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.8264pt\hbox{$\scriptstyle{\mathrm{pr}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
141.67679pt\raise-30.4874pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\coprod_{k\leq
i<l}\mathfrak{C}(k,l)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
73.91pt\raise-35.13321pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.8264pt\hbox{$\scriptstyle{\mathrm{pr}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
131.87103pt\raise-40.32071pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
131.87103pt\raise-40.32071pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathrm{N}_{i,j}}$}}}}}}}\ignorespaces}}}}\ignorespaces,$

so this data is the same as a map
$\mathrm{N}_{i,j}\rightarrow\mathrm{C}(x,y)$. We leave naturality to the
reader. ∎

######  3.11 Corollary.

The simplicial set $\mathrm{Q}_{i,j}$ is weakly contractible for all
$(i,j)\in{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}} \end{picture}}}^{\times 2}$.

###### Proof.

One readily checks that the nullhomotopy

$\mathrm{N}(\\{S\subseteq[i+1+j]\mid 0,i+1+j\in
S\\})*\Delta^{0}\rightarrow\mathrm{N}(\\{S\subseteq[i+1+j]\mid 0,i+1+j\in
S\\})$

contracting the $i+j$-cube to its terminal vertex $[i+1+j]$, factors over
$\mathrm{Q}_{i,j}$. ∎

We also observe that $\mathrm{Q}_{i,j}\cong\mathrm{Q}_{j,i}$ via the unique
isomorphism $\tau\colon[i]*[j]\cong[j]*[i]$, that flips $[i]$ and $[j]$ and
also reverses each internally, in formulae

$\tau(k_{l})=(i-k)_{r}\quad\text{and}\quad\tau(k_{r})=(j-k)_{l},$

where we have used the subscripts $l$ and $r$ to indicate the join factors.
One easily checks that this map descends to one between $\mathrm{Q}_{i,j}$ and
$\mathrm{Q}_{j,i}$. As far as the bicosimplicial structure is concerned denote
by $X^{\mathrm{fl}}$ the object obtained from
$X\colon{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}}\times{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}} \end{picture}}}\rightarrow B$ by
precomposing with the flip of the two simplex categories and by
$X^{\mathrm{lrev}}$ and $X^{\mathrm{rrev}}$ the objects obtained by
precomposing with the non-trivial automorphism of $\textstyle\Delta$ in either
the left or right factor, respectively, and write $X^{\mathrm{rev}}$ for the
composition of these two operations (and the analogous operation on
cosimplicial objects). Then one readily checks:

######  3.12 Corollary.

The construction above provides an isomorphism
$\mathrm{Q}^{\mathrm{fl}}\cong\mathrm{Q}^{\mathrm{rev}}$ of bicosimplicial
objects. In particular,
$\mathrm{Q}^{\mathrm{R}}\cong(\mathrm{Q}^{\mathrm{L}})^{\mathrm{rev}}$ and
$W^{\mathrm{rev}}\cong W$.

Let us stress again, that the superscript $\mathrm{rev}$ denotes the reversal
of the cosimplicial direction, not the simplicial one, for which we will use
the superscript $\mathrm{op}$.

###### Proof.

The first part is a simple check and the second part follows formally from the
coend formula for realisations together with the observations that

$\displaystyle\big{(}n\mapsto(\Delta^{n}\boxtimes\Delta^{0})^{\mathrm{fl,op}}\big{)}$
$\displaystyle=\big{(}n\mapsto\Delta^{0}\boxtimes(\Delta^{n})^{\mathrm{op}}\big{)}$
$\displaystyle=\big{(}n\mapsto\Delta^{0}\boxtimes\Delta^{n}\big{)}^{\mathrm{rev}}$

and similarly

$\big{(}n\mapsto(\operatorname{Cut}^{n})^{\mathrm{fl,op}}\big{)}=\big{(}n\mapsto\operatorname{Cut}^{n}\big{)}^{\mathrm{rev}},$

both of which are consequences of

$\big{(}n\mapsto(\Delta^{n})^{\mathrm{op}}\big{)}=\big{(}n\mapsto\Delta^{n}\big{)}^{\mathrm{rev}}.$

∎

Now for
$A\colon{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}} \end{picture}}}\rightarrow\mathrm{sSet}$
and $X\in\mathrm{sSet}$ one generally has

$\operatorname{Sing}_{A^{\mathrm{rev}}}(X)=\operatorname{Sing}_{A}(X)^{\mathrm{op}}$

naturally in both variables, so we find:

######  3.13 Corollary.

The construction above induces isomorphisms

$\operatorname{Hom}^{\mathrm{R}}_{\mathfrak{N}(\mathrm{C})}(x,y)\cong\operatorname{Hom}^{\mathrm{L}}_{\mathfrak{N}(\mathrm{C})}(x,y)^{\mathrm{op}}\quad\text{and}\quad\operatorname{Hom}_{\mathfrak{N}(\mathrm{C})}(x,y)\cong\operatorname{Hom}_{\mathfrak{N}(\mathrm{C})}(x,y)^{\mathrm{op}}.$

In particular, the middle mapping space in a coherent nerve really is
symmetric. We also invite the reader to check that the diagram

$\textstyle{\operatorname{Hom}^{\mathrm{R}}_{\mathfrak{N}(\mathrm{C})}(x,y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Hom}^{\mathrm{L}}_{\mathfrak{N}(\mathrm{C})}(x,y)^{\mathrm{op}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Hom}_{\mathfrak{N}(\mathrm{C})}(x,y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Hom}_{\mathfrak{N}(\mathrm{C})}(x,y)^{\mathrm{op}},}$

where the vertical maps are the canonical inclusions of the one-sided into the
middle morphism complexes, commutes.

##  4\. Proof of the main result

We start out by producing the comparison map
$\sigma\colon\mathrm{W}\rightarrow\Delta$, which in fact factors as
$\mathrm{W}\rightarrow\mathrm{Q}^{\mathrm{L}}\rightarrow\Delta$.222We think of
the map $\mathrm{W}\rightarrow\mathrm{Q}^{\mathrm{L}}$ as squeezing the Wurst
onto its Zipfel. To this end just observe that a natural transformation
$|\operatorname{Cut}^{n}|_{\mathrm{Q}}=\mathrm{W}_{n}\rightarrow\Delta^{n}$
corresponds to a natural map
$\operatorname{Cut}^{n}\rightarrow\operatorname{Sing}_{\mathrm{Q}}(\Delta^{n})$
of bisimplicial sets. But in bidegree $(i,j)$ these are given by

$\operatorname{Hom}_{\mathrm{sSet}}(\Delta^{i}*\Delta^{j},\Delta^{n})\longrightarrow\operatorname{Hom}_{\mathrm{sSet}}(\mathrm{Q}_{i,j},\Delta^{n})$

so it suffices to give maps $\mathrm{Q}_{i,j}\rightarrow\Delta^{i}*\Delta^{j}$
natural in $(i,j)\in\Delta^{\times 2}$. Since the target is a full simplex,
such maps can be specified on $0$-simplices, where we declare that
$S\subseteq[i]*[j]$ should go to $\mathrm{max}(S\cap[i])$, considered as a
$0$-simplex of $\Delta^{i}*\Delta^{j}$. The check that this is really well-
defined and natural we leave to the reader, but let us note, that it takes
values in $\Delta^{i}\subseteq\Delta^{i}*\Delta^{j}$, and therefore factors
over $\mathrm{Q}^{\mathrm{L}}$, compare 3.8 (3).

Composing with the equivalences from 3.12 yields an analogous map
$\mathrm{W}\rightarrow\mathrm{Q}^{\mathrm{R}}\rightarrow\Delta^{\mathrm{rev}}=\Delta^{\mathrm{op}}$
that takes $S\subseteq[i]*[j]$ to $\mathrm{min}(S\cap[j])$, but we shall not
make use of it.

To obtain our main result we now want to apply Reedy’s lemma (see 4.1 below)
to the transformation $\sigma\colon\mathrm{W}\rightarrow\Delta$ just
constructed. We are thus required to check that $\mathrm{W}_{n}$ is weakly
contractible for each
$n\in{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}} \end{picture}}}$ and that $\mathrm{W}$
is Reedy cofibrant, i.e. that
$|\partial\Delta^{n}|_{\mathrm{W}}\rightarrow|\Delta^{n}|_{\mathrm{W}}=\mathrm{W}_{n}$
is always a cofibration. While both statements can be checked directly, we
will deduce them by further application of Reedy’s lemma to the bicosimplicial
simplicial set $\mathrm{Q}$ using the equivalences
$\mathrm{W}_{n}=|\operatorname{Cut}^{n}|_{\mathrm{Q}}$.

Let us first state Reedy’s lemma, which we will need for both simplicial and
bisimplicial sets (it generally holds over any Reedy category, which both
$\textstyle\Delta$ and
${\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}} \end{picture}}}^{\times 2}$ are examples
of). To give a uniform statement, let us agree that by a simplex we shall mean
one of the $\Delta^{i}$ in the case of $\mathrm{sSet}$ and one of the
$\Delta^{i}\boxtimes\Delta^{j}$ in the case of $\mathrm{ssSet}$. Similarly, by
the boundary of a simplex we shall mean either $\partial\Delta^{i}$ or

$\partial(\Delta^{i}\boxtimes\Delta^{j})=(\partial(\Delta^{i-1})\boxtimes\Delta^{j})\cup(\Delta^{i}\boxtimes\partial\Delta^{j-1}),$

as appropriate. A (bi)cosimplicial simplicial set $X$ is called Reedy
cofibrant if $|\cdot|_{X}$ takes these boundary inclusions to cofibrations
(i.e. degreewise injections). Here is the version of Reedy’s result we shall
need:

######  4.1 Proposition (Reedy’s lemma).

Let $X$ be a Reedy cofibrant (bi)cosimplicial simplicial set, whose structure
maps are weak homotopy equivalences. Then $|\cdot|_{X}$ preserves degreewise
injections and weak homotopy equivalences, and in the cosimplicial case
$\operatorname{Sing}_{X}$ preserves homotopy equivalences between Kan
complexes.

If furthermore, $\eta\colon X\rightarrow Y$ is a pointwise weak equivalence
between two such (bi)cosimplicial simplicial sets, then

$\eta_{*}\colon|S|_{X}\rightarrow|S|_{Y}$

is a weak homotopy equivalence for every (bi)simplicial set $S$, and in the
cosimplicial case for any Kan complex $T$ the map

$\eta^{*}\colon\operatorname{Sing}_{Y}(T)\longrightarrow\operatorname{Sing}_{X}(T)$

is a homotopy equivalence of Kan complexes.

###### Proof sketch:.

The statement about $\eta_{*}$ is a simple skeletal induction using the
glueing lemma, and that $|\cdot|_{X}$ preserves cofibrations is similarly easy
(and both statements only use Reedy cofibrancy of $X$ and $Y$). The second
part implies that $\operatorname{Sing}_{X}$ preserves trivial fibrations.
Next, one shows that $|\cdot|_{X}$ preserves weak homotopy equivalences, for
example by considering the composite

$\operatorname{const}X_{0}\times\Delta\longrightarrow
X\times\Delta\xrightarrow{\mathrm{pr}}X,$

which gives the claim by an application of the first part, since

$S\longmapsto|S|_{\operatorname{const}X_{0}\times\Delta}=X_{0}\times S$

evidently preserves weak homotopy equivalences. Investing the fact that
cofibrations that are also weak equivalences are in fact anodyne, then shows
that $|\cdot|_{X}$ also preserves anodynes and it follows that
$\operatorname{Sing}_{X}$ preserves Kan fibrations, so in particular
$\operatorname{Sing}_{X}(T)$ is Kan again. Using this knowledge one easily
checks that the adjunction isomorphism

$\operatorname{Hom}_{\mathrm{sSet}}(|S|_{X},T)\cong\operatorname{Hom}_{\mathrm{sSet}}(S,\operatorname{Sing}_{X}(T))$

is compatible with the homotopy relation. This immediately implies that
$\operatorname{Sing}_{X}$ preserves homotopy equivalences between Kan
complexes. Finally, the diagram

$\textstyle{[S,\operatorname{Sing}_{Y}(T)]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta^{*}}$$\textstyle{[S,\operatorname{Sing}_{X}(T)]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{[|S|_{Y},T]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta_{*}}$$\textstyle{[|S|_{X},T]}$

commutes for every $S\in\mathrm{sSet}$ and by the previous parts has three
maps bijective, hence also the fourth. ∎

######  4.2 Remark.

  1. (1)

The only step in this result requiring non-trivial input (namely that trivial
cofibrations are anodyne) is that $\operatorname{Sing}_{X}(T)$ and
$\operatorname{Sing}_{Y}(T)$ are indeed Kan complexes (which, as far as we
know, is a necessary step for proving that they are even weakly equivalent).
In the case we want to apply the second part of the lemma, i.e. for
$X=\mathrm{W}$, $Y=\Delta$ and $T=\mathrm{C}(x,y)$ for some Kan enriched
category $\mathrm{C}$, this can be avoided: It is trivial for
$\operatorname{Sing}_{Y}(T)$ and follows for $\operatorname{Sing}_{X}(T)$,
since by design
$\operatorname{Sing}_{\mathrm{W}}(\mathrm{C}(x,y))=\operatorname{Hom}_{\mathfrak{N}(\mathrm{C})}(x,y)$
and morphism complexes in quasi-categories are Kan as a straight forward
consequence of Joyal’s lifting theorem [Jo02].

  2. (2)

Alternatively, by inspection of the proof above, one only needs to know that
$|\cdot|_{\mathrm{W}}$ preserves anodynes to conclude that
$\operatorname{Hom}_{\mathfrak{N}(\mathrm{C})}(x,y)$ is Kan without investing
either result. A saturation argument reduces this to the case of horn
inclusions, and here an explicit verification should be possible, but we shall
refrain from attempting it here.

  3. (3)

The statements about $\operatorname{Sing}_{X}$ are restricted to the
cosimplicial case (as opposed to the bicosimplicial one) to avoid a discussion
of various possible choices of fibrations and cofibrations in
$\mathrm{ssSet}$, as it is not required for our applications. By [GJ99,
Section IV.3.2] the argument above works equally well for example upon
replacing the words ‘anodyne’ and ‘Kan complex’ by ‘trivial Reedy cofibration’
and ‘Reedy fibrant bisimplicial set’, respectively.

We will now set out to show that $\mathrm{Q}$ satisfies the assumptions of
Reedy’s lemma. We already showed in 3.11 that $\mathrm{Q}$ is termwise
contractible, so it remains to check Reedy cofibrancy.

We will employ the following criterion:

######  4.3 Lemma.

Let $X$ by a (bi)cosimplicial simplicial set, such that for any pair of
codimension $1$ faces $K,K^{\prime}$ of a simplex $L$ the diagram

${{|}K\cap
K^{\prime}{|}_{X}}$${{|}K{|}_{X}}$${{|}K^{\prime}{|}_{X}}$${{|}L{|}_{X}}$

is cartesian. Then $X$ is Reedy cofibrant.

The point of the lemma is that all four terms in the square are actually
simplices of varying dimensions (before realisation) and therefore the diagram
itself is given by evaluating $X$ on a diagram, with no colimits occuring in
the computation of the four corners.

Note also, that the case $K=K^{\prime}$ in the condition above boils down to
precisely the statement that the boundary maps of $X$ are cofibrations (which
is automatic as the degeneracies give left inverses).

###### Proof.

It is easy to give a hands-on argument; here is an abstract one:

The bisimplicial set $\partial L\subseteq L$ is the levelwise image of the
tautological map $F=\coprod_{K}K\to L$, where the index runs through all
codimension $1$-faces of $L$. In the category of sets, images of an arbitrary
morphism $X\to Y$ can be computed by
$X\to\operatorname{Coeq}(X\times_{Y}X\rightrightarrows X)\to Y$, in the sense
that the composition is the original map, the first map surjective, and the
second one injective. Consequentially, this also holds in any
$\mathrm{Set}$-valued functor category, like $\mathrm{sSet}$. It follows that
$\partial L\rightarrow L$ agrees with the canonical map
$\operatorname{Coeq}(F\times_{L}F\rightrightarrows F)\rightarrow L$. We can
therefore compute

$\displaystyle|\partial L|_{X}$
$\displaystyle=\operatorname{Coeq}(|F\times_{L}F|_{X}\rightrightarrows|F|_{X})$
$\displaystyle=\operatorname{Coeq}(\coprod_{K,K^{\prime}}|K\times_{L}K^{\prime}|_{X}\rightrightarrows|F|_{X})$
$\displaystyle=\operatorname{Coeq}(\coprod_{K,K^{\prime}}|K|_{X}\times_{|L|_{X}}|K^{\prime}|_{X}\rightrightarrows|F|_{X})$
$\displaystyle=\operatorname{Coeq}(|F|_{X}\times_{|L|_{X}}|F|_{X}\rightrightarrows|F|_{X}),$

where we have used that pullbacks commute with colimits in the second and last
steps, and the assumption in the third one. By the considerations above, this
last expression agrees with the image of $|\partial L|_{X}$ in $|L|_{X}$,
which forces the induced map to be injective in each degree. ∎

######  4.4 Corollary.

The bicosimplicial simplicial set $\mathrm{Q}$ is Reedy cofibrant.

This statement is the only piece of combinatorics we did not manage to avoid;
alas the work has to be somewhere.

###### Proof.

Recall then that

$\mathrm{Q}_{i,j}=\mathrm{N}(\\{S\subseteq[i]*[j]\mid[i]\cap S,[j]\cap
S\neq\emptyset\\})/\sim$

where two chains of subsets $S_{0}\subseteq\ldots\subseteq S_{n}$ and
$S^{\prime}_{0}\subseteq\ldots\subseteq S^{\prime}_{n}$ are identified if
there exist $i_{0}\in[i]$ and $j_{0}\in[j]$ with $\\{i_{0},j_{0}\\}\subseteq
S_{0},S_{0}^{\prime}$ and
$S_{k}\cap[i_{0},j_{0}]=S_{k}^{\prime}\cap[i_{0},j_{0}]$ for all $k$.

We wish to apply the criterion from the previous lemma, which will require a
case distinction based on whether the two faces are taken in the same
simplicial direction or not. Consider the first case and, say, faces in the
first direction, the arguments for the second being dual. Then we need to show
that for any $0\leq l<k\leq i$ the intersection of $d_{k}\mathrm{Q}_{i-1,j}$
and $d_{l}\mathrm{Q}_{i-1,j}$ is no larger than
$d_{k}d_{l}\mathrm{Q}_{i-2,j}$. Denote by

$\pi\colon\mathrm{N}(\\{S\in[i+1+j]\mid[i]\cap S,[i+1,i+1+j]\cap
S\neq\emptyset\\})\longrightarrow\mathrm{Q}_{i,j}$

the projection. Then one easily checks that
$\pi^{-1}(d_{k}\mathrm{Q}_{i-1,j})$ consists of all those sequences
$S_{0}\subseteq\dots\subseteq S_{n}$ for which

$S_{0}\cap[k+1,i]\neq\emptyset\quad\text{or}\quad k\notin S_{n}.$

Similarly, one checks that both the preimage of the intersection and
$\pi^{-1}(d_{k}d_{l}\mathrm{Q}_{i-2,j})$ consist of those sequences with

  1. (1)

$S_{0}\cap[k+1,i]\neq\emptyset$,

  2. (2)

$S_{0}\cap[l+1,i]\neq\emptyset$, but $k\notin S_{n}$, or

  3. (3)

$k,l\notin S_{n}$.

For the mixed case, say regarding $d_{k}\mathrm{Q}_{i-1,j}$ and
$d_{l}\mathrm{Q}_{i,j-1}$ both the preimage of the intersection and the
preimage of $d_{k,l}\mathrm{Q}_{i-1,j-1}$ consist of those chains with

  1. (1)

$S_{0}\cap[k+1,i]\neq\emptyset$ and $S_{0}\cap[i+1,i+l]\neq\emptyset$,

  2. (2)

$S_{0}\cap[k+1,i]\neq\emptyset$ and $i+1+l\notin S_{n}$,

  3. (3)

$k\notin S_{n}$ and $S_{0}\cap[i+1,i+l]\neq\emptyset$, or finally

  4. (4)

$k,i+1+l\notin S_{n}$.

The claim follows. ∎

As the final ingredient we need:

######  4.5 Lemma.

The bisimplicial sets $\operatorname{Cut}^{n}$ are weakly contractible.

###### Proof.

Let
$J:{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}} \end{picture}}}^{\times
2}\rightarrow\mathrm{sSet}$ be the functor given by
$(i,j)\mapsto\Delta^{i}*\Delta^{j}$. From Reedy’s lemma we obtain weak
homotopy equivalences

$|\operatorname{Cut}^{n}|_{J}\longleftarrow|\operatorname{Cut}^{n}|_{J\times\Delta^{\boxtimes
2}}\longrightarrow|\operatorname{Cut}^{n}|_{\Delta^{\boxtimes
2}}=\mathrm{diag}(\operatorname{Cut}^{n}),$

where $\Delta^{\times 2}$ is the bicosimplicial simplicial set
$(i,j)\mapsto\Delta^{i}\times\Delta^{j}$. By the same argument as in 3.8 we
have $|\operatorname{Cut}^{n}|_{J}\cong\Delta^{n}\times\Delta^{1}$. ∎

###### Proof of the main result.

Since
$\operatorname{Hom}_{\mathfrak{N}(\mathrm{C})}(x,y)\cong\operatorname{Sing}_{\mathrm{W}}(\mathrm{C}(x,y))$
by construction of $\mathrm{W}$ an application of Reedy’s lemma to the map
$\sigma\colon\mathrm{W}\rightarrow\Delta$ constructed at the beginning of this
section will give the claim. We therefore need to check that $\mathrm{W}$ is
Reedy cofibrant and degreewise weakly contractible. To this end, recall that
$\mathrm{W}_{n}=|\operatorname{Cut}^{n}|_{\mathrm{Q}}$ naturally in
$n\in{\mathchoice{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\displaystyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\textstyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptstyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}}
\end{picture}}{\begin{picture}(0.0,0.0)\put(0.0,0.0){$\scriptscriptstyle\Delta$}
\put(0.52,0.77){\line(-13,-26){0.35}} \end{picture}}}$, or in other words
$|\cdot|_{\mathrm{W}}=|\cdot|_{\mathrm{Q}}\circ|\cdot|_{\operatorname{Cut}}$.
But both of these preserve degreewise injections,
$|\cdot|_{\operatorname{Cut}}$ by inspection, see 3.8, and
$|\cdot|_{\mathrm{Q}}$ by 4.4. Furthermore, by 3.11 another application of
Reedy’s lemma gives weak homotopy equivalences

$\mathrm{W}_{n}=|\operatorname{Cut}^{n}|_{\mathrm{Q}}\longleftarrow|\operatorname{Cut}^{n}|_{\mathrm{Q}\times\Delta^{\boxtimes
2}}\longrightarrow|\operatorname{Cut}^{n}|_{\Delta^{\boxtimes
2}}=\mathrm{diag}(\operatorname{Cut}^{n})$

the right hand term of which is contractible by the previous lemma. ∎

Note that 4.4 also implies that $\mathrm{Q}^{\mathrm{L}}$ is Reedy cofibrant
and thus we find that in fact all three maps in the diagram

$\textstyle{\mathrm{C}(x,y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\scriptstyle{\sigma}$$\textstyle{\operatorname{Hom}^{\mathrm{L}}_{\mathfrak{N}(\mathrm{C})}(x,y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Hom}_{\mathfrak{N}(\mathrm{C})}(x,y)}$

consist of homotopy equivalences. As mentioned it is a less elementary result
of Joyal that the lower horizontal map is an equivalence for any quasicategory
in place of the homotopy coherent nerve, see for example [Lu09, Section 4.2.1]
for an exposition.

## References

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  * [DS11] D.Dugger, D.Spivak, Rigidification of quasi-categories. Algebraic & Geometric Topology, 11 no. 1, 225 - 261 (2011)
  * [GJ99] P.Goerss, F.Jardine, Simplicial homotopy theory. Progress in Mathematics, 174, Birkhäuser Verlag (1999)
  * [Jo02] A.Joyal, Quasi-categories and Kan complexes. Journal of Pure and Applied Algebra, 175 no. 1-3, 207 - 222 (2002)
  * [Lu09] J.Lurie, Higher Topos Theory. Annals of Mathematics Studies, 170, Princeton University Press (2009)