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 Leonard Guetta committed Oct 22, 2020 1 \chapter{Homotopy and homology type of free 2-categories}  Leonard Guetta committed Jan 21, 2021 2 \chaptermark{Homology of free $2$-categories}  Leonard Guetta committed Oct 22, 2020 3 \section{Preliminaries: the case of free 1-categories}\label{section:prelimfreecat}  Leonard Guetta committed Oct 16, 2020 4 In this section, we review some homotopical results on free  Leonard Guetta committed Oct 14, 2020 5 ($1$-)categories that will be of great help in the sequel.  Leonard Guetta committed Jun 25, 2020 6 \begin{paragr}  Leonard Guetta committed Oct 14, 2020 7 8  A \emph{reflexive graph} $G$ consists of the data of two sets $G_0$ and $G_1$ together with  Leonard Guetta committed Oct 15, 2020 9  \begin{itemize}[label=-]  Leonard Guetta committed Jul 02, 2020 10  \item a source'' map $\src : G_1 \to G_0$,  Leonard Guetta committed Oct 14, 2020 11 12 13  \item a target'' map $\trgt : G_1 \to G_0$, \item a unit'' map $1_{(-)} : G_0 \to G_1$, \end{itemize}  Leonard Guetta committed Jul 02, 2020 14  such that for every $x \in G_0$,  Leonard Guetta committed Jun 25, 2020 15  $ Leonard Guetta committed Oct 14, 2020 16 17 18 19 20 21 22 23 24 25 26  \src(1_{x}) = \trgt (1_{x}) = x.$ The vocabulary of categories is used: elements of $G_0$ are \emph{objects} or \emph{$0$-cells}, elements of $G_1$ are \emph{arrows} or \emph{$1$-cells}, arrows of the form $1_{x}$ with $x$ an object are \emph{units}, etc. A \emph{morphism of reflexive graphs} $f : G \to G'$ consists of maps $f_0 : G_0 \to G'_0$ and $f_1 : G_1 \to G'_1$ that commute with sources, targets and units in an obvious sense. This defines the category $\Rgrph$ of reflexive graphs. Later we will make use of monomorphisms in the category $\Rgrph$; they are the morphisms $f : G \to G'$ that are injective on objects and on arrows, i.e. such that $f_0 : G_0 \to G_0'$ and $f_1 : G_1 \to G'_1$ are injective.  Leonard Guetta committed Jul 10, 2020 27 28  There is a underlying reflexive graph'' functor  Leonard Guetta committed Jun 25, 2020 29  $ Leonard Guetta committed Oct 14, 2020 30  U : \Cat \to \Rgrph,  Leonard Guetta committed Jun 25, 2020 31 32 33 $ which has a left adjoint $ Leonard Guetta committed Oct 14, 2020 34  L : \Rgrph \to \Cat.  Leonard Guetta committed Jun 25, 2020 35 $  Leonard Guetta committed Oct 14, 2020 36 37  For a reflexive graph $G$, the objects of $L(G)$ are exactly the objects of $G$ and an arrow $f$ of $L(G)$ is a chain  Leonard Guetta committed Jun 25, 2020 38  $ Leonard Guetta committed Oct 14, 2020 39 40 41 42  \begin{tikzcd} X_0 \ar[r,"f_1"]& X_1\ar[r,"f_2"]& X_2 \ar[r]& \cdots \ar[r] &X_{n-1} \ar[r,"f_n"]& X_{n} \end{tikzcd}  Leonard Guetta committed Jun 25, 2020 43 $  Leonard Guetta committed Oct 14, 2020 44 45 46 47  of arrows of $G$, such that \emph{none} of the $f_k$ are units. The integer $n$ is referred to as the \emph{length} of $f$ and is denoted by $\ell(f)$. Composition is given by concatenation of chains. \end{paragr}  Leonard Guetta committed Jun 25, 2020 48 \begin{lemma}  Leonard Guetta committed Oct 14, 2020 49 50  A category $C$ is free in the sense of \ref{def:freeoocat} if and only if there exists a reflexive graph $G$ such that  Leonard Guetta committed Jun 25, 2020 51  $ Leonard Guetta committed Oct 14, 2020 52  C \simeq L(G).  Leonard Guetta committed Jun 25, 2020 53 54 55 $ \end{lemma} \begin{proof}  Leonard Guetta committed Oct 14, 2020 56 57 58  If $C$ is free, consider the reflexive graph $G$ such that $G_0 = C_0$ and $G_1$ is the subset of $C_1$ whose elements are either generating $1$-cells of $C$ or units. It is straightforward to check that $C\simeq L(G)$.  Leonard Guetta committed Jun 25, 2020 59   Leonard Guetta committed Oct 14, 2020 60 61 62  Conversely, if $C \simeq L(G)$ for some reflexive graph $G$, then the description of the arrows of $L(G)$ given in the previous paragraph shows that $C$ is free and that its set of generating $1$-cells is (isomorphic to) the  Leonard Guetta committed Dec 27, 2020 63  set of non unital $1$-cells of $G$.  Leonard Guetta committed Jun 25, 2020 64 65 \end{proof} \begin{remark}  Leonard Guetta committed Oct 14, 2020 66 67 68 69 70 71  In other words, a category is free on a graph if and only if it is free on a reflexive graph. The difference between these two notions is at the level of morphisms: there are more morphisms of reflexive graphs because (generating) $1$\nbd{}cells may be sent to units. Hence, for a morphism of reflexive graphs $f : G \to G'$, the induced functor $L(f)$ is not necessarily rigid in the sense of Definition \ref{def:rigidmorphism}.  Leonard Guetta committed Jun 25, 2020 72 73 \end{remark} \begin{paragr}  Leonard Guetta committed Oct 25, 2020 74 75 76  There is another important description of the category $\Rgrph$. Write $\Delta_{\leq 1}$ for the full subcategory of $\Delta$ spanned by $[0]$ and $[1]$. The category $\Rgrph$ is nothing but $\Psh{\Delta_{\leq 1}}$, the  Leonard Guetta committed Oct 14, 2020 77 78 79  category of pre-sheaves on $\Delta_{\leq 1}$. In particular, the canonical inclusion $i : \Delta_{\leq 1} \rightarrow \Delta$ induces by pre-composition a functor  Leonard Guetta committed Jun 26, 2020 80  $ Leonard Guetta committed Oct 14, 2020 81  i^* : \Psh{\Delta} \to \Rgrph,  Leonard Guetta committed Jun 26, 2020 82 83 84 $ which, by the usual technique of Kan extensions, has a left adjoint $ Leonard Guetta committed Oct 14, 2020 85  i_! : \Rgrph \to \Psh{\Delta}.  Leonard Guetta committed Jun 26, 2020 86 $  Leonard Guetta committed Oct 14, 2020 87 88  For a graph $G$, the simplicial set $i_!(G)$ has $G_0$ as its set of $0$-simplices, $G_1$ as its set of $1$-simplices and all $k$-simplices are  Leonard Guetta committed Dec 27, 2020 89  degenerate for $k>1$. For future reference, we put here the following lemma.  Leonard Guetta committed Jun 26, 2020 90 \end{paragr}  Leonard Guetta committed Jun 26, 2020 91 \begin{lemma}\label{lemma:monopreserved}  Leonard Guetta committed Dec 27, 2020 92  The functor $i_! : \Rgrph \to \Psh{\Delta}$ preserves monomorphisms.  Leonard Guetta committed Jun 26, 2020 93 94 \end{lemma} \begin{proof}  Leonard Guetta committed Oct 25, 2020 95  What we need to show is that, given a morphism of simplicial sets  Leonard Guetta committed Jul 10, 2020 96  $ Leonard Guetta committed Oct 14, 2020 97  f : X \to Y,  Leonard Guetta committed Jul 10, 2020 98 $  Leonard Guetta committed Oct 14, 2020 99  if $f_0 : X_0 \to Y_0$ and $f_1 : X_1 \to Y_1$ are monomorphisms and if all  Leonard Guetta committed Dec 27, 2020 100  $n$\nbd{}simplices of $X$ are degenerate for $n\geq 2$, then $f$ is a  Leonard Guetta committed Oct 14, 2020 101  monomorphism. A proof of this assertion is contained in \cite[Paragraph  Leonard Guetta committed Nov 04, 2020 102  3.4]{gabriel1967calculus}. The key argument is the Eilenberg--Zilber Lemma  Leonard Guetta committed Oct 14, 2020 103  (Proposition 3.1 of op. cit.).  Leonard Guetta committed Jun 26, 2020 104 105 \end{proof} \begin{paragr}  Leonard Guetta committed Oct 14, 2020 106 107 108 109  Let us denote by $N : \Psh{\Delta} \to \Cat$ (instead of $N_1$ as in Paragraph \ref{paragr:nerve}) the usual nerve of categories and by $c : \Cat \to \Psh{\Delta}$ its left adjoint. Recall that for a (small) category $C$, an $n$-simplex of $N(C)$ is a chain  Leonard Guetta committed Jun 26, 2020 110  $ Leonard Guetta committed Oct 14, 2020 111 112 113 114  \begin{tikzcd} X_0 \ar[r,"f_1"]& X_1\ar[r,"f_2"]& X_2 \ar[r]& \cdots \ar[r] &X_{n-1} \ar[r,"f_n"]& X_{n} \end{tikzcd}  Leonard Guetta committed Jun 26, 2020 115 $  Leonard Guetta committed Dec 27, 2020 116  of arrows of $C$. Such an $n$-simplex is degenerate if and only if at least  Leonard Guetta committed Oct 14, 2020 117 118  one of the $f_k$ is a unit. It is straightforward to check that the composite of  Leonard Guetta committed Jun 26, 2020 119  $ Leonard Guetta committed Oct 14, 2020 120  \Cat \overset{N}{\rightarrow} \Psh{\Delta} \overset{i^*}{\rightarrow} \Rgrph  Leonard Guetta committed Jun 26, 2020 121 $  Leonard Guetta committed Oct 14, 2020 122 123  is nothing but the forgetful functor $U : \Cat \to \Rgrph$. Thus, the functor $L : \Rgrph \to \Cat$ is (isomorphic to) the composite of  Leonard Guetta committed Jun 26, 2020 124  $ Leonard Guetta committed Oct 14, 2020 125 126  \Rgrph \overset{i_!}{\rightarrow} \Psh{\Delta} \overset{c}{\rightarrow} \Cat.  Leonard Guetta committed Jun 26, 2020 127 $  Leonard Guetta committed Jun 26, 2020 128   Leonard Guetta committed Dec 27, 2020 129  We now review a construction due to Dwyer and Kan  Leonard Guetta committed Oct 25, 2020 130 131 132  (\cite{dwyer1980simplicial}). Let $G$ be a reflexive graph. For every $k\geq 1$, we define the simplicial set $N^k(G)$ as the sub-simplicial set of $N(L(G))$ whose $n$-simplices are chains  Leonard Guetta committed Jul 02, 2020 133  $ Leonard Guetta committed Oct 14, 2020 134 135 136 137  \begin{tikzcd} X_0 \ar[r,"f_1"]& X_1\ar[r,"f_2"]& X_2 \ar[r]& \cdots \ar[r] &X_{n-1} \ar[r,"f_n"]& X_{n} \end{tikzcd}  Leonard Guetta committed Jun 26, 2020 138 $  Leonard Guetta committed Jul 11, 2020 139 140  of arrows of $L(G)$ such that $ Leonard Guetta committed Dec 27, 2020 141  \sum_{1 \leq i \leq n}\ell(f_i) \leq k.  Leonard Guetta committed Jul 11, 2020 142 $ In particular, we have  Leonard Guetta committed Jun 26, 2020 143  $ Leonard Guetta committed Oct 14, 2020 144  N^1(G)=i_!(G)  Leonard Guetta committed Jun 26, 2020 145 146 147 $ and the transfinite composition of $ Leonard Guetta committed Oct 14, 2020 148 149  i_!(G) = N^1(G) \hookrightarrow N^2(G) \hookrightarrow \cdots \hookrightarrow N^{k}(G) \hookrightarrow N^{k+1}(G) \hookrightarrow \cdots  Leonard Guetta committed Jun 26, 2020 150 151 152 $ is easily seen to be the map $ Leonard Guetta committed Oct 14, 2020 153  \eta_{i_!(G)} : i_!(G) \to Nci_!(G),  Leonard Guetta committed Jun 26, 2020 154 $  Leonard Guetta committed Oct 14, 2020 155  where $\eta$ is the unit of the adjunction $c \dashv N$.  Leonard Guetta committed Jun 26, 2020 156 \end{paragr}  Leonard Guetta committed Nov 04, 2020 157 \begin{lemma}[Dwyer--Kan]\label{lemma:dwyerkan}  Leonard Guetta committed Oct 26, 2020 158  For every $k\geq 1$, the canonical inclusion map  Leonard Guetta committed Jun 26, 2020 159  $ Leonard Guetta committed Oct 14, 2020 160  N^{k}(G) \to N^{k+1}(G)  Leonard Guetta committed Jun 26, 2020 161 $  Leonard Guetta committed Jul 10, 2020 162  is a trivial cofibration of simplicial sets.  Leonard Guetta committed Jun 26, 2020 163 164 \end{lemma} \begin{proof}  Leonard Guetta committed Oct 14, 2020 165 166 167  Let $A_{k+1}=\mathrm{Im}(\partial_0)\cup\mathrm{Im}(\partial_{k+1})$ be the union of the first and last face of the standard $(k+1)$-simplex $\Delta_{k+1}$. Notice that the canonical inclusion  Leonard Guetta committed Jul 10, 2020 168  $ Leonard Guetta committed Oct 14, 2020 169  A_{k+1} \hookrightarrow \Delta_{k+1}  Leonard Guetta committed Jul 11, 2020 170 171 172 $ is a trivial cofibration. Let $I_{k+1}$ be the set of chains $ Leonard Guetta committed Oct 14, 2020 173 174 175 176  \begin{tikzcd} f = X_0 \ar[r,"f_1"]& X_1\ar[r,"f_2"]& X_2 \ar[r]& \cdots \ar[r] &X_{k-1} \ar[r,"f_k"]& X_{k}\ar[r,"f_{k+1}"]& X_{k+1} \end{tikzcd}  Leonard Guetta committed Jul 11, 2020 177 178 179 $ of arrows of $L(G)$ such that for every $1 \leq i \leq k+1$ $ Leonard Guetta committed Oct 14, 2020 180  \ell(f_i)=1,  Leonard Guetta committed Jul 10, 2020 181 $  Leonard Guetta committed Oct 14, 2020 182 183  i.e.\ each $f_i$ is a non-unit arrow of $G$. For every $f \in I_{k+1}$, we define a morphism $\varphi_f : A_{k+1} \to N^{k}(G)$ in the following fashion:  Leonard Guetta committed Jul 11, 2020 184  \begin{itemize}  Leonard Guetta committed Oct 14, 2020 185 186  \item[-]$\varphi_{f}\vert_{\mathrm{Im}(\partial_0)}$ is the $k$-simplex of $N^{k}(G)$  Leonard Guetta committed Jul 11, 2020 187  $ Leonard Guetta committed Oct 14, 2020 188 189 190  \begin{tikzcd} X_1\ar[r,"f_2"]& X_2 \ar[r]& \cdots \ar[r] &X_{k} \ar[r,"f_{k+1}"]& X_{k+1},  Leonard Guetta committed Jul 11, 2020 191 192  \end{tikzcd}$  Leonard Guetta committed Oct 14, 2020 193 194 195 196 197 198  \item[-] $\varphi_{f}\vert_{\mathrm{Im}(\partial_{k+1})}$ is the $k$-simplex of $N^{k}(G)$ $\begin{tikzcd} X_0 \ar[r,"f_1"]& X_1\ar[r,"f_2"]& X_2 \ar[r]& \cdots \ar[r] &X_{k-1} \ar[r,"f_k"]& X_{k}.  Leonard Guetta committed Jul 11, 2020 199 200 201  \end{tikzcd}$ \end{itemize}  Leonard Guetta committed Oct 25, 2020 202  All in all, we have a cocartesian square  Leonard Guetta committed Jul 10, 2020 203  $ Leonard Guetta committed Oct 14, 2020 204 205 206  \begin{tikzcd} \displaystyle \coprod_{f \in I_{k+1}}A_{k+1} \ar[d] \ar[r,"(\varphi_f)_f"] & N^{k}(G)\ar[d] \\ \displaystyle \coprod_{f \in I_{k+1}}\Delta_{k+1} \ar[r] & N^{k+1}(G),  Leonard Guetta committed Oct 25, 2020 207  \ar[from=1-1,to=2-2,phantom,"\ulcorner",very near end]  Leonard Guetta committed Oct 14, 2020 208  \end{tikzcd}  Leonard Guetta committed Jul 10, 2020 209 $  Leonard Guetta committed Jul 11, 2020 210  which proves that the right vertical arrow is a trivial cofibration.  Leonard Guetta committed Jun 26, 2020 211 \end{proof}  Leonard Guetta committed Oct 09, 2020 212 From this lemma, we deduce the following proposition.  Leonard Guetta committed Jun 26, 2020 213 214 215 \begin{proposition} Let $G$ be a reflexive graph. The map $ Leonard Guetta committed Oct 14, 2020 216  \eta_{i_!(G)} : i_!(G) \to Nci_!(G),  Leonard Guetta committed Jun 26, 2020 217 $  Leonard Guetta committed Oct 14, 2020 218 219  where $\eta$ is the unit of the adjunction $c \dashv N$, is a trivial cofibration of simplicial sets.  Leonard Guetta committed Jul 02, 2020 220 \end{proposition}  Leonard Guetta committed Jun 26, 2020 221 \begin{proof}  Leonard Guetta committed Oct 14, 2020 222 223  This follows from the fact that trivial cofibrations are stable by transfinite composition.  Leonard Guetta committed Jun 26, 2020 224 225 226 227 228 \end{proof} From the previous proposition, we deduce the following very useful corollary. \begin{corollary}\label{cor:hmtpysquaregraph} Let $ Leonard Guetta committed Oct 14, 2020 229 230 231 232  \begin{tikzcd} A \ar[d,"\alpha"] \ar[r,"\beta"] &B \ar[d,"\delta"] \\ C \ar[r,"\gamma"]& D \end{tikzcd}  Leonard Guetta committed Jun 26, 2020 233 $  Leonard Guetta committed Oct 14, 2020 234  be a cocartesian square in $\Rgrph$. If either $\alpha$ or $\beta$ is a  Leonard Guetta committed Jan 06, 2021 235  monomorphism, then the induced square of $\Cat$  Leonard Guetta committed Jun 26, 2020 236  $ Leonard Guetta committed Oct 14, 2020 237 238 239 240  \begin{tikzcd} L(A) \ar[d,"L(\alpha)"] \ar[r,"L(\beta)"]& L(B) \ar[d,"L(\delta)"] \\ L(C) \ar[r,"L(\gamma)"]& L(D) \end{tikzcd}  Leonard Guetta committed Jun 26, 2020 241 $  Leonard Guetta committed Jan 06, 2021 242  is a Thomason homotopy cocartesian.  Leonard Guetta committed Jun 26, 2020 243 244 245 246 \end{corollary} \begin{proof} Since the nerve $N$ induces an equivalence of op-prederivators $ Leonard Guetta committed Oct 14, 2020 247  \Ho(\Cat^{\Th}) \to \Ho(\Psh{\Delta}),  Leonard Guetta committed Jun 26, 2020 248 249 250 $ it suffices to prove that the induced square of simplicial sets $ Leonard Guetta committed Oct 14, 2020 251 252 253 254  \begin{tikzcd} NL(A) \ar[d,"NL(\alpha)"] \ar[r,"NL(\beta)"]& NL(B) \ar[d,"NL(\delta)"] \\ NL(C) \ar[r,"NL(\gamma)"]& NL(D) \end{tikzcd}  Leonard Guetta committed Jun 26, 2020 255 $  Leonard Guetta committed Oct 14, 2020 256 257 258 259 260 261 262 263  is homotopy cocartesian. But, since $L \simeq c\circ i_!$, it follows from Lemma \ref{cor:hmtpysquaregraph} that this last square is weakly equivalent to the square of simplicial sets $\begin{tikzcd} i_!(A) \ar[d,"i_!(\alpha)"] \ar[r,"i_!(\beta)"] &i_!(B) \ar[d,"i_!(\delta)"] \\ i_!(C) \ar[r,"i_!(\gamma)"]& i_!(D). \end{tikzcd}  Leonard Guetta committed Jun 26, 2020 264 $  Leonard Guetta committed Oct 25, 2020 265 266 267 268 269  This square is cocartesian because $i_!$ is a left adjoint. Since $i_!$ preserves monomorphisms (Lemma \ref{lemma:monopreserved}), the result follows from the fact that the monomorphisms are the cofibrations of the standard Quillen model structure on simplicial sets and from Lemma \ref{lemma:hmtpycocartesianreedy}.  Leonard Guetta committed Jun 26, 2020 270 \end{proof}  Leonard Guetta committed Jul 08, 2020 271 \begin{paragr}  Leonard Guetta committed Oct 25, 2020 272 273  By working a little more, we obtain the more general result stated in the proposition below. Let us say that a morphism of reflexive  274  graphs $\alpha : A \to B$ is \emph{quasi-injective on arrows} when  Leonard Guetta committed Oct 25, 2020 275  for all $f$ and $g$ arrows of $A$, if  Leonard Guetta committed Oct 14, 2020 276 277 278 279  $\alpha(f)=\alpha(g),$ then either $f=g$ or $f$ and $g$ are both units. In other words, $\alpha$  Leonard Guetta committed Oct 25, 2020 280  never sends a non-unit arrow to a unit arrow and $\alpha$ never identifies two  Leonard Guetta committed Oct 14, 2020 281 282 283 284  non-unit arrows. It follows that if $\alpha$ is quasi-injective on arrows and injective on objects, then it is also injective on arrows and hence, a monomorphism of $\Rgrph$. \end{paragr}  Leonard Guetta committed Jul 09, 2020 285 \begin{proposition}\label{prop:hmtpysquaregraphbetter}  Leonard Guetta committed Jun 28, 2020 286 287  Let $ Leonard Guetta committed Oct 14, 2020 288 289 290  \begin{tikzcd} A \ar[d,"\alpha"] \ar[r,"\beta"] &B \ar[d,"\delta"] \\ C \ar[r,"\gamma"]& D  Leonard Guetta committed Oct 25, 2020 291  \ar[from=1-1,to=2-2,phantom,"\ulcorner",very near end]  Leonard Guetta committed Oct 14, 2020 292  \end{tikzcd}  Leonard Guetta committed Jun 28, 2020 293 $  Leonard Guetta committed Oct 14, 2020 294 295  be a cocartesian square in $\Rgrph$. Suppose that the following two conditions are satisfied  Leonard Guetta committed Jul 02, 2020 296  \begin{enumerate}[label=\alph*)]  Leonard Guetta committed Jun 28, 2020 297  \item Either $\alpha$ or $\beta$ is injective on objects.  Leonard Guetta committed Oct 14, 2020 298  \item Either $\alpha$ or $\beta$ is quasi-injective on arrows.  Leonard Guetta committed Jul 02, 2020 299  \end{enumerate}  Leonard Guetta committed Jan 05, 2021 300  Then, the induced square of $\Cat$  Leonard Guetta committed Jun 28, 2020 301  $ Leonard Guetta committed Oct 14, 2020 302 303 304 305  \begin{tikzcd} L(A) \ar[d,"L(\alpha)"] \ar[r,"L(\beta)"] &L(B) \ar[d,"L(\delta)"] \\ L(C) \ar[r,"L(\gamma)"] &L(D) \end{tikzcd}  Leonard Guetta committed Jun 28, 2020 306 $  Leonard Guetta committed Jan 05, 2021 307  is Thomason homotopy cocartesian square.  Leonard Guetta committed Jun 28, 2020 308 309 \end{proposition} \begin{proof}  Leonard Guetta committed Oct 14, 2020 310 311 312 313  The case where $\alpha$ or $\beta$ is both injective on objects and quasi-injective on arrows is Corollary \ref{cor:hmtpysquaregraph}. Hence, we only have to treat the case when $\alpha$ is injective on objects and $\beta$ is quasi-injective on arrows; the remaining case being symmetric.  Leonard Guetta committed Jul 02, 2020 314   315  Let use denote by $E$ the set of objects of $B$ that are in the image of  Leonard Guetta committed Jan 04, 2021 316  $\beta$. We consider this set as well as the set $A_0$ of objects of $A$ as discrete reflexive graphs, i.e.\ reflexive graphs  Leonard Guetta committed Dec 27, 2020 317  with no non-unit arrows. Now, let $G$ be the reflexive graph defined by the  Leonard Guetta committed Oct 14, 2020 318  following cocartesian square  Leonard Guetta committed Jul 02, 2020 319  $ Leonard Guetta committed Oct 14, 2020 320  \begin{tikzcd}  Leonard Guetta committed Jan 04, 2021 321  A_0\ar[r] \ar[d] & E \ar[d]\\  Leonard Guetta committed Oct 14, 2020 322 323  A \ar[r] & G, \ar[from=1-1,to=2-2,phantom,very near end,"\ulcorner"] \end{tikzcd}  Leonard Guetta committed Jul 02, 2020 324 $  Leonard Guetta committed Jan 04, 2021 325 326  where the morphism $A_0 \to A$ is the canonical inclusion, and the morphism $A_0 \to E$ is induced by the restriction of $\beta$ on objects. In other words, $G$ is  Leonard Guetta committed Oct 14, 2020 327 328  obtained from $A$ by collapsing the objects that are identified through $\beta$. It admits the following explicit description: $G_0$ is (isomorphic  Leonard Guetta committed Dec 27, 2020 329  to) $E$ and the set of non-unit arrows of $G$ is (isomorphic to) the set of  Leonard Guetta committed Jan 07, 2021 330 331  non-unit arrows of $A$; the source (resp.\ target) of a non-unit arrow $f$ of $G$ is the source (resp.\ target) of $\beta(f)$. This completely describes $G$.  Leonard Guetta committed Oct 14, 2020 332 333  % Notice also for later reference that the morphism $\coprod_{x \in E}F_x % \to A$ is a monomorphism, i.e. injective on objects and arrows.  Leonard Guetta committed Jul 09, 2020 334 335 336  Now, we have the following solid arrow commutative diagram $ Leonard Guetta committed Oct 14, 2020 337  \begin{tikzcd}  Leonard Guetta committed Jan 04, 2021 338  A_0 \ar[r] \ar[d] & E \ar[ddr,bend left]\ar[d]&\\  Leonard Guetta committed Oct 14, 2020 339 340 341  A \ar[drr,bend right,"\beta"'] \ar[r] & G \ar[dr, dotted]&\\ &&B, \ar[from=1-1,to=2-2,phantom,very near end,"\ulcorner"] \end{tikzcd}  Leonard Guetta committed Jul 09, 2020 342 $  Leonard Guetta committed Oct 14, 2020 343  where the arrow $E \to B$ is the canonical inclusion. Hence, by universal  Leonard Guetta committed Dec 27, 2020 344  property, the dotted arrow exists and makes the whole diagram commute. A  Leonard Guetta committed Oct 14, 2020 345 346  thorough verification easily shows that the morphism $G \to B$ is a monomorphism of $\Rgrph$.  Leonard Guetta committed Jul 09, 2020 347   Leonard Guetta committed Dec 27, 2020 348 349  By forming successive cocartesian squares and combining with the square obtained earlier, we obtain a diagram of three cocartesian squares:  Leonard Guetta committed Oct 14, 2020 350 351  $\begin{tikzcd}[row sep = large]  Leonard Guetta committed Jan 04, 2021 352  A_0\ar[r] \ar[d] & E \ar[d]&\\  Leonard Guetta committed Oct 14, 2020 353 354 355 356 357 358 359 360  A \ar[d,"\alpha"] \ar[r] & G \ar[d] \ar[r] & B \ar[d,"\delta"]\\ C \ar[r] & H \ar[r] & D. \ar[from=1-1,to=2-2,phantom,"\ulcorner" very near end,"\text{\textcircled{\tiny \textbf{1}}}" near start, description] \ar[from=2-1,to=3-2,phantom,"\ulcorner" very near end,"\text{\textcircled{\tiny \textbf{2}}}", description] \ar[from=2-2,to=3-3,phantom,"\ulcorner" very near end,"\text{\textcircled{\tiny \textbf{3}}}", description] \end{tikzcd}  Leonard Guetta committed Jul 09, 2020 361 $  Leonard Guetta committed Oct 14, 2020 362 363 364 365 366 367 368  What we want to prove is that the image by the functor $L$ of the pasting of squares \textcircled{\tiny \textbf{2}} and \textcircled{\tiny \textbf{3}} is homotopy cocartesian. Since the morphism $G \to B$ is a monomorphism, we deduce from Corollary \ref{cor:hmtpysquaregraph} that the image by the functor $L$ of square \textcircled{\tiny \textbf{3}} is homotopy cocartesian. Hence, in virtue of Lemma \ref{lemma:pastinghmtpycocartesian}, all we have to show is that the image by $L$ of square \textcircled{\tiny \textbf{2}} is homotopy  Leonard Guetta committed Jan 04, 2021 369  cocartesian. On the other hand, the morphisms  Leonard Guetta committed Oct 14, 2020 370  $ Leonard Guetta committed Jan 04, 2021 371  A_0 \to A  Leonard Guetta committed Oct 14, 2020 372 $  Leonard Guetta committed Jan 04, 2021 373 374 375 376 377  and $A_0 \to C$ are monomorphisms and thus, using Corollary  378 379 380  \ref{cor:hmtpysquaregraph}, we deduce that the image by $L$ of square \textcircled{\tiny \textbf{1}} and of the pasting of squares \textcircled{\tiny \textbf{1}} and \textcircled{\tiny \textbf{2}}  Leonard Guetta committed Jan 04, 2021 381  are homotopy cocartesian. By Lemma \ref{lemma:pastinghmtpycocartesian} again, this proves that the image by $L$ of  382  square \textcircled{\tiny \textbf{2}} is homotopy cocartesian.  Leonard Guetta committed Jun 28, 2020 383 \end{proof}  Leonard Guetta committed Oct 14, 2020 384 385 We now apply Corollary \ref{cor:hmtpysquaregraph} and Proposition \ref{prop:hmtpysquaregraphbetter} to a few examples.  Leonard Guetta committed Oct 15, 2020 386 387 \begin{example}[Identifying two objects]\label{example:identifyingobjects} Let $C$ be a free category, $A$ and $B$ be two objects of $C$ with $A\neq B$ and let $C'$ be  Leonard Guetta committed Dec 27, 2020 388  the category obtained from $C$ by identifying $A$ and $B$, i.e.\ defined by  Leonard Guetta committed Oct 15, 2020 389 390 391 392 393  the following cocartesian square $\begin{tikzcd} \sS_0 \ar[d] \ar[r,"{\langle A,B \rangle}"] & C \ar[d] \\ \sD_0 \ar[r] & C'.  394  \ar[from=1-1,to=2-2,phantom,"\ulcorner",very near end]  Leonard Guetta committed Oct 15, 2020 395 396 397 398  \end{tikzcd}$ Then, this square is Thomason homotopy cocartesian. Indeed, it is obviously the image by the functor $L$ of a cocartesian square of $\Rgrph$ and the top  Leonard Guetta committed Oct 25, 2020 399  morphism is a monomorphism. Hence, we can apply Corollary \ref{cor:hmtpysquaregraph}.  Leonard Guetta committed Oct 15, 2020 400  \end{example}  Leonard Guetta committed Jun 26, 2020 401 \begin{example}[Adding a generator]  Leonard Guetta committed Oct 25, 2020 402  Let $C$ be a free category, $A$ and $B$ two objects of $C$ (possibly equal)  Leonard Guetta committed Oct 14, 2020 403  and let $C'$ be the category obtained from $C$ by adding a generator $A \to  Leonard Guetta committed Dec 27, 2020 404  B$, i.e.\ defined by the following cocartesian square:  Leonard Guetta committed Jun 26, 2020 405  $ Leonard Guetta committed Oct 14, 2020 406  \begin{tikzcd}  Leonard Guetta committed Oct 15, 2020 407  \sS_0 \ar[d,"i_1"] \ar[r,"{\langle A, B \rangle}"] & C \ar[d] \\  Leonard Guetta committed Oct 14, 2020 408  \sD_1 \ar[r] & C'.  409  \ar[from=1-1,to=2-2,phantom,"\ulcorner",very near end]  Leonard Guetta committed Oct 14, 2020 410  \end{tikzcd}  Leonard Guetta committed Jun 26, 2020 411 $  Leonard Guetta committed Jan 05, 2021 412  Then, this square is Thomason homotopy cocartesian. Indeed, it obviously is the image of a square of $\Rgrph$ by  413 414  the functor $L$ and the morphism $i_1 : \sS_0 \to \sD_1$ comes from a monomorphism of $\Rgrph$. Hence, we can apply Corollary  Leonard Guetta committed Oct 14, 2020 415  \ref{cor:hmtpysquaregraph}.  Leonard Guetta committed Jun 26, 2020 416 417 \end{example} \begin{remark}  Leonard Guetta committed Jan 05, 2021 418 419 420  Since $i_1 : \sS_0 \to \sD_1$ is a folk cofibration% , since a Thomason homotopy % cocartesian square in $\Cat$ is also so in $\oo\Cat$ and since every free category is obtained by recursively adding generators  Leonard Guetta committed Oct 14, 2020 421  starting from a set of objects (seen as a $0$-category), the previous example  Leonard Guetta committed Jan 05, 2021 422  yields another proof that \emph{free} (1\nbd{})categories are \good{} (which we  Leonard Guetta committed Oct 14, 2020 423  already knew since we have seen that \emph{all} (1-)categories are \good{}).  Leonard Guetta committed Jun 26, 2020 424 \end{remark}  Leonard Guetta committed Oct 09, 2020 425 \begin{example}[Identifying two generators]  Leonard Guetta committed Oct 25, 2020 426  Let $C$ be a free category and let $f,g : A \to B$ be parallel generating arrows of  Leonard Guetta committed Oct 14, 2020 427  $C$ such that $f\neq g$. Now consider the category $C'$ obtained from $C$ by  Leonard Guetta committed Dec 27, 2020 428  identifying'' $f$ and $g$, i.e. defined by the following cocartesian  Leonard Guetta committed Oct 14, 2020 429  square  Leonard Guetta committed Jun 27, 2020 430  $ Leonard Guetta committed Oct 14, 2020 431 432 433  \begin{tikzcd} \sS_1\ar[d] \ar[r,"{\langle f, g \rangle}"] &C \ar[d] \\ \sD_1 \ar[r] & C',  434  \ar[from=1-1,to=2-2,phantom,"\ulcorner",very near end]  Leonard Guetta committed Oct 14, 2020 435  \end{tikzcd}  Leonard Guetta committed Jun 27, 2020 436 $  Leonard Guetta committed Oct 14, 2020 437 438  where the morphism $\sS_1 \to \sD_1$ is the one that sends the two generating arrows of $\sS_1$ to the unique generating arrow of $\sD_1$. Then this square  Leonard Guetta committed Jan 05, 2021 439  is Thomason homotopy cocartesian.  Leonard Guetta committed Oct 14, 2020 440 441 442 443 444 445 446  Indeed, it is the image by the functor $L$ of a cocartesian square in $\Rgrph$, the morphism $\sS_1 \to \sD_1$ is injective on objects and the morphism $\sS_1 \to C$ is quasi-injective on arrows. Hence, we can apply Proposition \ref{prop:hmtpysquaregraphbetter}. Note that since we did \emph{not} suppose that $A\neq B$, the top morphism of the previous square is not necessarily a monomorphism and we cannot always apply Corollary \ref{cor:hmtpysquaregraph}.  Leonard Guetta committed Jun 27, 2020 447 \end{example}  Leonard Guetta committed Oct 08, 2020 448 \begin{example}[Killing a generator]\label{example:killinggenerator}  Leonard Guetta committed Dec 27, 2020 449  Let $C$ be a free category and let $f : A \to B$ be one of its generating arrows  Leonard Guetta committed Oct 14, 2020 450  such that $A \neq B$. Now consider the category $C'$ obtained from $C$ by  Leonard Guetta committed Dec 27, 2020 451  killing'' $f$, i.e. defined by the following cocartesian square:  Leonard Guetta committed Jun 27, 2020 452  $ Leonard Guetta committed Oct 14, 2020 453 454 455  \begin{tikzcd} \sD_1 \ar[d] \ar[r,"\langle f \rangle"] & C \ar[d] \\ \sD_0 \ar[r] & C'.  456  \ar[from=1-1,to=2-2,phantom,"\ulcorner",very near end]  Leonard Guetta committed Oct 14, 2020 457  \end{tikzcd}  Leonard Guetta committed Jun 27, 2020 458 $  459 460 461 462 463  Then, this above square is Thomason homotopy cocartesian. Indeed, it obviously is the image of a cocartesian square in $\Rgrph$ by the functor $L$ and since the source and target of $f$ are different, the top map comes from a monomorphism of $\Rgrph$. Hence, we can apply Corollary \ref{cor:hmtpysquaregraph}.  Leonard Guetta committed Jun 27, 2020 464 465 \end{example} \begin{remark}  Leonard Guetta committed Oct 14, 2020 466 467 468  Note that in the previous example, we see that it was useful to consider the category of reflexive graphs and not only the category of graphs because the map $\sD_1 \to \sD_0$ does not come from a morphism in the category of graphs.  Leonard Guetta committed Jun 26, 2020 469   Leonard Guetta committed Oct 14, 2020 470  Note also that the hypothesis that $A\neq B$ was fundamental in the previous  Leonard Guetta committed Oct 25, 2020 471  example as for $A=B$ the square is \emph{not} Thomason homotopy cocartesian.  Leonard Guetta committed Oct 14, 2020 472 \end{remark}  Leonard Guetta committed Jul 11, 2020 473   Leonard Guetta committed Oct 09, 2020 474 \section{Preliminaries: bisimplicial sets}  Leonard Guetta committed Jul 11, 2020 475 \begin{paragr}  Leonard Guetta committed Oct 14, 2020 476 477 478 479 480 481  A \emph{bisimplicial set} is a presheaf over the category $\Delta \times \Delta$, $X : \Delta^{\op} \times \Delta^{\op} \to \Set.$ In a similar fashion as for simplicial sets (\ref{paragr:simpset}), for $n,m  482  \geq 0$, we use the notations  Leonard Guetta committed Oct 14, 2020 483 484 485 486 487 488 489 490 491 492  \begin{align*} X_{n,m} &:= X([n],[m]) \\ \partial_i^h &:=X(\delta^i,\mathrm{id}) : X_{n+1,m} \to X_{n,m}\\ \partial_j^v &:=X(\mathrm{id},\delta^j) : X_{n,m+1} \to X_{n,m}\\ s_i^h &:=X(\sigma^i,\mathrm{id}): X_{n,m} \to X_{n+1,m}\\ s_j^v&:=X(\mathrm{id},\sigma^j) : X_{n,m} \to X_{n,m+1}. \end{align*} The maps $\partial_i^h$ and $s_i^h$ will be referred to as the \emph{horizontal} face and degeneracy operators; and $\partial_i^v$ and $s_i^v$ as the \emph{vertical} face and degeneracy operators.  Leonard Guetta committed Jul 21, 2020 493   Leonard Guetta committed Oct 14, 2020 494  Note that for every $n\geq 0$, we have simplicial sets  Leonard Guetta committed Jul 14, 2020 495  \begin{align*}  Leonard Guetta committed Oct 14, 2020 496 497 498 499 500 501 502 503 504  X_{\bullet,n} : \Delta^{\op} &\to \Set \\ [k] &\mapsto X_{k,n} \end{align*} and \begin{align*} X_{n,\bullet} : \Delta^{\op} &\to \Set \\ [k] &\mapsto X_{n,k}. \end{align*} The category of bisimplicial sets is denoted by $\Psh{\Delta\times\Delta}$.  Leonard Guetta committed Jul 11, 2020 505   Leonard Guetta committed Oct 14, 2020 506 507  \iffalse Moreover for every $n \geq 0$, if we fix the first variable to $n$, we obtain a simplicial set  Leonard Guetta committed Jul 11, 2020 508  \begin{align*}  Leonard Guetta committed Jul 21, 2020 509  X_{n,\bullet} : \Delta^{\op} &\to \Set \\  Leonard Guetta committed Jul 11, 2020 510 511 512 513  [m] &\mapsto X_{n,m}. \end{align*} Similarly, if we fix the second variable to $n$, we obtain a simplicial \begin{align*}  Leonard Guetta committed Jul 21, 2020 514  X_{\bullet,n} : \Delta^{\op} &\to \Set \\  Leonard Guetta committed Jul 11, 2020 515 516  [m] &\mapsto X_{m,n}. \end{align*}  Leonard Guetta committed Oct 09, 2020 517  The correspondences  Leonard Guetta committed Jul 11, 2020 518  $ Leonard Guetta committed Oct 14, 2020 519  n \mapsto X_{n,\bullet} \,\text{ and }\, n\mapsto X_{\bullet,n}  Leonard Guetta committed Jul 11, 2020 520 $  Leonard Guetta committed Oct 14, 2020 521 522  actually define functors $\Delta \to \Psh{\Delta}$. They correspond to the two currying'' operations  Leonard Guetta committed Jul 11, 2020 523  $ Leonard Guetta committed Oct 14, 2020 524  \Psh{\Delta\times\Delta} \to \underline{\Hom}(\Delta^{\op},\Psh{\Delta}),  Leonard Guetta committed Jul 11, 2020 525 $  Leonard Guetta committed Oct 14, 2020 526 527 528  which are isomorphisms of categories. In other words, the category of bisimplicial sets can be identified with the category of functors $\underline{\Hom}(\Delta^{\op},\Psh{\Delta})$ in two canonical ways. \fi  Leonard Guetta committed Jul 11, 2020 529 530 \end{paragr} \begin{paragr}  Leonard Guetta committed Jul 13, 2020 531 532 533 534 535 536 537  The functor \begin{align*} \delta : \Delta &\to \Delta\times\Delta \\ [n] &\mapsto ([n],[n]) \end{align*} induces by pre-composition a functor $ Leonard Guetta committed Oct 14, 2020 538 539 540 541 542 543 544  \delta^* : \Psh{\Delta\times\Delta} \to \Psh{\Delta}.$ By the usual calculus of Kan extensions, $\delta^*$ admits a left adjoint $\delta_!$ and a right adjoint $\delta_*$ $\delta_! \dashv \delta^* \dashv \delta_*.$  545  We say that a morphism $f : X \to Y$ of bisimplicial sets is a \emph{diagonal  Leonard Guetta committed Oct 16, 2020 546  weak equivalence} (resp.\ \emph{diagonal fibration}) when $\delta^*(f)$ is a  547  weak equivalence (resp.\ fibration) of simplicial sets. By  Leonard Guetta committed Oct 14, 2020 548 549 550 551 552 553 554 555  definition, $\delta^*$ induces a morphism of op-prederivators $\overline{\delta^*} : \Ho(\Psh{\Delta\times \Delta}^{\mathrm{diag}}) \to \Ho(\Psh{\Delta}).$ Recall from \cite[Proposition 1.2]{moerdijk1989bisimplicial} that the category of bisimplicial sets can be equipped with a model structure whose weak equivalences are the diagonal weak equivalences and whose fibrations are the  Leonard Guetta committed Oct 25, 2020 556  diagonal fibrations. We shall refer to this model  Leonard Guetta committed Oct 14, 2020 557  structure as the \emph{diagonal model structure}.  Leonard Guetta committed Jul 13, 2020 558 \end{paragr}  Leonard Guetta committed Jul 22, 2020 559 \begin{proposition}\label{prop:diageqderivator}  Leonard Guetta committed Oct 14, 2020 560 561  Consider that $\Psh{\Delta\times\Delta}$ is equipped with the diagonal model structure. Then, the adjunction  Leonard Guetta committed Jul 13, 2020 562  $ Leonard Guetta committed Oct 14, 2020 563 564 565  \begin{tikzcd} \delta_! : \Psh{\Delta} \ar[r,shift left] & \Psh{\Delta\times\Delta} \ar[l,shift left]: \delta^*,  Leonard Guetta committed Jul 13, 2020 566 567 568 569 570  \end{tikzcd}$ is a Quillen equivalence. \end{proposition} \begin{proof}  Leonard Guetta committed Oct 05, 2020 571 572 573 574  By definition $\delta^*$ preserves weak equivalences and fibrations and thus, the adjunction is a Quillen adjunction. The fact that $\delta^*$ induces an equivalence at the level of homotopy categories is \cite[Proposition 1.2]{moerdijk1989bisimplicial}.  Leonard Guetta committed Jul 13, 2020 575 \end{proof}  Leonard Guetta committed Oct 05, 2020 576 \begin{paragr}  Leonard Guetta committed Oct 14, 2020 577 578 579 580 581 582  In particular, the morphism of op-prederivators $\overline{\delta^*} : \Ho(\Psh{\Delta\times \Delta}^{\mathrm{diag}}) \to \Ho(\Psh{\Delta})$ is actually an equivalence of op-prederivators.  Leonard Guetta committed Oct 05, 2020 583 \end{paragr}  Leonard Guetta committed Oct 14, 2020 584 Diagonal weak equivalences are not the only interesting weak equivalences for  Leonard Guetta committed Oct 16, 2020 585 bisimplicial sets.  Leonard Guetta committed Jul 13, 2020 586 \begin{paragr}  Leonard Guetta committed Oct 14, 2020 587 588 589  A morphism $f : X \to Y$ of bisimplicial sets is a \emph{vertical (resp.\ horizontal) weak equivalence} when for every $n \geq 0$, the induced morphism of simplicial sets  Leonard Guetta committed Jul 13, 2020 590  $ Leonard Guetta committed Oct 14, 2020 591  f_{\bullet,n} : X_{\bullet,n} \to Y_{\bullet,n}  Leonard Guetta committed Jul 13, 2020 592 593 $ (resp.  Leonard Guetta committed Oct 14, 2020 594 595  $f_{n,\bullet} : X_{n,\bullet} \to Y_{n,\bullet})  Leonard Guetta committed Jul 13, 2020 596 $  Leonard Guetta committed Oct 05, 2020 597  is a weak equivalence of simplicial sets. Recall now a very useful lemma.  Leonard Guetta committed Jul 13, 2020 598 599 \end{paragr} \begin{lemma}\label{bisimpliciallemma}  Leonard Guetta committed Oct 14, 2020 600 601  Let $f : X \to Y$ be a morphism of bisimplicial sets. If $f$ is a vertical or horizontal weak equivalence then it is a diagonal weak equivalence.  Leonard Guetta committed Jul 13, 2020 602 603 \end{lemma} \begin{proof}  Leonard Guetta committed Oct 14, 2020 604 605  See for example \cite[Chapter XII,4.3]{bousfield1972homotopy} or \cite[Proposition 2.1.7]{cisinski2004localisateur}.  Leonard Guetta committed Jul 13, 2020 606 607 \end{proof} \begin{paragr}  Leonard Guetta committed Oct 14, 2020 608  In particular, the identity functor of the category of bisimplicial sets  Leonard Guetta committed Oct 25, 2020 609  induces the morphisms of op-prederivators:  Leonard Guetta committed Oct 14, 2020 610 611 612 613  $\Ho(\Psh{\Delta\times\Delta}^{\mathrm{vert}}) \to \Ho(\Psh{\Delta\times\Delta}^{\mathrm{diag}})$  Leonard Guetta committed Jul 13, 2020 614  and  Leonard Guetta committed Oct 14, 2020 615 616 617 618  $\Ho(\Psh{\Delta\times\Delta}^{\mathrm{hor}}) \to \Ho(\Psh{\Delta\times\Delta}^{\mathrm{diag}}).$  Leonard Guetta committed Jul 11, 2020 619 \end{paragr}  Leonard Guetta committed Oct 06, 2020 620 \begin{proposition}\label{prop:bisimplicialcocontinuous}  Leonard Guetta committed Oct 05, 2020 621  The morphisms of op-prederivators  Leonard Guetta committed Oct 14, 2020 622 623 624 625  $\Ho(\Psh{\Delta\times\Delta}^{\mathrm{vert}}) \to \Ho(\Psh{\Delta\times\Delta}^{\mathrm{diag}})$  Leonard Guetta committed Oct 05, 2020 626  and  Leonard Guetta committed Oct 14, 2020 627 628 629 630  $\Ho(\Psh{\Delta\times\Delta}^{\mathrm{hor}}) \to \Ho(\Psh{\Delta\times\Delta}^{\mathrm{diag}})$  Leonard Guetta committed Oct 05, 2020 631 632 633 634 635 636 637  are homotopy cocontinuous. \end{proposition} \begin{proof} Recall that the category of bisimplicial sets can be equipped with a model structure where the weak equivalences are the vertical (resp.\ horizontal) weak equivalences and the cofibrations are the monomorphisms (see for example \cite[Chapter IV]{goerss2009simplicial} or \cite{cisinski2004localisateur}).  Leonard Guetta committed Oct 14, 2020 638 639 640 641 642  We respectively refer to these model structures as the \emph{vertical model structure} and \emph{horizontal model structure}. Since the functor $\delta^* : \Psh{\Delta\times\Delta} \to \Psh{\Delta}$ preserves monomorphisms, it follows from Lemma \ref{bisimpliciallemma} that the adjunction  Leonard Guetta committed Oct 05, 2020 643 644 645 646  $\begin{tikzcd} \delta^* : \Psh{\Delta\times\Delta} \ar[r,shift left] & \ar[l,shift left] \Psh{\Delta} : \delta_*  Leonard Guetta committed Oct 14, 2020 647 648 649 650 651 652 653 654 655 656 657 658 659 660  \end{tikzcd}$ is a Quillen adjunction when $\Psh{\Delta\times\Delta}$ is equipped with either the vertical model structure or the horizontal model structure. In particular, the induced morphisms of op-prederivators $\overline{\delta^*} : \Ho(\Psh{\Delta\times\Delta}^{\mathrm{vert}}) \to \Ho(\Psh{\Delta})$ and $\overline{\delta^*} : \Ho(\Psh{\Delta\times\Delta}^{\mathrm{hor}}) \to \Ho(\Psh{\Delta})$  Leonard Guetta committed Jan 20, 2021 661  are homotopy cocontinuous. Now, the obvious identity  Leonard Guetta committed Oct 14, 2020 662  $\delta^*=\delta^* \circ \mathrm{id}_{\Psh{\Delta\times\Delta}}$ implies that  Leonard Guetta committed Oct 16, 2020 663  we have commutative triangles  Leonard Guetta committed Oct 14, 2020 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695  $\begin{tikzcd} \Ho(\Psh{\Delta\times\Delta}^{\mathrm{vert}}) \ar[r] \ar[rd,"\overline{\delta^*}"']& \Ho(\Psh{\Delta\times\Delta}^{\mathrm{diag}}) \ar[d,"\overline{\delta^*}"] \\ &\Ho(\Psh{\Delta}) \end{tikzcd}$ and $\begin{tikzcd} \Ho(\Psh{\Delta\times\Delta}^{\mathrm{hor}}) \ar[r] \ar[rd,"\overline{\delta^*}"']& \Ho(\Psh{\Delta\times\Delta}^{\mathrm{diag}}) \ar[d,"\overline{\delta^*}"] \\ &\Ho(\Psh{\Delta}). \end{tikzcd}$ The result follows then from the fact that $\overline{\delta^*} : \Ho(\Psh{\Delta\times\Delta}^{\mathrm{diag}}) \to \Ho(\Psh{\Delta})$ is an equivalence of op-prederivators. \end{proof} In practice, we will use the following corollary. \begin{corollary}\label{cor:bisimplicialsquare} Let $\begin{tikzcd} A \ar[r,"u"]\ar[d,"f"] & B \ar[d,"g"] \\ C \ar[r,"v"] & D \end{tikzcd}$  696  be a commutative square in the category of bisimplicial sets satisfying at least one of the two  Leonard Guetta committed Oct 14, 2020 697 698 699  following conditions: \begin{enumerate}[label=(\alph*)] \item For every $n\geq 0$, the square of simplicial sets  Leonard Guetta committed Oct 06, 2020 700 701  $\begin{tikzcd}  Leonard Guetta committed Oct 14, 2020 702 703  A_{\bullet,n} \ar[r,"{u_{\bullet,n}}"]\ar[d,"{f_{\bullet,n}}"] & B_{\bullet,n} \ar[d,"{g_{\bullet,n}}"] \\ C_{\bullet,n} \ar[r,"{v_{\bullet,n}}"] & D_{\bullet,n}  Leonard Guetta committed Oct 06, 2020 704 705  \end{tikzcd}$  Leonard Guetta committed Oct 14, 2020 706 707 708  is homotopy cocartesian. \item For every $n\geq 0$, the square of simplicial sets $ Leonard Guetta committed Oct 06, 2020 709  \begin{tikzcd}  Leonard Guetta committed Oct 14, 2020 710 711  A_{n,\bullet} \ar[r,"{u_{n,\bullet}}"]\ar[d,"{f_{n,\bullet}}"] & B_{n,\bullet} \ar[d,"{g_{n,\bullet}}"] \\ C_{n,\bullet} \ar[r,"{v_{n,\bullet}}"] & D_{n,\bullet}  Leonard Guetta committed Oct 06, 2020 712 713  \end{tikzcd}$  Leonard Guetta committed Oct 14, 2020 714  is homotopy cocartesian.  Leonard Guetta committed Jul 23, 2020 715 716  \end{enumerate} Then, the square  Leonard Guetta committed Oct 14, 2020 717 718 719 720 721  $\begin{tikzcd} \delta^*(A) \ar[r,"\delta^*(u)"]\ar[d,"\delta^*(f)"] & \delta^*(B) \ar[d,"\delta^*(g)"] \\ \delta^*(C) \ar[r,"\delta^*(v)"] & \delta^*(D) \end{tikzcd}  Leonard Guetta committed Jul 13, 2020 722 723 724 725 $ is a homotopy cocartesian square of simplicial sets. \end{corollary} \begin{proof}  Leonard Guetta committed Oct 06, 2020 726 727 728  From \cite[Corollary 10.3.10(i)]{groth2013book} we know that the square of bisimplicial sets $ Leonard Guetta committed Oct 14, 2020 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744  \begin{tikzcd} A \ar[r,"u"]\ar[d,"f"] & B \ar[d,"g"] \\ C \ar[r,"v"] & D \end{tikzcd}$ is homotopy cocartesian with respect to the vertical weak equivalences if and only if for every $n\geq 0$, the square $\begin{tikzcd} A_{\bullet,n} \ar[r,"{u_{\bullet,n}}"]\ar[d,"{f_{\bullet,n}}"] & B_{\bullet,n} \ar[d,"{g_{\bullet,n}}"] \\ C_{\bullet,n} \ar[r,"{v_{\bullet,n}}"] & D_{\bullet,n} \end{tikzcd}$ is a homotopy cocartesian square of simplicial sets and similarly for horizontal weak equivalences. The result follows then from Proposition \ref{prop:bisimplicialcocontinuous}.  Leonard Guetta committed Oct 06, 2020 745 \end{proof}  Leonard Guetta committed Oct 20, 2020 746 \section{Bisimplicial nerve for 2-categories}\label{section:bisimplicialnerve}  Leonard Guetta committed Oct 06, 2020 747 748 749 We shall now describe a nerve'' for $2$-categories with values in bisimplicial sets and recall a few results that shows that this nerve is, in some sense, equivalent to the nerve defined in \ref{paragr:nerve}.  Leonard Guetta committed Jul 14, 2020 750 751 \begin{notation} \begin{itemize}  Leonard Guetta committed Oct 14, 2020 752  \item[-] Once again, we write $N : \Cat \to \Psh{\Delta}$ instead of $N_1$ for  Leonard Guetta committed Oct 25, 2020 753  the usual nerve of categories. Moreover, using the usual notation for the  Leonard Guetta committed Oct 14, 2020 754  set of $k$-simplices of a simplicial set, if $C$ is a (small) category, then  Leonard Guetta committed Jul 14, 2020 755  $ Leonard Guetta committed Oct 14, 2020 756  N(C)_k  Leonard Guetta committed Jul 14, 2020 757 758 $ is the set of $k$-simplices of the nerve of $C$.  Leonard Guetta committed Oct 14, 2020 759 760 761 762 763 764  \item[-] Similarly, we write $N : 2\Cat \to \Psh{\Delta}$ instead of $N_2$ for the nerve of $2$-categories. This makes sense since the nerve for categories is the restriction of the nerve for $2$-categories. \item[-] For $2$-categories, we refer to the $\comp_0$-composition of $2$-cells as the \emph{horizontal composition} and the $\comp_1$-composition of $2$-cells as the \emph{vertical composition}.  Leonard Guetta committed Jul 14, 2020 765 766  \item[-] For a $2$-category $C$ and $x$ and $y$ objects of $C$, we denote by $ Leonard Guetta committed Oct 14, 2020 767  C(x,y)  Leonard Guetta committed Jul 14, 2020 768 $  Leonard Guetta committed Oct 14, 2020 769 770 771 772  the category whose objects are the $1$-cells of $C$ with $x$ as source and $y$ as target, and whose arrows are the $2$-cells of $C$ with $x$ as $0$-source and $y$ as $0$-target. Composition is induced by vertical composition in $C$.  Leonard Guetta committed Jul 14, 2020 773 774 775  \end{itemize} \end{notation} \begin{paragr}  Leonard Guetta committed Oct 25, 2020 776  Every $2$-category $C$ defines a simplicial object in $\Cat$,  Leonard Guetta committed Oct 16, 2020 777  $S(C): \Delta^{\op} \to \Cat,$ where, for each $n \geq 0$,  Leonard Guetta committed Jul 21, 2020 778  $ Leonard Guetta committed Oct 16, 2020 779  S_n(C):= \coprod_{(x_0,\cdots,x_n)\in \Ob(C)^{\times (n+1)}}C(x_0,x_1)  Leonard Guetta committed Oct 14, 2020 780 781  \times \cdots \times C(x_{n-1},x_n).$  Leonard Guetta committed Jan 05, 2021 782 783  Note that for $n=0$, the above formula reads $S_0(C)=C_0$. For $n>0$, the face operators $\partial_i : S_{n}(C) \to S_{n-1}(C)$ are induced by horizontal composition for \$0 < i