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 Leonard Guetta committed Jun 13, 2020 1 \chapter{Homotopy and homology type of free $2$-categories}  Leonard Guetta committed Oct 20, 2020 2 \section{Preliminaries: the case of free $1$-categories}\label{section:prelimfreecat}  Leonard Guetta committed Oct 16, 2020 3 In this section, we review some homotopical results on free  Leonard Guetta committed Oct 14, 2020 4 ($1$-)categories that will be of great help in the sequel.  Leonard Guetta committed Jun 25, 2020 5 \begin{paragr}  Leonard Guetta committed Oct 14, 2020 6 7  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 8  \begin{itemize}[label=-]  Leonard Guetta committed Jul 02, 2020 9  \item a source'' map $\src : G_1 \to G_0$,  Leonard Guetta committed Oct 14, 2020 10 11 12  \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 13  such that for every $x \in G_0$,  Leonard Guetta committed Jun 25, 2020 14  $ Leonard Guetta committed Oct 14, 2020 15 16 17 18 19 20 21 22 23 24 25  \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 26 27  There is a underlying reflexive graph'' functor  Leonard Guetta committed Jun 25, 2020 28  $ Leonard Guetta committed Oct 14, 2020 29  U : \Cat \to \Rgrph,  Leonard Guetta committed Jun 25, 2020 30 31 32 $ which has a left adjoint $ Leonard Guetta committed Oct 14, 2020 33  L : \Rgrph \to \Cat.  Leonard Guetta committed Jun 25, 2020 34 $  Leonard Guetta committed Oct 14, 2020 35 36  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 37  $ Leonard Guetta committed Oct 14, 2020 38 39 40 41  \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 42 $  Leonard Guetta committed Oct 14, 2020 43 44 45 46  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 47 \begin{lemma}  Leonard Guetta committed Oct 14, 2020 48 49  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 50  $ Leonard Guetta committed Oct 14, 2020 51  C \simeq L(G).  Leonard Guetta committed Jun 25, 2020 52 53 54 $ \end{lemma} \begin{proof}  Leonard Guetta committed Oct 14, 2020 55 56 57  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 58   Leonard Guetta committed Oct 14, 2020 59 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 non unital $1$-cells of $G$.  Leonard Guetta committed Jun 25, 2020 63 64 \end{proof} \begin{remark}  Leonard Guetta committed Oct 14, 2020 65 66 67 68 69 70  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 71 72 \end{remark} \begin{paragr}  Leonard Guetta committed Oct 14, 2020 73 74 75 76 77 78  There is another important description of the category $\Rgrph$. Consider $\Delta_{\leq 1}$ the full subcategory of $\Delta$ spanned by $[0]$ and $[1]$. Then, the category $\Rgrph$ is nothing but $\Psh{\Delta_{\leq 1}}$, the 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 79  $ Leonard Guetta committed Oct 14, 2020 80  i^* : \Psh{\Delta} \to \Rgrph,  Leonard Guetta committed Jun 26, 2020 81 82 83 $ which, by the usual technique of Kan extensions, has a left adjoint $ Leonard Guetta committed Oct 14, 2020 84  i_! : \Rgrph \to \Psh{\Delta}.  Leonard Guetta committed Jun 26, 2020 85 $  Leonard Guetta committed Oct 14, 2020 86 87  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 Oct 16, 2020 88  degenerated for $k>1$. For future reference, we put here the following lemma.  Leonard Guetta committed Jun 26, 2020 89 \end{paragr}  Leonard Guetta committed Jun 26, 2020 90 \begin{lemma}\label{lemma:monopreserved}  Leonard Guetta committed Jun 26, 2020 91 92 93  The functor $i_! : \Rgrph \to \Psh{\Delta}$ preserves monomorphism. \end{lemma} \begin{proof}  Leonard Guetta committed Jul 10, 2020 94 95  What we need to show is that, given a morphism of presheaves $ Leonard Guetta committed Oct 14, 2020 96  f : X \to Y,  Leonard Guetta committed Jul 10, 2020 97 $  Leonard Guetta committed Oct 14, 2020 98 99 100 101 102  if $f_0 : X_0 \to Y_0$ and $f_1 : X_1 \to Y_1$ are monomorphisms and if all $n$-simplices of $X$ are degenerated for $n\geq 2$, then $f$ is a monomorphism. A proof of this assertion is contained in \cite[Paragraph 3.4]{gabriel1967calculus}. The key argument is the Eilenberg-Zilber Lemma (Proposition 3.1 of op. cit.).  Leonard Guetta committed Jun 26, 2020 103 104 \end{proof} \begin{paragr}  Leonard Guetta committed Oct 14, 2020 105 106 107 108  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 109  $ Leonard Guetta committed Oct 14, 2020 110 111 112 113  \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 114 $  Leonard Guetta committed Oct 14, 2020 115 116 117  of arrows of $C$. Such an $n$-simplex is degenerated if and only if at least one of the $f_k$ is a unit. It is straightforward to check that the composite of  Leonard Guetta committed Jun 26, 2020 118  $ Leonard Guetta committed Oct 14, 2020 119  \Cat \overset{N}{\rightarrow} \Psh{\Delta} \overset{i^*}{\rightarrow} \Rgrph  Leonard Guetta committed Jun 26, 2020 120 $  Leonard Guetta committed Oct 14, 2020 121 122  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 123  $ Leonard Guetta committed Oct 14, 2020 124 125  \Rgrph \overset{i_!}{\rightarrow} \Psh{\Delta} \overset{c}{\rightarrow} \Cat.  Leonard Guetta committed Jun 26, 2020 126 $  Leonard Guetta committed Jun 26, 2020 127 128 129  We now review a construction of Dwyer and Kan from \cite{dwyer1980simplicial}.  Leonard Guetta committed Oct 14, 2020 130 131 132  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 Oct 14, 2020 141  \sum_{1 \leq i \leq n}\ell(f_i) \leq n.  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 Jun 26, 2020 157 158 159 \begin{lemma}[Dwyer-Kan]\label{lemma:dwyerkan} For every $k\leq 1$, the canonical inclusion map $ 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 $N^{k}(G)$ $\begin{tikzcd} X_0 \ar[r,"f_1"]& X_1\ar[r,"f_2"]& X_2 \ar[r]& \cdots \ar[r] &X_{k-1} \ar[r,"f_k"]& X_{k}.  Leonard Guetta committed Jul 11, 2020 199 200 201 202  \end{tikzcd}$ \end{itemize} Now, we have a cocartesian square  Leonard Guetta committed Jul 10, 2020 203  $ Leonard Guetta committed Oct 14, 2020 204 205 206 207  \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), \end{tikzcd}  Leonard Guetta committed Jul 10, 2020 208 $  Leonard Guetta committed Jul 11, 2020 209  which proves that the right vertical arrow is a trivial cofibration.  Leonard Guetta committed Jun 26, 2020 210 \end{proof}  Leonard Guetta committed Oct 09, 2020 211 From this lemma, we deduce the following proposition.  Leonard Guetta committed Jun 26, 2020 212 213 214 \begin{proposition} Let $G$ be a reflexive graph. The map $ Leonard Guetta committed Oct 14, 2020 215  \eta_{i_!(G)} : i_!(G) \to Nci_!(G),  Leonard Guetta committed Jun 26, 2020 216 $  Leonard Guetta committed Oct 14, 2020 217 218  where $\eta$ is the unit of the adjunction $c \dashv N$, is a trivial cofibration of simplicial sets.  Leonard Guetta committed Jul 02, 2020 219 \end{proposition}  Leonard Guetta committed Jun 26, 2020 220 \begin{proof}  Leonard Guetta committed Oct 14, 2020 221 222  This follows from the fact that trivial cofibrations are stable by transfinite composition.  Leonard Guetta committed Jun 26, 2020 223 224 225 226 227 \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 228 229 230 231  \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 232 $  Leonard Guetta committed Oct 14, 2020 233 234  be a cocartesian square in $\Rgrph$. If either $\alpha$ or $\beta$ is a monomorphism, then the induced square  Leonard Guetta committed Jun 26, 2020 235  $ Leonard Guetta committed Oct 14, 2020 236 237 238 239  \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 240 $  Leonard Guetta committed Oct 14, 2020 241 242  is a \emph{homotopy} cocartesian square of $\Cat$ when equipped with Thomason equivalences.  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 14, 2020 265 266 267 268  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 monomorphisms are cofibrations of simplicial sets for the standard Quillen model structure and Lemma \ref{lemma:hmtpycocartesianreedy}.  Leonard Guetta committed Jun 26, 2020 269 \end{proof}  Leonard Guetta committed Jul 08, 2020 270 \begin{paragr}  Leonard Guetta committed Oct 14, 2020 271 272 273 274 275 276 277 278 279 280 281 282 283  Actually, by working a little more, we obtain a more general result, which is stated in the proposition below. Let us say that a morphism of reflexive graphs, $\alpha : A \to B$, is \emph{quasi-injective on arrows} when for all $f$ and $g$ arrows of $A$, if $\alpha(f)=\alpha(g),$ then either $f=g$ or $f$ and $g$ are both units. In other words, $\alpha$ never send a non-unit arrow to a unit arrow and $\alpha$ never identifies two 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 284 \begin{proposition}\label{prop:hmtpysquaregraphbetter}  Leonard Guetta committed Jun 28, 2020 285 286  Let $ Leonard Guetta committed Oct 14, 2020 287 288 289 290  \begin{tikzcd} A \ar[d,"\alpha"] \ar[r,"\beta"] &B \ar[d,"\delta"] \\ C \ar[r,"\gamma"]& D \end{tikzcd}  Leonard Guetta committed Jun 28, 2020 291 $  Leonard Guetta committed Oct 14, 2020 292 293  be a cocartesian square in $\Rgrph$. Suppose that the following two conditions are satisfied  Leonard Guetta committed Jul 02, 2020 294  \begin{enumerate}[label=\alph*)]  Leonard Guetta committed Jun 28, 2020 295  \item Either $\alpha$ or $\beta$ is injective on objects.  Leonard Guetta committed Oct 14, 2020 296  \item Either $\alpha$ or $\beta$ is quasi-injective on arrows.  Leonard Guetta committed Jul 02, 2020 297  \end{enumerate}  Leonard Guetta committed Jun 28, 2020 298 299  Then, the square $ Leonard Guetta committed Oct 14, 2020 300 301 302 303  \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 304 $  Leonard Guetta committed Oct 14, 2020 305 306  is a \emph{homotopy} cocartesian square of $\Cat$ when equipped with Thomason equivalences.  Leonard Guetta committed Jun 28, 2020 307 308 \end{proposition} \begin{proof}  Leonard Guetta committed Oct 14, 2020 309 310 311 312  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 313   Leonard Guetta committed Oct 14, 2020 314 315 316 317 318 319  Let use denote by $E$ the set of objects of $B$ that lies in the image of $\beta$. For each element $x$ of $E$, we denote by $F_x$ the fiber'' of $x$, that is the set of objects of $A$ that $\beta$ sends to $x$. We consider the set $E$ and each $F_x$ as discrete reflexive graphs, i.e. reflexive graphs with no non-unital arrow. Now, let $G$ be the reflexive graph defined with the following cocartesian square  Leonard Guetta committed Jul 02, 2020 320  $ Leonard Guetta committed Oct 14, 2020 321 322 323 324  \begin{tikzcd} \displaystyle\coprod_{x \in E}F_x \ar[r] \ar[d] & E \ar[d]\\ A \ar[r] & G, \ar[from=1-1,to=2-2,phantom,very near end,"\ulcorner"] \end{tikzcd}  Leonard Guetta committed Jul 02, 2020 325 $  Leonard Guetta committed Oct 14, 2020 326 327 328 329 330 331 332 333 334 335  where the morphism $\coprod_{x \in E}F_x \to A$ is induced by the inclusion of each $F_x$ in $A$, and the morphism $\coprod_{x \in E}F_x \to E$ is the only one that sends an element $a \in F_x$ to $x$. In other words, $G$ is obtained from $A$ by collapsing the objects that are identified through $\beta$. It admits the following explicit description: $G_0$ is (isomorphic to) $E$ and the set of non-units arrows of $G$ is (isomorphic to) the set of non-units 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 describe $G$. % 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 336 337 338  Now, we have the following solid arrow commutative diagram $ Leonard Guetta committed Oct 14, 2020 339 340 341 342 343  \begin{tikzcd} \displaystyle\coprod_{x \in E}F_x \ar[r] \ar[d] & E \ar[ddr,bend left]\ar[d]&\\ 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 344 $  Leonard Guetta committed Oct 14, 2020 345 346 347 348  where the arrow $E \to B$ is the canonical inclusion. Hence, by universal property, the dotted arrow exists and makes the whole diagram commutes. A thorough verification easily shows that the morphism $G \to B$ is a monomorphism of $\Rgrph$.  Leonard Guetta committed Jul 09, 2020 349   Leonard Guetta committed Oct 14, 2020 350 351 352 353 354 355 356 357 358 359 360 361 362  By forming successive cocartesian square and combining with the square obtained earlier, we obtain a diagram of three cocartesian square: $\begin{tikzcd}[row sep = large] \displaystyle\coprod_{x \in E}F_x \ar[r] \ar[d] & E \ar[d]&\\ 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 363 $  Leonard Guetta committed Oct 14, 2020 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381  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 cocartesian. On the other hand, we know that both morphisms $\coprod_{x \in E}F_x \to A \text{ and } A \to C$ are injective on arrows, but since $\coprod_{x \in E}F_x$ does not have non-units arrows, both these morphisms are actually monomorphisms. Hence, using Corollary \ref{cor:hmtpysquaregraph}, we deduce that image by $L$ of square \textcircled{\tiny \textbf{1}} and of the pasting of squares \textcircled{\tiny \textbf{1}} and \textcircled{\tiny \textbf{2}} are homotopy cocartesian. This proves that the image by $L$ of square \textcircled{\tiny \textbf{2}} is homotopy cocartesian.  Leonard Guetta committed Jun 28, 2020 382 \end{proof}  Leonard Guetta committed Oct 14, 2020 383 384 We now apply Corollary \ref{cor:hmtpysquaregraph} and Proposition \ref{prop:hmtpysquaregraphbetter} to a few examples.  Leonard Guetta committed Oct 15, 2020 385 386 387 388 389 390 391 392 393 394 395 396 397 398 \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 the category obtained from $C$ by identifying $A$ and $B$, i.e.\ defined with the following cocartesian square $\begin{tikzcd} \sS_0 \ar[d] \ar[r,"{\langle A,B \rangle}"] & C \ar[d] \\ \sD_0 \ar[r] & C'. \end{tikzcd}$ 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 morphism is a monomorphism and we can apply Corollary \ref{cor:hmtpysquaregraph}. \end{example}  Leonard Guetta committed Jun 26, 2020 399 \begin{example}[Adding a generator]  Leonard Guetta committed Oct 14, 2020 400 401  Let $C$ be a free category, $A$ and $B$ (possibly equal) two objects of $C$ and let $C'$ be the category obtained from $C$ by adding a generator $A \to  Leonard Guetta committed Oct 15, 2020 402  B$, i.e.\ defined with the following cocartesian square:  Leonard Guetta committed Jun 26, 2020 403  $ Leonard Guetta committed Oct 14, 2020 404  \begin{tikzcd}  Leonard Guetta committed Oct 15, 2020 405  \sS_0 \ar[d,"i_1"] \ar[r,"{\langle A, B \rangle}"] & C \ar[d] \\  Leonard Guetta committed Oct 14, 2020 406 407  \sD_1 \ar[r] & C'. \end{tikzcd}  Leonard Guetta committed Jun 26, 2020 408 $  Leonard Guetta committed Oct 14, 2020 409 410  Then, this square is homotopy cocartesian in $\Cat$ (when equipped with the Thomason equivalences). Indeed, it obviously is the image of a square of  Leonard Guetta committed Oct 15, 2020 411  $\Rgrph$ by the functor $L$ and the morphism $i_1 : \sS_0 \to \sD_1$  Leonard Guetta committed Oct 14, 2020 412 413  comes from a monomorphism of $\Rgrph$. Hence, we can apply Corollary \ref{cor:hmtpysquaregraph}.  Leonard Guetta committed Jun 26, 2020 414 415 \end{example} \begin{remark}  Leonard Guetta committed Oct 14, 2020 416 417 418 419  Since every free category is obtained by recursively adding generators starting from a set of objects (seen as a $0$-category), the previous example yields another proof that \emph{free} (1-)categories are \good{} (which we already knew since we have seen that \emph{all} (1-)categories are \good{}).  Leonard Guetta committed Jun 26, 2020 420 \end{remark}  Leonard Guetta committed Oct 09, 2020 421 \begin{example}[Identifying two generators]  Leonard Guetta committed Oct 14, 2020 422 423 424 425  Let $C$ be a free category and $f,g : A \to B$ parallel generating arrows of $C$ such that $f\neq g$. Now consider the category $C'$ obtained from $C$ by identifying'' $f$ and $g$, i.e. defined with the following cocartesian square  Leonard Guetta committed Jun 27, 2020 426  $ Leonard Guetta committed Oct 14, 2020 427 428 429 430  \begin{tikzcd} \sS_1\ar[d] \ar[r,"{\langle f, g \rangle}"] &C \ar[d] \\ \sD_1 \ar[r] & C', \end{tikzcd}  Leonard Guetta committed Jun 27, 2020 431 $  Leonard Guetta committed Oct 14, 2020 432 433 434 435 436 437 438 439 440 441  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 is homotopy cocartesian in $\Cat$ (when equipped with Thomason equivalences). 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 442 \end{example}  Leonard Guetta committed Oct 08, 2020 443 \begin{example}[Killing a generator]\label{example:killinggenerator}  Leonard Guetta committed Oct 14, 2020 444 445 446  Let $C$ be a free category and let $f : A \to B$ one of its generating arrow such that $A \neq B$. Now consider the category $C'$ obtained from $C$ by killing'' $f$, i.e. defined with the following cocartesian square:  Leonard Guetta committed Jun 27, 2020 447  $ Leonard Guetta committed Oct 14, 2020 448 449 450 451  \begin{tikzcd} \sD_1 \ar[d] \ar[r,"\langle f \rangle"] & C \ar[d] \\ \sD_0 \ar[r] & C'. \end{tikzcd}  Leonard Guetta committed Jun 27, 2020 452 $  Leonard Guetta committed Oct 14, 2020 453 454 455 456  Then, this above square is homotopy cocartesian in $\Cat$ (equipped with Thomason equivalences). Indeed, it obviously is the image of a 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 $\Rgrph$.  Leonard Guetta committed Jun 27, 2020 457 458 \end{example} \begin{remark}  Leonard Guetta committed Oct 14, 2020 459 460 461  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 462   Leonard Guetta committed Oct 14, 2020 463 464 465  Note also that the hypothesis that $A\neq B$ was fundamental in the previous example as for $A=B$ the square is \emph{not} homotopy cocartesian. \end{remark}  Leonard Guetta committed Jul 11, 2020 466   Leonard Guetta committed Oct 09, 2020 467 \section{Preliminaries: bisimplicial sets}  Leonard Guetta committed Jul 11, 2020 468 \begin{paragr}  Leonard Guetta committed Oct 14, 2020 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485  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 $n,m \geq 0$, we use the notation \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 486   Leonard Guetta committed Oct 14, 2020 487  Note that for every $n\geq 0$, we have simplicial sets  Leonard Guetta committed Jul 14, 2020 488  \begin{align*}  Leonard Guetta committed Oct 14, 2020 489 490 491 492 493 494 495 496 497  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 498   Leonard Guetta committed Oct 14, 2020 499 500  \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 501  \begin{align*}  Leonard Guetta committed Jul 21, 2020 502  X_{n,\bullet} : \Delta^{\op} &\to \Set \\  Leonard Guetta committed Jul 11, 2020 503 504 505 506  [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 507  X_{\bullet,n} : \Delta^{\op} &\to \Set \\  Leonard Guetta committed Jul 11, 2020 508 509  [m] &\mapsto X_{m,n}. \end{align*}  Leonard Guetta committed Oct 09, 2020 510  The correspondences  Leonard Guetta committed Jul 11, 2020 511  $ Leonard Guetta committed Oct 14, 2020 512  n \mapsto X_{n,\bullet} \,\text{ and }\, n\mapsto X_{\bullet,n}  Leonard Guetta committed Jul 11, 2020 513 $  Leonard Guetta committed Oct 14, 2020 514 515  actually define functors $\Delta \to \Psh{\Delta}$. They correspond to the two currying'' operations  Leonard Guetta committed Jul 11, 2020 516  $ Leonard Guetta committed Oct 14, 2020 517  \Psh{\Delta\times\Delta} \to \underline{\Hom}(\Delta^{\op},\Psh{\Delta}),  Leonard Guetta committed Jul 11, 2020 518 $  Leonard Guetta committed Oct 14, 2020 519 520 521  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 522 523 \end{paragr} \begin{paragr}  Leonard Guetta committed Jul 13, 2020 524 525 526 527 528 529 530  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 531 532 533 534 535 536 537  \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 $\delta_*$ $\delta_! \dashv \delta^* \dashv \delta_*.$  Leonard Guetta committed Oct 05, 2020 538  We say that a morphism a bisimplicial sets, $f : X \to Y$, is a \emph{diagonal  Leonard Guetta committed Oct 16, 2020 539  weak equivalence} (resp.\ \emph{diagonal fibration}) when $\delta^*(f)$ is a  Leonard Guetta committed Oct 14, 2020 540 541 542 543 544 545 546 547 548 549 550  weak equivalence of simplicial sets (resp.\ fibration of simplicial sets). By 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 diagonal fibrations in an obvious sense. We shall refer to this model structure as the \emph{diagonal model structure}.  Leonard Guetta committed Jul 13, 2020 551 \end{paragr}  Leonard Guetta committed Jul 22, 2020 552 \begin{proposition}\label{prop:diageqderivator}  Leonard Guetta committed Oct 14, 2020 553 554  Consider that $\Psh{\Delta\times\Delta}$ is equipped with the diagonal model structure. Then, the adjunction  Leonard Guetta committed Jul 13, 2020 555  $ Leonard Guetta committed Oct 14, 2020 556 557 558  \begin{tikzcd} \delta_! : \Psh{\Delta} \ar[r,shift left] & \Psh{\Delta\times\Delta} \ar[l,shift left]: \delta^*,  Leonard Guetta committed Jul 13, 2020 559 560 561 562 563  \end{tikzcd}$ is a Quillen equivalence. \end{proposition} \begin{proof}  Leonard Guetta committed Oct 05, 2020 564 565 566 567  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 568 \end{proof}  Leonard Guetta committed Oct 05, 2020 569 \begin{paragr}  Leonard Guetta committed Oct 14, 2020 570 571 572 573 574 575  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 576 \end{paragr}  Leonard Guetta committed Oct 14, 2020 577 Diagonal weak equivalences are not the only interesting weak equivalences for  Leonard Guetta committed Oct 16, 2020 578 bisimplicial sets.  Leonard Guetta committed Jul 13, 2020 579 \begin{paragr}  Leonard Guetta committed Oct 14, 2020 580 581 582  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 583  $ Leonard Guetta committed Oct 14, 2020 584  f_{\bullet,n} : X_{\bullet,n} \to Y_{\bullet,n}  Leonard Guetta committed Jul 13, 2020 585 586 $ (resp.  Leonard Guetta committed Oct 14, 2020 587 588  $f_{n,\bullet} : X_{n,\bullet} \to Y_{n,\bullet})  Leonard Guetta committed Jul 13, 2020 589 $  Leonard Guetta committed Jul 11, 2020 590   Leonard Guetta committed Oct 05, 2020 591  is a weak equivalence of simplicial sets. Recall now a very useful lemma.  Leonard Guetta committed Jul 13, 2020 592 593 \end{paragr} \begin{lemma}\label{bisimpliciallemma}  Leonard Guetta committed Oct 14, 2020 594 595  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 596 597 \end{lemma} \begin{proof}  Leonard Guetta committed Oct 14, 2020 598 599  See for example \cite[Chapter XII,4.3]{bousfield1972homotopy} or \cite[Proposition 2.1.7]{cisinski2004localisateur}.  Leonard Guetta committed Jul 13, 2020 600 601 \end{proof} \begin{paragr}  Leonard Guetta committed Oct 14, 2020 602 603 604 605 606 607  In particular, the identity functor of the category of bisimplicial sets induces morphisms of op-prederivators: $\Ho(\Psh{\Delta\times\Delta}^{\mathrm{vert}}) \to \Ho(\Psh{\Delta\times\Delta}^{\mathrm{diag}})$  Leonard Guetta committed Jul 13, 2020 608  and  Leonard Guetta committed Oct 14, 2020 609 610 611 612  $\Ho(\Psh{\Delta\times\Delta}^{\mathrm{hor}}) \to \Ho(\Psh{\Delta\times\Delta}^{\mathrm{diag}}).$  Leonard Guetta committed Jul 11, 2020 613 \end{paragr}  Leonard Guetta committed Oct 06, 2020 614 \begin{proposition}\label{prop:bisimplicialcocontinuous}  Leonard Guetta committed Oct 05, 2020 615  The morphisms of op-prederivators  Leonard Guetta committed Oct 14, 2020 616 617 618 619  $\Ho(\Psh{\Delta\times\Delta}^{\mathrm{vert}}) \to \Ho(\Psh{\Delta\times\Delta}^{\mathrm{diag}})$  Leonard Guetta committed Oct 05, 2020 620  and  Leonard Guetta committed Oct 14, 2020 621 622 623 624  $\Ho(\Psh{\Delta\times\Delta}^{\mathrm{hor}}) \to \Ho(\Psh{\Delta\times\Delta}^{\mathrm{diag}})$  Leonard Guetta committed Oct 05, 2020 625 626 627 628 629 630 631  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 632 633 634 635 636  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 637 638 639 640  $\begin{tikzcd} \delta^* : \Psh{\Delta\times\Delta} \ar[r,shift left] & \ar[l,shift left] \Psh{\Delta} : \delta_*  Leonard Guetta committed Oct 14, 2020 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656  \end{tikzcd}$ 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 $\overline{\delta^*} : \Ho(\Psh{\Delta\times\Delta}^{\mathrm{hor}}) \to \Ho(\Psh{\Delta})$ are homotopy cocontinuous. On the other hand, the obvious identity $\delta^*=\delta^* \circ \mathrm{id}_{\Psh{\Delta\times\Delta}}$ implies that  Leonard Guetta committed Oct 16, 2020 657  we have commutative triangles  Leonard Guetta committed Oct 14, 2020 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693  $\begin{tikzcd} \Ho(\Psh{\Delta\times\Delta}^{\mathrm{vert}}) \ar[r] \ar[rd,"\overline{\delta^*}"']& \Ho(\Psh{\Delta\times\Delta}^{\mathrm{diag}}) \ar[d,"\overline{\delta^*}"] \\ &\Ho(\Psh{\Delta}) \end{tikzcd}$ 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 $\begin{tikzcd} A \ar[r,"u"]\ar[d,"f"] & B \ar[d,"g"] \\ C \ar[r,"v"] & D \end{tikzcd}$ be a square in the category of bisimplicial sets satisfying either of the following conditions: \begin{enumerate}[label=(\alph*)] \item For every $n\geq 0$, the square of simplicial sets  Leonard Guetta committed Oct 06, 2020 694 695  $\begin{tikzcd}  Leonard Guetta committed Oct 14, 2020 696 697  A_{\bullet,n} \ar[r,"{u_{\bullet,n}}"]\ar[d,"{f_{\bullet,n}}"] & B_{\bullet,n} \ar[d,"{g_{\bullet,n}}"] \\ C_{\bullet,n} \ar[r,"{v_{\bullet,n}}"] & D_{\bullet,n}  Leonard Guetta committed Oct 06, 2020 698 699  \end{tikzcd}$  Leonard Guetta committed Oct 14, 2020 700 701 702  is homotopy cocartesian. \item For every $n\geq 0$, the square of simplicial sets $ Leonard Guetta committed Oct 06, 2020 703  \begin{tikzcd}  Leonard Guetta committed Oct 14, 2020 704 705  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 706 707  \end{tikzcd}$  Leonard Guetta committed Oct 14, 2020 708  is homotopy cocartesian.  Leonard Guetta committed Jul 23, 2020 709 710  \end{enumerate} Then, the square  Leonard Guetta committed Oct 14, 2020 711 712 713 714 715  $\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 716 717 718 719 $ is a homotopy cocartesian square of simplicial sets. \end{corollary} \begin{proof}  Leonard Guetta committed Oct 06, 2020 720 721 722  From \cite[Corollary 10.3.10(i)]{groth2013book} we know that the square of bisimplicial sets $ Leonard Guetta committed Oct 14, 2020 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738  \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 739 \end{proof}  Leonard Guetta committed Oct 20, 2020 740 \section{Bisimplicial nerve for 2-categories}\label{section:bisimplicialnerve}  Leonard Guetta committed Oct 06, 2020 741 742 743 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 744 745 \begin{notation} \begin{itemize}  Leonard Guetta committed Oct 14, 2020 746 747 748  \item[-] Once again, we write $N : \Cat \to \Psh{\Delta}$ instead of $N_1$ for the usual nerve of categories. Moreover, using the usual convention for the set of $k$-simplices of a simplicial set, if $C$ is a (small) category, then  Leonard Guetta committed Jul 14, 2020 749  $ Leonard Guetta committed Oct 14, 2020 750  N(C)_k  Leonard Guetta committed Jul 14, 2020 751 752 $ is the set of $k$-simplices of the nerve of $C$.  Leonard Guetta committed Oct 14, 2020 753 754 755 756 757 758  \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 759 760  \item[-] For a $2$-category $C$ and $x$ and $y$ objects of $C$, we denote by $ Leonard Guetta committed Oct 14, 2020 761  C(x,y)  Leonard Guetta committed Jul 14, 2020 762 $  Leonard Guetta committed Oct 14, 2020 763 764 765 766  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 767 768 769  \end{itemize} \end{notation} \begin{paragr}  Leonard Guetta committed Jul 21, 2020 770  Each $2$-category $C$ defines a simplicial object in $\Cat$,  Leonard Guetta committed Oct 16, 2020 771  $S(C): \Delta^{\op} \to \Cat,$ where, for each $n \geq 0$,  Leonard Guetta committed Jul 21, 2020 772  $ Leonard Guetta committed Oct 16, 2020 773  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 774 775  \times \cdots \times C(x_{n-1},x_n).$  Leonard Guetta committed Oct 16, 2020 776 777 778  Note that for $n=0$, the above formula reads $H_0(C)=C_0$. The face operators $\partial_i : S_{n}(C) \to S_{n-1}(C)$ are induced by horizontal composition and the degeneracy operators $s_i : S_{n}(C) \to S_{n+1}(C)$ are  Leonard Guetta committed Oct 14, 2020 779  induced by the units for the horizontal composition.  Leonard Guetta committed Jul 21, 2020 780   Leonard Guetta committed Oct 16, 2020 781  Post-composing $S(C)$ with the nerve functor $N : \Cat \to \Psh{\Delta}$, we  Leonard Guetta committed Oct 14, 2020 782  obtain a functor  Leonard Guetta committed Jul 21, 2020 783  $ Leonard Guetta committed Oct 16, 2020 784  NS(C) : \Delta^{\op} \to \Psh{\Delta}.  Leonard Guetta committed Jul 21, 2020 785 786 787 $ \end{paragr} \begin{remark}  Leonard Guetta committed Oct 16, 2020 788 789  When $C$ is a $1$-category, the simplicial object $S(C)$ is nothing but the usual nerve of $C$ where, for each $n\geq 0$, $S_n(C)$ is seen as a discrete  Leonard Guetta committed Oct 14, 2020 790 791  category. \end{remark}  Leonard Guetta committed Jul 21, 2020 792 \begin{definition}  Leonard Guetta committed Oct 14, 2020 793 794  The \emph{bisimplicial nerve} of a $2$-category $C$ is the bisimplicial set $\binerve(C)$ defined as  Leonard Guetta committed Jul 21, 2020 795  $ Leonard Guetta committed Oct 16, 2020 796  \binerve(C)_{n,m}:=N(S_n(C))_m,  Leonard Guetta committed Jul 21, 2020 797 $  Leonard Guetta committed Oct 14, 2020 798  for all $n,m \geq 0$.  Leonard Guetta committed Jul 21, 2020 799 \end{definition}  Leonard Guetta committed Jul 22, 2020 800 \begin{paragr}\label{paragr:formulabisimplicialnerve}  Leonard Guetta committed Oct 14, 2020 801  In other words, the bisimplicial nerve of $C$ is obtained by un-curryfying''  Leonard Guetta committed Oct 16, 2020 802  the functor $NS(C) : \Delta^{op} \to \Psh{\Delta}$.  Leonard Guetta committed Jul 22, 2020 803   Leonard Guetta committed Jul 21, 2020 804  Since the nerve $N$ commutes with products and sums, we obtain the formula  Leonard Guetta committed Jul 23, 2020 805  \label{fomulabinerve}  Leonard Guetta committed Oct 14, 2020 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831  \binerve(C)_{n,m} = \coprod_{(x_0,\cdots,x_n)\in \Ob(C)^{\times (n+1)}}N(C(x_0,x_1))_m \times \cdots \times N(C(x_{n-1},x_n))_m. More intuitively, an element of $\binerve(C)_{n,m}$ consists of a pasting scheme'' in $C$ that looks like $m \underbrace{\left\{\begin{tikzcd}[column sep=huge,ampersand replacement=\&] \bullet \ar[r,"\vdots"]\ar[r,bend left=90,looseness=1.4,""{name=A,below}] \ar[r,bend left=35,""{name=B,above}] \ar[r,bend right=35,"\vdots",""{name=G,below}]\ar[r,bend right=90,looseness=1.4,""{name=H,above}] \& \bullet\ar[r,"\vdots"]\ar[r,bend left=90,looseness=1.4,""{name=C,below}] \ar[r,bend left=35,""{name=D,above}] \ar[r,bend right=35,"\vdots",""{name=I,below}]\ar[r,bend right=90,looseness=1.4,""{name=J,above}] \&\bullet\ar[r,phantom,description,"\cdots"]\&\bullet\ar[r,"\vdots"]\ar[r,bend left=90,looseness=1.4,""{name=E,below}] \ar[r,bend left=35,""{name=F,above}] \ar[r,bend right=35,"\vdots",""{name=K,below}]\ar[r,bend right=90,looseness=1.4,""{name=L,above}] \&\bullet \ar[from=A,to=B,Rightarrow] \ar[from=C,to=D,Rightarrow] \ar[from=E,to=F,Rightarrow] \ar[from=G,to=H,Rightarrow] \ar[from=I,to=J,Rightarrow] \ar[from=K,to=L,Rightarrow] \end{tikzcd}\right.}_{ n }.$  Leonard Guetta committed Jul 23, 2020 832 \end{paragr}  Leonard Guetta committed Oct 09, 2020 833 In the definition of the bisimplicial nerve of a $2$\nbd{}category we gave, one  Leonard Guetta committed Oct 14, 2020 834 direction of the bisimplicial set is privileged over the other. We now give  Leonard Guetta committed Oct 16, 2020 835 an equivalent definition of the bisimplicial nerve which uses the other direction.  Leonard Guetta committed Jul 23, 2020 836 \begin{paragr}  Leonard Guetta committed Oct 14, 2020 837  Let $C$ be a $2$\nbd{}category. For every $k \geq 1$, we define a  Leonard Guetta committed Oct 16, 2020 838  $1$\nbd{}category $V_k(C)$ in the following fashion:  Leonard Guetta committed Oct 06, 2020 839  \begin{itemize}[label=-]  Leonard Guetta committed Oct 16, 2020 840  \item The objects of $V_k(C)$ are the objects of $C$.  Leonard Guetta committed Jul 23, 2020 841 842  \item A morphism $\alpha$ is a sequence $ Leonard Guetta committed Oct 14, 2020 843  \alpha=(\alpha_1,\alpha_2,\cdots,\alpha_k)  Leonard Guetta committed Jul 23, 2020 844 $  Leonard Guetta committed Oct 14, 2020 845 846  of $2$-cells of $C$ that are vertically composable, i.e.\ such that for every $1 \leq i \leq k-1$,  Leonard Guetta committed Jul 23, 2020 847  $ Leonard Guetta committed Oct 14, 2020 848  \src(\alpha_i)=\trgt(\alpha_{i+1}).  Leonard Guetta committed Jul 23, 2020 849 850 851 $ The source and target of alpha are given by $ Leonard Guetta committed Oct 14, 2020 852 853  \src(\alpha):=\src_0(\alpha_1)\text{ and }\trgt(\alpha):=\trgt_0(\alpha_1).  Leonard Guetta committed Jul 23, 2020 854 $  Leonard Guetta committed Oct 14, 2020 855 856  (Note that we could have used any of the $\alpha_i$ instead of $\alpha_1$ since they all have the same $0$\nbd{}source and $0$\nbd{}target.)  Leonard Guetta committed Jul 23, 2020 857 858  \item Composition is given by $ Leonard Guetta committed Oct 14, 2020 859  (\alpha_1,\alpha_2,\cdots,\alpha_k)\circ(\beta_1,\beta_2,\cdots,\beta_k):=(\alpha_1\comp_0\beta_1,\alpha_2\comp_0\beta_2,\cdots,\alpha_k\comp_0\beta_k)  Leonard Guetta committed Jul 23, 2020 860 861 862 $ and the unit on an object $x$ is the sequence $ Leonard Guetta committed Oct 14, 2020 863  (1^2_x,\cdots, 1^2_x).  Leonard Guetta committed Jul 23, 2020 864 865 $ \end{itemize}  Leonard Guetta committed Oct 16, 2020 866  For $k=0$, we define $V_0(C)$ to be the category obtained from $C$ by simply  Leonard Guetta committed Oct 06, 2020 867  forgetting the $2$\nbd{}cells (which is nothing but $\tau^{s}_{\leq 1}(C)$  Leonard Guetta committed Oct 14, 2020 868  with the notations of \ref{paragr:defncat}). The correspondence $n \mapsto  Leonard Guetta committed Oct 16, 2020 869  V_n(C)$ defines to a simplicial object in $\Cat$  Leonard Guetta committed Jul 22, 2020 870  $ Leonard Guetta committed Oct 14, 2020 871  V(C) : \Delta^{\op} \to \Cat,  Leonard Guetta committed Jul 22, 2020 872 $  Leonard Guetta committed Oct 14, 2020 873 874  where the face operators are induced by the vertical composition and the degeneracy operators are induced by the units for the vertical composition.  Leonard Guetta committed Jul 22, 2020 875 \end{paragr}  Leonard Guetta committed Jul 23, 2020 876 877 878 \begin{lemma}\label{lemma:binervehorizontal} Let $C$ be a $2$-category. For every $n \geq 0$, we have $ Leonard Guetta committed Oct 16, 2020 879  N(V_m(C))_n=(\binerve(C))_{n,m}.  Leonard Guetta committed Jul 23, 2020 880 $  Leonard Guetta committed Oct 14, 2020 881 \end{lemma}  Leonard Guetta committed Jul 22, 2020 882 \begin{proof}  Leonard Guetta committed Oct 14, 2020 883 884  This is simply a reformulation of the formula given in Paragraph \ref{paragr:formulabisimplicialnerve}.  Leonard Guetta committed Jul 22, 2020 885 886 \end{proof} \begin{paragr}  Leonard Guetta committed Jul 23, 2020 887  The bisimplicial nerve canonically defines a functor  Leonard Guetta committed Jul 22, 2020 888  $ Leonard Guetta committed Oct 14, 2020 889  \binerve : 2\Cat \to \Psh{\Delta\times\Delta}  Leonard Guetta committed Jul 22, 2020 890 $  Leonard Guetta committed Oct 14, 2020 891 892 893  which enables us to compare the homotopy theory of $2\Cat$ with the homotopy theory of bisimplicial sets. \end{paragr}  Leonard Guetta committed Jul 23, 2020 894 \begin{lemma}\label{lemma:binervthom}  Leonard Guetta committed Oct 14, 2020 895 896  A $2$-functor $F : C \to D$ is a Thomason equivalence if and only if $\binerve(F)$ is a diagonal weak equivalence of bisimplicial sets.  Leonard Guetta committed Jul 23, 2020 897 898 \end{lemma} \begin{proof}  Leonard Guetta committed Oct 06, 2020 899  It follows from what is shown in \cite[Section 2]{bullejos2003geometry} that  Leonard Guetta committed Oct 09, 2020 900  there is a zigzag of weak equivalence of simplicial sets  Leonard Guetta committed Jul 23, 2020 901  $ Leonard Guetta committed Oct 14, 2020 902  \delta^*(\binerve(C)) \leftarrow \cdots \rightarrow N(C)  Leonard Guetta committed Jul 23, 2020 903 $  Leonard Guetta committed Oct 06, 2020 904 905  which is natural in $C$. This implies what we wanted to show. See also \cite[Théorème 3.13]{ara2020comparaison}.  Leonard Guetta committed Jul 23, 2020 906 \end{proof}  Leonard Guetta committed Oct 14, 2020 907 908 From this lemma, we deduce two useful criteria to detect Thomason equivalences of $2$-categories.  Leonard Guetta committed Jul 22, 2020 909 \begin{corollary}\label{cor:criterionThomeqI}  Leonard Guetta committed Oct 14, 2020 910 911  Let $F : C \to D$ be a $2$-functor. If \begin{enumerate}[label=\alph*)]  Leonard Guetta committed Oct 08, 2020 912  \item $F_0 : C_0 \to D_0$ is a bijection,  Leonard Guetta committed Jul 23, 2020 913  \end{enumerate}  Leonard Guetta committed Oct 14, 2020 914 915 916 917 918 919 920 921 922  and \begin{enumerate}[resume*] \item for all objects $x,y$ of $C$, the functor $C(x,y) \to D(F(x),F(y))$ induced by $F$ is a Thomason equivalence of $1$-categories, \end{enumerate} then $F$ is a Thomason equivalence of $2$-categories.  Leonard Guetta committed Jul 22, 2020 923 924 925 926 \end{corollary} \begin{proof} By definition, for every $2$-category $C$ and every $m \geq 0$, we have $ Leonard Guetta committed Oct 16, 2020 927  (\binerve(C))_{\bullet,m} = NS(C).  Leonard Guetta committed Jul 22, 2020 928 $  Leonard Guetta committed Oct 14, 2020 929 930 931  The result follows then from Lemma \ref{bisimpliciallemma} and the fact that weak equivalences of simplicial sets are stable by coproducts and finite products.  Leonard Guetta committed Jul 23, 2020 932 \end{proof}  Leonard Guetta committed Oct 08, 2020 933 \begin{corollary}\label{cor:criterionThomeqII}  Leonard Guetta committed Oct 14, 2020 934  Let $F : C \to D$ be a $2$-functor. If for every $k \geq 0$,  Leonard Guetta committed Oct 16, 2020 935  $V_k(F) : V_k(C) \to V_k(D)$ is a Thomason equivalence of $1$-categories,  Leonard Guetta committed Oct 14, 2020 936  then $F$ is a Thomason equivalence of $2$-categories.  Leonard Guetta committed Jul 23, 2020 937 938 939 940 \end{corollary} \begin{proof} From Lemma \ref{lemma:binervehorizontal}, we now that for every $m \geq 0$, $ Leonard Guetta committed Oct 16, 2020 941  \binerve(C)_{\bullet,m}=N(V_m(C)).  Leonard Guetta committed Jul 23, 2020 942 943 $ The result follows them from Lemma \ref{bisimpliciallemma}.  Leonard Guetta committed Jul 22, 2020 944 945 \end{proof} \begin{paragr}  Leonard Guetta committed Oct 14, 2020 946 947 948 949 950 951 952 953 954 955 956 957 958 959 960  It also follows from Lemma \ref{lemma:binervthom} that the bisimplicial nerve induces a morphism of op\nbd{}prederivators $\overline{\binerve} : \Ho(2\Cat^{\Th}) \to \Ho(\Psh{\Delta\times\Delta}).$ As we shall soon see, this morphism is an \emph{equivalence} of op-prederivators. First, consider the triangle of functors $\begin{tikzcd} 2\Cat \ar[rr,"\binerve"] \ar[dr,"N"'] & & \Psh{\Delta\times\Delta} \ar[ld,"\delta^*"] \\ &\Psh{\Delta}. \end{tikzcd}$ This triangle is \emph{not} commutative but it becomes commutative (up to an isomorphism) after localization.  Leonard Guetta committed Oct 06, 2020 961   Leonard Guetta committed Jul 22, 2020 962 963 964 965 \end{paragr} \begin{proposition}\label{prop:streetvsbisimplicial} The triangle of morphisms of op-prederivators $ Leonard Guetta committed Oct 14, 2020 966 967 968 969  \begin{tikzcd} \Ho(2\Cat^{\Th}) \ar[rr,"\overline{\binerve}"] \ar[dr,"\overline{N}"'] & & \Ho(\Psh{\Delta\times\Delta}^{\mathrm{diag}}) \ar[ld,"\overline{\delta^*}"] \\ &\Ho(\Psh{\Delta}) \end{tikzcd}  Leonard Guetta committed Jul 22, 2020 970 971 972 973 $ is commutative up to a canonical isomorphism. \end{proposition} \begin{proof}  Leonard Guetta committed Oct 14, 2020 974 975  It is a consequence of the results contained in \cite[Section 2]{bullejos2003geometry}.  Leonard Guetta committed Jul 22, 2020 976 \end{proof}  Leonard Guetta committed Jul 23, 2020 977 \begin{paragr}  Leonard Guetta committed Oct 14, 2020 978 979 980 981  Since $\overline{\delta^*}$ and $\overline{N}$ are equivalences of op-prederivators (Proposition \ref{prop:diageqderivator} and Theorem \ref{thm:gagna} respectively), it follows from the previous proposition that the morphism  Leonard Guetta committed Jul 22, 2020 982  \[  Leonard Guetta committed Oct 14, 2020 983 984  \overline{\binerve} : \Ho(2\Cat^{\Th