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\chapter{Homotopy and homology type of free $2$-categories}
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\section{Preliminaries: the case of free $1$-categories}\label{section:prelimfreecat}
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In this section, we review some homotopical results on free
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($1$-)categories that will be of great help in the sequel.
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\begin{paragr}
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  A \emph{reflexive graph} $G$ consists of the data of two sets $G_0$ and $G_1$
  together with
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  \begin{itemize}[label=-]
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  \item a ``source'' map $\src : G_1 \to G_0$,
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  \item a ``target'' map $\trgt : G_1 \to G_0$,
  \item a ``unit'' map $1_{(-)} : G_0 \to G_1$,
  \end{itemize}
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  such that for every $x \in G_0$,
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  \[
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    \src(1_{x}) = \trgt (1_{x}) = x.
  \]
  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.
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  There is a ``underlying reflexive graph'' functor
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  \[
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    U : \Cat \to \Rgrph,
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  \]
  which has a left adjoint
  \[
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    L : \Rgrph \to \Cat.
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  \]
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  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
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  \[
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    \begin{tikzcd}
      X_0 \ar[r,"f_1"]& X_1\ar[r,"f_2"]& X_2 \ar[r]& \cdots \ar[r] &X_{n-1}
      \ar[r,"f_n"]& X_{n}
    \end{tikzcd}
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  \]
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  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}
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\begin{lemma}
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  A category $C$ is free in the sense of \ref{def:freeoocat} if and only if
  there exists a reflexive graph $G$ such that
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  \[
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    C \simeq L(G).
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  \]
\end{lemma}
\begin{proof}
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  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 $C\simeq L(G)$.
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  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 $G$.
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\end{proof}
\begin{remark}
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  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}.
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\end{remark}
\begin{paragr}
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  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
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  \[
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    i^* : \Psh{\Delta} \to \Rgrph,
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  \]
  which, by the usual technique of Kan extensions, has a left adjoint
  \[
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    i_! : \Rgrph \to \Psh{\Delta}.
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  \]
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  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
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  degenerated for $k>1$. For future reference, we put here the following lemma.
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\end{paragr}
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\begin{lemma}\label{lemma:monopreserved}
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  The functor $i_! : \Rgrph \to \Psh{\Delta}$ preserves monomorphism.
\end{lemma}
\begin{proof}
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  What we need to show is that, given a morphism of presheaves
  \[
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    f : X \to Y,
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  \]
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  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.).
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\end{proof}
\begin{paragr}
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  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
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  \[
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    \begin{tikzcd}
      X_0 \ar[r,"f_1"]& X_1\ar[r,"f_2"]& X_2 \ar[r]& \cdots \ar[r] &X_{n-1}
      \ar[r,"f_n"]& X_{n}
    \end{tikzcd}
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  \]
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  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
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  \[
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    \Cat \overset{N}{\rightarrow} \Psh{\Delta} \overset{i^*}{\rightarrow} \Rgrph
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  \]
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  is nothing but the forgetful functor $U : \Cat \to \Rgrph$. Thus, the functor
  $L : \Rgrph \to \Cat$ is (isomorphic to) the composite of
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  \[
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    \Rgrph \overset{i_!}{\rightarrow} \Psh{\Delta} \overset{c}{\rightarrow}
    \Cat.
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  \]
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  We now review a construction of Dwyer and Kan from \cite{dwyer1980simplicial}.

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  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
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  \[
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    \begin{tikzcd}
      X_0 \ar[r,"f_1"]& X_1\ar[r,"f_2"]& X_2 \ar[r]& \cdots \ar[r] &X_{n-1}
      \ar[r,"f_n"]& X_{n}
    \end{tikzcd}
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  \]
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  of arrows of $L(G)$ such that
  \[
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    \sum_{1 \leq i \leq n}\ell(f_i) \leq n.
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  \] In particular, we have
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  \[
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    N^1(G)=i_!(G)
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  \]
  and the transfinite composition of
  \[
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    i_!(G) = N^1(G) \hookrightarrow N^2(G) \hookrightarrow \cdots
    \hookrightarrow N^{k}(G) \hookrightarrow N^{k+1}(G) \hookrightarrow \cdots
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  \]
  is easily seen to be the map
  \[
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    \eta_{i_!(G)} : i_!(G) \to Nci_!(G),
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  \]
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  where $\eta$ is the unit of the adjunction $c \dashv N$.
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\end{paragr}
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\begin{lemma}[Dwyer-Kan]\label{lemma:dwyerkan}
  For every $k\leq 1$, the canonical inclusion map
  \[
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    N^{k}(G) \to N^{k+1}(G)
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  \]
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  is a trivial cofibration of simplicial sets.
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\end{lemma}
\begin{proof}
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  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
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  \[
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    A_{k+1} \hookrightarrow \Delta_{k+1}
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  \]
  is a trivial cofibration. Let $I_{k+1}$ be the set of chains
  \[
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    \begin{tikzcd}
      f = X_0 \ar[r,"f_1"]& X_1\ar[r,"f_2"]& X_2 \ar[r]& \cdots \ar[r] &X_{k-1}
      \ar[r,"f_k"]& X_{k}\ar[r,"f_{k+1}"]& X_{k+1}
    \end{tikzcd}
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  \]
  of arrows of $L(G)$ such that for every $1 \leq i \leq k+1$
  \[
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    \ell(f_i)=1,
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  \]
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  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:
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  \begin{itemize}
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  \item[-]$\varphi_{f}\vert_{\mathrm{Im}(\partial_0)}$ is the $k$-simplex of
    $N^{k}(G)$
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    \[
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      \begin{tikzcd}
        X_1\ar[r,"f_2"]& X_2 \ar[r]& \cdots \ar[r] &X_{k} \ar[r,"f_{k+1}"]&
        X_{k+1},
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      \end{tikzcd}
    \]
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  \item[-] $\varphi_{f}\vert_{\mathrm{Im}(\partial_{k+1})}$ is the $k$-simplex
    of $N^{k}(G)$
    \[
      \begin{tikzcd}
        X_0 \ar[r,"f_1"]& X_1\ar[r,"f_2"]& X_2 \ar[r]& \cdots \ar[r] &X_{k-1}
        \ar[r,"f_k"]& X_{k}.
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      \end{tikzcd}
    \]
  \end{itemize}
  Now, we have a cocartesian square
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  \[
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    \begin{tikzcd}
      \displaystyle \coprod_{f \in I_{k+1}}A_{k+1} \ar[d] \ar[r,"(\varphi_f)_f"] & N^{k}(G)\ar[d] \\
      \displaystyle \coprod_{f \in I_{k+1}}\Delta_{k+1} \ar[r] & N^{k+1}(G),
    \end{tikzcd}
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  \]
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  which proves that the right vertical arrow is a trivial cofibration.
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\end{proof}
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From this lemma, we deduce the following proposition.
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\begin{proposition}
  Let $G$ be a reflexive graph. The map
  \[
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    \eta_{i_!(G)} : i_!(G) \to Nci_!(G),
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  \]
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  where $\eta$ is the unit of the adjunction $c \dashv N$, is a trivial
  cofibration of simplicial sets.
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\end{proposition}
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\begin{proof}
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  This follows from the fact that trivial cofibrations are stable by transfinite
  composition.
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\end{proof}
From the previous proposition, we deduce the following very useful corollary.
\begin{corollary}\label{cor:hmtpysquaregraph}
  Let
  \[
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    \begin{tikzcd}
      A \ar[d,"\alpha"] \ar[r,"\beta"] &B \ar[d,"\delta"] \\
      C \ar[r,"\gamma"]& D
    \end{tikzcd}
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  \]
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  be a cocartesian square in $\Rgrph$. If either $\alpha$ or $\beta$ is a
  monomorphism, then the induced square
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  \[
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    \begin{tikzcd}
      L(A) \ar[d,"L(\alpha)"] \ar[r,"L(\beta)"]& L(B) \ar[d,"L(\delta)"] \\
      L(C) \ar[r,"L(\gamma)"]& L(D)
    \end{tikzcd}
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  \]
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  is a \emph{homotopy} cocartesian square of $\Cat$ when equipped with Thomason
  equivalences.
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\end{corollary}
\begin{proof}
  Since the nerve $N$ induces an equivalence of op-prederivators
  \[
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    \Ho(\Cat^{\Th}) \to \Ho(\Psh{\Delta}),
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  \]
  it suffices to prove that the induced square of simplicial sets
  \[
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    \begin{tikzcd}
      NL(A) \ar[d,"NL(\alpha)"] \ar[r,"NL(\beta)"]& NL(B) \ar[d,"NL(\delta)"] \\
      NL(C) \ar[r,"NL(\gamma)"]& NL(D)
    \end{tikzcd}
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  \]
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  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
  \[
    \begin{tikzcd}
      i_!(A) \ar[d,"i_!(\alpha)"] \ar[r,"i_!(\beta)"] &i_!(B) \ar[d,"i_!(\delta)"] \\
      i_!(C) \ar[r,"i_!(\gamma)"]& i_!(D).
    \end{tikzcd}
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  \]
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  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}.
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\end{proof}
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\begin{paragr}
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  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}
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\begin{proposition}\label{prop:hmtpysquaregraphbetter}
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  Let
  \[
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    \begin{tikzcd}
      A \ar[d,"\alpha"] \ar[r,"\beta"] &B \ar[d,"\delta"] \\
      C \ar[r,"\gamma"]& D
    \end{tikzcd}
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  \]
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  be a cocartesian square in $\Rgrph$. Suppose that the following two conditions
  are satisfied
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  \begin{enumerate}[label=\alph*)]
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  \item Either $\alpha$ or $\beta$ is injective on objects.
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  \item Either $\alpha$ or $\beta$ is quasi-injective on arrows.
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  \end{enumerate}
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  Then, the square
  \[
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    \begin{tikzcd}
      L(A) \ar[d,"L(\alpha)"] \ar[r,"L(\beta)"] &L(B) \ar[d,"L(\delta)"] \\
      L(C) \ar[r,"L(\gamma)"] &L(D)
    \end{tikzcd}
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  \]
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  is a \emph{homotopy} cocartesian square of $\Cat$ when equipped with Thomason
  equivalences.
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\end{proposition}
\begin{proof}
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  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.
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  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
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  \[
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    \begin{tikzcd}
      \displaystyle\coprod_{x \in E}F_x \ar[r] \ar[d] & E \ar[d]\\
      A \ar[r] & G, \ar[from=1-1,to=2-2,phantom,very near end,"\ulcorner"]
    \end{tikzcd}
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  \]
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  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.
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  Now, we have the following solid arrow commutative diagram
  \[
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    \begin{tikzcd}
      \displaystyle\coprod_{x \in E}F_x \ar[r] \ar[d] & E  \ar[ddr,bend left]\ar[d]&\\
      A  \ar[drr,bend right,"\beta"'] \ar[r] & G \ar[dr, dotted]&\\
      &&B, \ar[from=1-1,to=2-2,phantom,very near end,"\ulcorner"]
    \end{tikzcd}
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  \]
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  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 $\Rgrph$.
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  By forming successive cocartesian square and combining with the square
  obtained earlier, we obtain a diagram of three cocartesian square:
  \[
    \begin{tikzcd}[row sep = large]
      \displaystyle\coprod_{x \in E}F_x \ar[r] \ar[d] & E \ar[d]&\\
      A \ar[d,"\alpha"] \ar[r] & G \ar[d] \ar[r] & B \ar[d,"\delta"]\\
      C \ar[r] & H \ar[r] & D. \ar[from=1-1,to=2-2,phantom,"\ulcorner" very near
      end,"\text{\textcircled{\tiny \textbf{1}}}" near start, description]
      \ar[from=2-1,to=3-2,phantom,"\ulcorner" very near
      end,"\text{\textcircled{\tiny \textbf{2}}}", description]
      \ar[from=2-2,to=3-3,phantom,"\ulcorner" very near
      end,"\text{\textcircled{\tiny \textbf{3}}}", description]
    \end{tikzcd}
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  \]
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  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.
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\end{proof}
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We now apply Corollary \ref{cor:hmtpysquaregraph} and Proposition
\ref{prop:hmtpysquaregraphbetter} to a few examples.
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\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}
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\begin{example}[Adding a generator]
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  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 $A \to
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  B$, i.e.\ defined with the following cocartesian square:
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  \[
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    \begin{tikzcd}
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      \sS_0 \ar[d,"i_1"] \ar[r,"{\langle A, B \rangle}"] & C \ar[d] \\
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      \sD_1 \ar[r] & C'.
    \end{tikzcd}
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  \]
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  Then, this square is homotopy cocartesian in $\Cat$ (when equipped with the
  Thomason equivalences). Indeed, it obviously is the image of a square of
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  $\Rgrph$ by the functor $L$ and the morphism $i_1 : \sS_0 \to \sD_1$
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  comes from a monomorphism of $\Rgrph$. Hence, we can apply Corollary
  \ref{cor:hmtpysquaregraph}.
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\end{example}
\begin{remark}
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  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{}).
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\end{remark}
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\begin{example}[Identifying two generators]
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  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
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  \[
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    \begin{tikzcd}
      \sS_1\ar[d] \ar[r,"{\langle f, g \rangle}"] &C \ar[d] \\
      \sD_1 \ar[r] & C',
    \end{tikzcd}
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  \]
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  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}.
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\end{example}
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\begin{example}[Killing a generator]\label{example:killinggenerator}
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  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:
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  \[
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    \begin{tikzcd}
      \sD_1 \ar[d] \ar[r,"\langle f \rangle"] & C \ar[d] \\
      \sD_0 \ar[r] & C'.
    \end{tikzcd}
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  \]
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  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 $\Rgrph$.
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\end{example}
\begin{remark}
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  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.
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  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}
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\section{Preliminaries: bisimplicial sets}
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\begin{paragr}
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  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.
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  Note that for every $n\geq 0$, we have simplicial sets
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  \begin{align*}
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    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 $\Psh{\Delta\times\Delta}$.
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  \iffalse Moreover for every $n \geq 0$, if we fix the first variable to $n$,
  we obtain a simplicial set
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  \begin{align*}
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    X_{n,\bullet} : \Delta^{\op} &\to \Set \\
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    [m] &\mapsto X_{n,m}.
  \end{align*}
  Similarly, if we fix the second variable to $n$, we obtain a simplicial
  \begin{align*}
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    X_{\bullet,n} : \Delta^{\op} &\to \Set \\
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    [m] &\mapsto X_{m,n}.
  \end{align*}
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  The correspondences
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  \[
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    n \mapsto X_{n,\bullet} \,\text{ and }\, n\mapsto X_{\bullet,n}
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  \]
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  actually define functors $\Delta \to \Psh{\Delta}$. They correspond to the two
  ``currying'' operations
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  \[
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    \Psh{\Delta\times\Delta} \to \underline{\Hom}(\Delta^{\op},\Psh{\Delta}),
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  \]
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  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
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\end{paragr}
\begin{paragr}
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  The functor
  \begin{align*}
    \delta : \Delta &\to \Delta\times\Delta \\
    [n] &\mapsto ([n],[n])
  \end{align*}
  induces by pre-composition a functor
  \[
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    \delta^* : \Psh{\Delta\times\Delta} \to \Psh{\Delta}.
  \]
  By the usual calculus of Kan extensions, $\delta^*$ admits a left adjoint
  $\delta_!$ and a right adjoint $\delta_*$
  \[
    \delta_! \dashv \delta^* \dashv \delta_*.
  \]
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  We say that a morphism a bisimplicial sets, $f : X \to Y$, is a \emph{diagonal
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    weak equivalence} (resp.\ \emph{diagonal fibration}) when $\delta^*(f)$ is a
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  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}.
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\end{paragr}
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\begin{proposition}\label{prop:diageqderivator}
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  Consider that $\Psh{\Delta\times\Delta}$ is equipped with the diagonal model
  structure. Then, the adjunction
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  \[
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    \begin{tikzcd}
      \delta_! : \Psh{\Delta} \ar[r,shift left] & \Psh{\Delta\times\Delta}
      \ar[l,shift left]: \delta^*,
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    \end{tikzcd}
  \]
  is a Quillen equivalence.
\end{proposition}
\begin{proof}
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  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}.
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\end{proof}
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\begin{paragr}
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  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.
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\end{paragr}
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Diagonal weak equivalences are not the only interesting weak equivalences for
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bisimplicial sets.
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\begin{paragr}
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  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
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  \[
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    f_{\bullet,n} : X_{\bullet,n} \to Y_{\bullet,n}
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  \]
  (resp.
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  \[
    f_{n,\bullet} : X_{n,\bullet} \to Y_{n,\bullet})
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  \]
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  is a weak equivalence of simplicial sets. Recall now a very useful lemma.
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\end{paragr}
\begin{lemma}\label{bisimpliciallemma}
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  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.
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\end{lemma}
\begin{proof}
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  See for example \cite[Chapter XII,4.3]{bousfield1972homotopy} or
  \cite[Proposition 2.1.7]{cisinski2004localisateur}.
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\end{proof}
\begin{paragr}
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  In particular, the identity functor of the category of bisimplicial sets
  induces morphisms of op-prederivators:
  \[
    \Ho(\Psh{\Delta\times\Delta}^{\mathrm{vert}}) \to
    \Ho(\Psh{\Delta\times\Delta}^{\mathrm{diag}})
  \]
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  and
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  \[
    \Ho(\Psh{\Delta\times\Delta}^{\mathrm{hor}}) \to
    \Ho(\Psh{\Delta\times\Delta}^{\mathrm{diag}}).
  \]
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\end{paragr}
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\begin{proposition}\label{prop:bisimplicialcocontinuous}
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  The morphisms of op-prederivators
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  \[
    \Ho(\Psh{\Delta\times\Delta}^{\mathrm{vert}}) \to
    \Ho(\Psh{\Delta\times\Delta}^{\mathrm{diag}})
  \]
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  and
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  \[
    \Ho(\Psh{\Delta\times\Delta}^{\mathrm{hor}}) \to
    \Ho(\Psh{\Delta\times\Delta}^{\mathrm{diag}})
  \]
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  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}).
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  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
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  \[
    \begin{tikzcd}
      \delta^* : \Psh{\Delta\times\Delta} \ar[r,shift left] & \ar[l,shift left]
      \Psh{\Delta} : \delta_*
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    \end{tikzcd}
  \]
  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
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  we have commutative triangles
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  \[
    \begin{tikzcd}
      \Ho(\Psh{\Delta\times\Delta}^{\mathrm{vert}}) \ar[r]
      \ar[rd,"\overline{\delta^*}"']&
      \Ho(\Psh{\Delta\times\Delta}^{\mathrm{diag}})
      \ar[d,"\overline{\delta^*}"] \\
      &\Ho(\Psh{\Delta})
    \end{tikzcd}
  \]
  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
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    \[
      \begin{tikzcd}
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        A_{\bullet,n} \ar[r,"{u_{\bullet,n}}"]\ar[d,"{f_{\bullet,n}}"] & B_{\bullet,n} \ar[d,"{g_{\bullet,n}}"] \\
        C_{\bullet,n} \ar[r,"{v_{\bullet,n}}"] & D_{\bullet,n}
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      \end{tikzcd}
    \]
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    is homotopy cocartesian.
  \item For every $n\geq 0$, the square of simplicial sets
    \[
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      \begin{tikzcd}
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        A_{n,\bullet} \ar[r,"{u_{n,\bullet}}"]\ar[d,"{f_{n,\bullet}}"] & B_{n,\bullet} \ar[d,"{g_{n,\bullet}}"] \\
        C_{n,\bullet} \ar[r,"{v_{n,\bullet}}"] & D_{n,\bullet}
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      \end{tikzcd}
    \]
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    is homotopy cocartesian.
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  \end{enumerate}
  Then, the square
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  \[
    \begin{tikzcd}
      \delta^*(A) \ar[r,"\delta^*(u)"]\ar[d,"\delta^*(f)"] & \delta^*(B) \ar[d,"\delta^*(g)"] \\
      \delta^*(C) \ar[r,"\delta^*(v)"] & \delta^*(D)
    \end{tikzcd}
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  \]
  is a homotopy cocartesian square of simplicial sets.
\end{corollary}
\begin{proof}
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  From \cite[Corollary 10.3.10(i)]{groth2013book} we know that the square of
  bisimplicial sets
  \[
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    \begin{tikzcd}
      A \ar[r,"u"]\ar[d,"f"] & B \ar[d,"g"] \\
      C \ar[r,"v"] & D
    \end{tikzcd}
  \]
  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}.
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\end{proof}
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\section{Bisimplicial nerve for 2-categories}\label{section:bisimplicialnerve}
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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}.
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\begin{notation}
  \begin{itemize}
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  \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
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    \[
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      N(C)_k
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    \]
    is the set of $k$-simplices of the nerve of $C$.
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  \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}.
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  \item[-] For a $2$-category $C$ and $x$ and $y$ objects of $C$, we denote by
    \[
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      C(x,y)
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    \]
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    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 $C$.
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  \end{itemize}
\end{notation}
\begin{paragr}
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  Each $2$-category $C$ defines a simplicial object in $\Cat$,
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  \[S(C): \Delta^{\op} \to \Cat,\] where, for each $n \geq 0$,
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  \[
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    S_n(C):= \coprod_{(x_0,\cdots,x_n)\in \Ob(C)^{\times (n+1)}}C(x_0,x_1)
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    \times \cdots \times C(x_{n-1},x_n).
  \]
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  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
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  induced by the units for the horizontal composition.
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  Post-composing $S(C)$ with the nerve functor $N : \Cat \to \Psh{\Delta}$, we
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  obtain a functor
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  \[
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    NS(C) : \Delta^{\op} \to \Psh{\Delta}.
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  \]
\end{paragr}
\begin{remark}
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  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
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  category.
\end{remark}
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\begin{definition}
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  The \emph{bisimplicial nerve} of a $2$-category $C$ is the bisimplicial set
  $\binerve(C)$ defined as
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  \[
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    \binerve(C)_{n,m}:=N(S_n(C))_m,
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  \]
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  for all $n,m \geq 0$.
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\end{definition}
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\begin{paragr}\label{paragr:formulabisimplicialnerve}
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  In other words, the bisimplicial nerve of $C$ is obtained by ``un-curryfying''
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  the functor $NS(C) : \Delta^{op} \to \Psh{\Delta}$.
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  Since the nerve $N$ commutes with products and sums, we obtain the formula
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  \begin{equation}\label{fomulabinerve}
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    \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.
  \end{equation}
  More intuitively, an element of $\binerve(C)_{n,m}$ consists of a ``pasting
  scheme'' in $C$ that looks like
  \[
    m \underbrace{\left\{\begin{tikzcd}[column sep=huge,ampersand
          replacement=\&] \bullet \ar[r,"\vdots"]\ar[r,bend
          left=90,looseness=1.4,""{name=A,below}] \ar[r,bend
          left=35,""{name=B,above}] \ar[r,bend
          right=35,"\vdots",""{name=G,below}]\ar[r,bend
          right=90,looseness=1.4,""{name=H,above}] \&
          \bullet\ar[r,"\vdots"]\ar[r,bend
          left=90,looseness=1.4,""{name=C,below}] \ar[r,bend
          left=35,""{name=D,above}] \ar[r,bend
          right=35,"\vdots",""{name=I,below}]\ar[r,bend
          right=90,looseness=1.4,""{name=J,above}]
          \&\bullet\ar[r,phantom,description,"\cdots"]\&\bullet\ar[r,"\vdots"]\ar[r,bend
          left=90,looseness=1.4,""{name=E,below}] \ar[r,bend
          left=35,""{name=F,above}] \ar[r,bend
          right=35,"\vdots",""{name=K,below}]\ar[r,bend
          right=90,looseness=1.4,""{name=L,above}] \&\bullet
          \ar[from=A,to=B,Rightarrow] \ar[from=C,to=D,Rightarrow]
          \ar[from=E,to=F,Rightarrow] \ar[from=G,to=H,Rightarrow]
          \ar[from=I,to=J,Rightarrow] \ar[from=K,to=L,Rightarrow]
        \end{tikzcd}\right.}_{ n }.
  \]
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\end{paragr}
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In the definition of the bisimplicial nerve of a $2$\nbd{}category we gave, one
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direction of the bisimplicial set is privileged over the other. We now give
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an equivalent definition of the bisimplicial nerve which uses the other direction.
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\begin{paragr}
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  Let $C$ be a $2$\nbd{}category. For every $k \geq 1$, we define a
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  $1$\nbd{}category $V_k(C)$ in the following fashion:
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  \begin{itemize}[label=-]
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  \item The objects of $V_k(C)$ are the objects of $C$.
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  \item A morphism $\alpha$ is a sequence
    \[
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      \alpha=(\alpha_1,\alpha_2,\cdots,\alpha_k)
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    \]
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    of $2$-cells of $C$ that are vertically composable, i.e.\ such that for
    every $1 \leq i \leq k-1$,
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    \[
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      \src(\alpha_i)=\trgt(\alpha_{i+1}).
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    \]
    The source and target of alpha are given by
    \[
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      \src(\alpha):=\src_0(\alpha_1)\text{ and
      }\trgt(\alpha):=\trgt_0(\alpha_1).
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    \]
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    (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.)
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  \item Composition is given by
    \[
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      (\alpha_1,\alpha_2,\cdots,\alpha_k)\circ(\beta_1,\beta_2,\cdots,\beta_k):=(\alpha_1\comp_0\beta_1,\alpha_2\comp_0\beta_2,\cdots,\alpha_k\comp_0\beta_k)
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    \]
    and the unit on an object $x$ is the sequence
    \[
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      (1^2_x,\cdots, 1^2_x).
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    \]
  \end{itemize}
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  For $k=0$, we define $V_0(C)$ to be the category obtained from $C$ by simply
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  forgetting the $2$\nbd{}cells (which is nothing but $\tau^{s}_{\leq 1}(C)$
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  with the notations of \ref{paragr:defncat}). The correspondence $n \mapsto
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  V_n(C)$ defines to a simplicial object in $\Cat$
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  \[
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    V(C) : \Delta^{\op} \to \Cat,
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  \]
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  where the face operators are induced by the vertical composition and the
  degeneracy operators are induced by the units for the vertical composition.
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\end{paragr}
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\begin{lemma}\label{lemma:binervehorizontal}
  Let $C$ be a $2$-category. For every $n \geq 0$, we have
  \[
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    N(V_m(C))_n=(\binerve(C))_{n,m}.
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  \]
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\end{lemma}
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\begin{proof}
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  This is simply a reformulation of the formula given in Paragraph
  \ref{paragr:formulabisimplicialnerve}.
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\end{proof}
\begin{paragr}
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  The bisimplicial nerve canonically defines a functor
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  \[
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    \binerve : 2\Cat \to \Psh{\Delta\times\Delta}
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  \]
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  which enables us to compare the homotopy theory of $2\Cat$ with the homotopy
  theory of bisimplicial sets.
\end{paragr}
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\begin{lemma}\label{lemma:binervthom}
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  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.
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\end{lemma}
\begin{proof}
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  It follows from what is shown in \cite[Section 2]{bullejos2003geometry} that
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  there is a zigzag of weak equivalence of simplicial sets
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  \[
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    \delta^*(\binerve(C)) \leftarrow \cdots \rightarrow N(C)
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  \]
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  which is natural in $C$. This implies what we wanted to show. See also
  \cite[Théorème 3.13]{ara2020comparaison}.
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\end{proof}
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From this lemma, we deduce two useful criteria to detect Thomason equivalences
of $2$-categories.
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\begin{corollary}\label{cor:criterionThomeqI}
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  Let $F : C \to D$ be a $2$-functor. If
  \begin{enumerate}[label=\alph*)]
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dodo    
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  \item $F_0 : C_0 \to D_0$ is a bijection,
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  \end{enumerate}
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  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.
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\end{corollary}
\begin{proof}
  By definition, for every $2$-category $C$ and every $m \geq 0$, we have
  \[
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    (\binerve(C))_{\bullet,m} = NS(C).
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  \]
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  The result follows then from Lemma \ref{bisimpliciallemma} and the fact that
  weak equivalences of simplicial sets are stable by coproducts and finite
  products.
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\end{proof}
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\begin{corollary}\label{cor:criterionThomeqII}
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  Let $F : C \to D$ be a $2$-functor. If for every $k \geq 0$,
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  \[V_k(F) : V_k(C) \to V_k(D)\] is a Thomason equivalence of $1$-categories,
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  then $F$ is a Thomason equivalence of $2$-categories.
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\end{corollary}
\begin{proof}
  From Lemma \ref{lemma:binervehorizontal}, we now that for every $m \geq 0$,
  \[
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    \binerve(C)_{\bullet,m}=N(V_m(C)).
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  \]
  The result follows them from Lemma \ref{bisimpliciallemma}.
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\end{proof}
\begin{paragr}
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  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.
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\end{paragr}
\begin{proposition}\label{prop:streetvsbisimplicial}
  The triangle of morphisms of op-prederivators
  \[
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    \begin{tikzcd}
      \Ho(2\Cat^{\Th}) \ar[rr,"\overline{\binerve}"] \ar[dr,"\overline{N}"'] & & \Ho(\Psh{\Delta\times\Delta}^{\mathrm{diag}}) \ar[ld,"\overline{\delta^*}"] \\
      &\Ho(\Psh{\Delta})
    \end{tikzcd}
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  \]
  is commutative up to a canonical isomorphism.
\end{proposition}
\begin{proof}
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  It is a consequence of the results contained in \cite[Section
  2]{bullejos2003geometry}.
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\end{proof}
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\begin{paragr}
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  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
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  \[
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    \overline{\binerve} : \Ho(2\Cat^{\Th