2cat.tex

\chapter{Homotopy and homology type of free 2-categories}
\chaptermark{Homology of free $2$-categories}
\section{Preliminaries: the case of free 1-categories}\label{section:prelimfreecat}
In this section, we review some homotopical results on free
($1$-)categories that will be of great help in the sequel.
\begin{paragr}
  A \emph{reflexive graph} $G$ consists of the data of two sets $G_0$ and $G_1$
  together with
  \begin{itemize}[label=-]
  \item a ``source'' map $\src : G_1 \to G_0$,
  \item a ``target'' map $\trgt : G_1 \to G_0$,
  \item a ``unit'' map $1_{(-)} : G_0 \to G_1$,
  \end{itemize}
  such that for every $x \in G_0$,
  \[
    \src(1_{x}) = \trgt (1_{x}) = x.
  \]
  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.
  
  There is a ``underlying reflexive graph'' functor
  \[
    U : \Cat \to \Rgrph,
  \]
  which has a left adjoint
  \[
    L : \Rgrph \to \Cat.
  \]
  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
  \[
    \begin{tikzcd}
      X_0 \ar[r,"f_1"]& X_1\ar[r,"f_2"]& X_2 \ar[r]& \cdots \ar[r] &X_{n-1}
      \ar[r,"f_n"]& X_{n}
    \end{tikzcd}
  \]
  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}
\begin{lemma}
  A category $C$ is free in the sense of \ref{def:freeoocat} if and only if
  there exists a reflexive graph $G$ such that
  \[
    C \simeq L(G).
  \]
\end{lemma}
\begin{proof}
  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 $C\simeq L(G)$.

  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
  set of non unital $1$-cells of $G$.
\end{proof}
\begin{remark}
  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}.
\end{remark}
\begin{paragr}
  There is another important description of the category $\Rgrph$. Write
  $\Delta_{\leq 1}$ for the full subcategory of $\Delta$ spanned by $[0]$ and $[1]$.
  The category $\Rgrph$ is nothing but $\Psh{\Delta_{\leq 1}}$, the
  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
  \[
    i^* : \Psh{\Delta} \to \Rgrph,
  \]
  which, by the usual technique of Kan extensions, has a left adjoint
  \[
    i_! : \Rgrph \to \Psh{\Delta}.
  \]
  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
  degenerate for $k>1$. For future reference, we put here the following lemma.
\end{paragr}
\begin{lemma}\label{lemma:monopreserved}
  The functor $i_! : \Rgrph \to \Psh{\Delta}$ preserves monomorphisms.
\end{lemma}
\begin{proof}
  What we need to show is that, given a morphism of simplicial sets
  \[
    f : X \to Y,
  \]
  if $f_0 : X_0 \to Y_0$ and $f_1 : X_1 \to Y_1$ are monomorphisms and if all
  $n$\nbd{}simplices of $X$ are degenerate 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.).
\end{proof}
\begin{paragr}
  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
  \[
    \begin{tikzcd}
      X_0 \ar[r,"f_1"]& X_1\ar[r,"f_2"]& X_2 \ar[r]& \cdots \ar[r] &X_{n-1}
      \ar[r,"f_n"]& X_{n}
    \end{tikzcd}
  \]
  of arrows of $C$. Such an $n$-simplex is degenerate if and only if at least
  one of the $f_k$ is a unit. It is straightforward to check that the composite
  of
  \[
    \Cat \overset{N}{\rightarrow} \Psh{\Delta} \overset{i^*}{\rightarrow} \Rgrph
  \]
  is nothing but the forgetful functor $U : \Cat \to \Rgrph$. Thus, the functor
  $L : \Rgrph \to \Cat$ is (isomorphic to) the composite of
  \[
    \Rgrph \overset{i_!}{\rightarrow} \Psh{\Delta} \overset{c}{\rightarrow}
    \Cat.
  \]

  We now review a construction due to Dwyer and Kan
  (\cite{dwyer1980simplicial}). Let $G$ be a reflexive graph. For
  every $k\geq 1$, we define the simplicial set $N^k(G)$ as the
  sub-simplicial set of $N(L(G))$ whose $n$-simplices are chains
  \[
    \begin{tikzcd}
      X_0 \ar[r,"f_1"]& X_1\ar[r,"f_2"]& X_2 \ar[r]& \cdots \ar[r] &X_{n-1}
      \ar[r,"f_n"]& X_{n}
    \end{tikzcd}
  \]
  of arrows of $L(G)$ such that
  \[
    \sum_{1 \leq i \leq n}\ell(f_i) \leq k.
  \] In particular, we have
  \[
    N^1(G)=i_!(G)
  \]
  and the transfinite composition of
  \[
    i_!(G) = N^1(G) \hookrightarrow N^2(G) \hookrightarrow \cdots
    \hookrightarrow N^{k}(G) \hookrightarrow N^{k+1}(G) \hookrightarrow \cdots
  \]
  is easily seen to be the map
  \[
    \eta_{i_!(G)} : i_!(G) \to Nci_!(G),
  \]
  where $\eta$ is the unit of the adjunction $c \dashv N$.
\end{paragr}
\begin{lemma}[Dwyer--Kan]\label{lemma:dwyerkan}
  For every $k\geq 1$, the canonical inclusion map
  \[
    N^{k}(G) \to N^{k+1}(G)
  \]
  is a trivial cofibration of simplicial sets.
\end{lemma}
\begin{proof}
  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
  \[
    A_{k+1} \hookrightarrow \Delta_{k+1}
  \]
  is a trivial cofibration. Let $I_{k+1}$ be the set of chains
  \[
    \begin{tikzcd}
      f = X_0 \ar[r,"f_1"]& X_1\ar[r,"f_2"]& X_2 \ar[r]& \cdots \ar[r] &X_{k-1}
      \ar[r,"f_k"]& X_{k}\ar[r,"f_{k+1}"]& X_{k+1}
    \end{tikzcd}
  \]
  of arrows of $L(G)$ such that for every $1 \leq i \leq k+1$
  \[
    \ell(f_i)=1,
  \]
  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:
  \begin{itemize}
  \item[-]$\varphi_{f}\vert_{\mathrm{Im}(\partial_0)}$ is the $k$-simplex of
    $N^{k}(G)$
    \[
      \begin{tikzcd}
        X_1\ar[r,"f_2"]& X_2 \ar[r]& \cdots \ar[r] &X_{k} \ar[r,"f_{k+1}"]&
        X_{k+1},
      \end{tikzcd}
    \]
  \item[-] $\varphi_{f}\vert_{\mathrm{Im}(\partial_{k+1})}$ is the $k$-simplex
    of $N^{k}(G)$
    \[
      \begin{tikzcd}
        X_0 \ar[r,"f_1"]& X_1\ar[r,"f_2"]& X_2 \ar[r]& \cdots \ar[r] &X_{k-1}
        \ar[r,"f_k"]& X_{k}.
      \end{tikzcd}
    \]
  \end{itemize}
  All in all, we have a cocartesian square
  \[
    \begin{tikzcd}
      \displaystyle \coprod_{f \in I_{k+1}}A_{k+1} \ar[d] \ar[r,"(\varphi_f)_f"] & N^{k}(G)\ar[d] \\
      \displaystyle \coprod_{f \in I_{k+1}}\Delta_{k+1} \ar[r] & N^{k+1}(G),
      \ar[from=1-1,to=2-2,phantom,"\ulcorner",very near end]
    \end{tikzcd}
  \]
  which proves that the right vertical arrow is a trivial cofibration.
\end{proof}
From this lemma, we deduce the following proposition.
\begin{proposition}
  Let $G$ be a reflexive graph. The map
  \[
    \eta_{i_!(G)} : i_!(G) \to Nci_!(G),
  \]
  where $\eta$ is the unit of the adjunction $c \dashv N$, is a trivial
  cofibration of simplicial sets.
\end{proposition}
\begin{proof}
  This follows from the fact that trivial cofibrations are stable by transfinite
  composition.
\end{proof}
From the previous proposition, we deduce the following very useful corollary.
\begin{corollary}\label{cor:hmtpysquaregraph}
  Let
  \[
    \begin{tikzcd}
      A \ar[d,"\alpha"] \ar[r,"\beta"] &B \ar[d,"\delta"] \\
      C \ar[r,"\gamma"]& D
    \end{tikzcd}
  \]
  be a cocartesian square in $\Rgrph$. If either $\alpha$ or $\beta$ is a
  monomorphism, then the induced square of $\Cat$
  \[
    \begin{tikzcd}
      L(A) \ar[d,"L(\alpha)"] \ar[r,"L(\beta)"]& L(B) \ar[d,"L(\delta)"] \\
      L(C) \ar[r,"L(\gamma)"]& L(D)
    \end{tikzcd}
  \]
  is a Thomason homotopy cocartesian.
\end{corollary}
\begin{proof}
  Since the nerve $N$ induces an equivalence of op-prederivators
  \[
    \Ho(\Cat^{\Th}) \to \Ho(\Psh{\Delta}),
  \]
  it suffices to prove that the induced square of simplicial sets
  \[
    \begin{tikzcd}
      NL(A) \ar[d,"NL(\alpha)"] \ar[r,"NL(\beta)"]& NL(B) \ar[d,"NL(\delta)"] \\
      NL(C) \ar[r,"NL(\gamma)"]& NL(D)
    \end{tikzcd}
  \]
  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
  \[
    \begin{tikzcd}
      i_!(A) \ar[d,"i_!(\alpha)"] \ar[r,"i_!(\beta)"] &i_!(B) \ar[d,"i_!(\delta)"] \\
      i_!(C) \ar[r,"i_!(\gamma)"]& i_!(D).
    \end{tikzcd}
  \]
  This square is cocartesian because $i_!$ is a left adjoint. Since
  $i_!$ preserves monomorphisms (Lemma \ref{lemma:monopreserved}), the
  result follows from the fact that the monomorphisms are the
  cofibrations of the standard Quillen model structure on simplicial
  sets and from Lemma \ref{lemma:hmtpycocartesianreedy}.
\end{proof}
\begin{paragr}
  By working a little more, we obtain the more general result 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 sends 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}
\begin{proposition}\label{prop:hmtpysquaregraphbetter}
  Let
  \[
    \begin{tikzcd}
      A \ar[d,"\alpha"] \ar[r,"\beta"] &B \ar[d,"\delta"] \\
      C \ar[r,"\gamma"]& D
      \ar[from=1-1,to=2-2,phantom,"\ulcorner",very near end]
    \end{tikzcd}
  \]
  be a cocartesian square in $\Rgrph$. Suppose that the following two conditions
  are satisfied
  \begin{enumerate}[label=\alph*)]
  \item Either $\alpha$ or $\beta$ is injective on objects.
  \item Either $\alpha$ or $\beta$ is quasi-injective on arrows.
  \end{enumerate}
  Then, the induced square of $\Cat$
  \[
    \begin{tikzcd}
      L(A) \ar[d,"L(\alpha)"] \ar[r,"L(\beta)"] &L(B) \ar[d,"L(\delta)"] \\
      L(C) \ar[r,"L(\gamma)"] &L(D)
    \end{tikzcd}
  \]
  is Thomason homotopy cocartesian square.
\end{proposition}
\begin{proof}
  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.

  Let use denote by $E$ the set of objects of $B$ that are in the image of
  $\beta$. We consider this set as well as the set $A_0$ of objects of $A$ as discrete reflexive graphs, i.e.\ reflexive graphs
  with no non-unit arrows. Now, let $G$ be the reflexive graph defined by the
  following cocartesian square
  \[
    \begin{tikzcd}
      A_0\ar[r] \ar[d] & E \ar[d]\\
      A \ar[r] & G, \ar[from=1-1,to=2-2,phantom,very near end,"\ulcorner"]
    \end{tikzcd}
  \]
  where the morphism \[ A_0 \to A\] is the canonical inclusion, and the
  morphism \[A_0 \to E\] is induced by the restriction of $\beta$ on objects. In other words, $G$ is
  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-unit arrows of $G$ is (isomorphic to) the set of
  non-unit arrows of $A$; the source (resp.\ target) of a non-unit arrow $f$ of
  $G$ is the source (resp.\ target) of $\beta(f)$. This completely describes $G$.
  % 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.

  Now, we have the following solid arrow commutative diagram
  \[
    \begin{tikzcd}
      A_0 \ar[r] \ar[d] & E  \ar[ddr,bend left]\ar[d]&\\
      A  \ar[drr,bend right,"\beta"'] \ar[r] & G \ar[dr, dotted]&\\
      &&B, \ar[from=1-1,to=2-2,phantom,very near end,"\ulcorner"]
    \end{tikzcd}
  \]
  where the arrow $E \to B$ is the canonical inclusion. Hence, by universal
  property, the dotted arrow exists and makes the whole diagram commute. A
  thorough verification easily shows that the morphism $G \to B$ is a
  monomorphism of $\Rgrph$.

  By forming successive cocartesian squares and combining with the square
  obtained earlier, we obtain a diagram of three cocartesian squares:
  \[
    \begin{tikzcd}[row sep = large]
      A_0\ar[r] \ar[d] & E \ar[d]&\\
      A \ar[d,"\alpha"] \ar[r] & G \ar[d] \ar[r] & B \ar[d,"\delta"]\\
      C \ar[r] & H \ar[r] & D. \ar[from=1-1,to=2-2,phantom,"\ulcorner" very near
      end,"\text{\textcircled{\tiny \textbf{1}}}" near start, description]
      \ar[from=2-1,to=3-2,phantom,"\ulcorner" very near
      end,"\text{\textcircled{\tiny \textbf{2}}}", description]
      \ar[from=2-2,to=3-3,phantom,"\ulcorner" very near
      end,"\text{\textcircled{\tiny \textbf{3}}}", description]
    \end{tikzcd}
  \]
  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, the morphisms
  \[
    A_0 \to A 
  \]
  and
  \[
    A_0 \to C
  \]
  are monomorphisms and thus, using Corollary
  \ref{cor:hmtpysquaregraph}, we deduce that the image by $L$ of square
  \textcircled{\tiny \textbf{1}} and of the pasting of squares
  \textcircled{\tiny \textbf{1}} and \textcircled{\tiny \textbf{2}}
  are homotopy cocartesian. By Lemma \ref{lemma:pastinghmtpycocartesian} again, this proves that the image by $L$ of
  square \textcircled{\tiny \textbf{2}} is homotopy cocartesian.
\end{proof}
We now apply Corollary \ref{cor:hmtpysquaregraph} and Proposition
\ref{prop:hmtpysquaregraphbetter} to a few examples.
\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 by
  the following cocartesian square
  \[
    \begin{tikzcd}
      \sS_0 \ar[d] \ar[r,"{\langle A,B \rangle}"] & C \ar[d] \\
      \sD_0 \ar[r] & C'.
      \ar[from=1-1,to=2-2,phantom,"\ulcorner",very near end]
    \end{tikzcd}
  \]
  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. Hence, we can apply Corollary \ref{cor:hmtpysquaregraph}.
  \end{example}
\begin{example}[Adding a generator]
  Let $C$ be a free category, $A$ and $B$ two objects of $C$ (possibly equal)
  and let $C'$ be the category obtained from $C$ by adding a generator $A \to
  B$, i.e.\ defined by the following cocartesian square:
  \[
    \begin{tikzcd}
      \sS_0 \ar[d,"i_1"] \ar[r,"{\langle A, B \rangle}"] & C \ar[d] \\
      \sD_1 \ar[r] & C'.
      \ar[from=1-1,to=2-2,phantom,"\ulcorner",very near end]
    \end{tikzcd}
  \]
  Then, this square is Thomason homotopy cocartesian. Indeed, it obviously is the image of a square of $\Rgrph$ by
  the functor $L$ and the morphism $i_1 : \sS_0 \to \sD_1$ comes from
  a monomorphism of $\Rgrph$. Hence, we can apply Corollary
  \ref{cor:hmtpysquaregraph}.
\end{example}
\begin{remark}
  Since $i_1 : \sS_0 \to \sD_1$ is a folk cofibration% , since a Thomason homotopy
  % cocartesian square in $\Cat$ is also so in $\oo\Cat$
  and since every free category is obtained by recursively adding generators
  starting from a set of objects (seen as a $0$-category), the previous example
  yields another proof that \emph{free} (1\nbd{})categories are \good{} (which we
  already knew since we have seen that \emph{all} (1-)categories are \good{}).
\end{remark}
\begin{example}[Identifying two generators]
  Let $C$ be a free category and let $f,g : A \to B$ be 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 by the following cocartesian
  square
  \[
    \begin{tikzcd}
      \sS_1\ar[d] \ar[r,"{\langle f, g \rangle}"] &C \ar[d] \\
      \sD_1 \ar[r] & C',
      \ar[from=1-1,to=2-2,phantom,"\ulcorner",very near end]
    \end{tikzcd}
  \]
  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 Thomason homotopy cocartesian.
  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}.
\end{example}
\begin{example}[Killing a generator]\label{example:killinggenerator}
  Let $C$ be a free category and let $f : A \to B$ be one of its generating arrows
  such that $A \neq B$. Now consider the category $C'$ obtained from $C$ by
  ``killing'' $f$, i.e. defined by the following cocartesian square:
  \[
    \begin{tikzcd}
      \sD_1 \ar[d] \ar[r,"\langle f \rangle"] & C \ar[d] \\
      \sD_0 \ar[r] & C'.
      \ar[from=1-1,to=2-2,phantom,"\ulcorner",very near end]
    \end{tikzcd}
  \]
  Then, this above square is Thomason homotopy cocartesian. Indeed, it
  obviously is the image of a cocartesian square in $\Rgrph$ by the
  functor $L$ and since the source and target of $f$ are different,
  the top map comes from a monomorphism of $\Rgrph$. Hence, we can
  apply Corollary \ref{cor:hmtpysquaregraph}.
\end{example}
\begin{remark}
  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.

  Note also that the hypothesis that $A\neq B$ was fundamental in the previous
  example as for $A=B$ the square is \emph{not} Thomason homotopy cocartesian.
\end{remark}

\section{Preliminaries: bisimplicial sets}
\begin{paragr}
  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 notations
  \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.
 
  Note that for every $n\geq 0$, we have simplicial sets
  \begin{align*}
    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 $\Psh{\Delta\times\Delta}$.

  \iffalse Moreover for every $n \geq 0$, if we fix the first variable to $n$,
  we obtain a simplicial set
  \begin{align*}
    X_{n,\bullet} : \Delta^{\op} &\to \Set \\
    [m] &\mapsto X_{n,m}.
  \end{align*}
  Similarly, if we fix the second variable to $n$, we obtain a simplicial
  \begin{align*}
    X_{\bullet,n} : \Delta^{\op} &\to \Set \\
    [m] &\mapsto X_{m,n}.
  \end{align*}
  The correspondences
  \[
    n \mapsto X_{n,\bullet} \,\text{ and }\, n\mapsto X_{\bullet,n}
  \]
  actually define functors $\Delta \to \Psh{\Delta}$. They correspond to the two
  ``currying'' operations
  \[
    \Psh{\Delta\times\Delta} \to \underline{\Hom}(\Delta^{\op},\Psh{\Delta}),
  \]
  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
\end{paragr}
\begin{paragr}
  The functor
  \begin{align*}
    \delta : \Delta &\to \Delta\times\Delta \\
    [n] &\mapsto ([n],[n])
  \end{align*}
  induces by pre-composition a functor
  \[
    \delta^* : \Psh{\Delta\times\Delta} \to \Psh{\Delta}.
  \]
  By the usual calculus of Kan extensions, $\delta^*$ admits a left adjoint
  $\delta_!$ and a right adjoint $\delta_*$
  \[
    \delta_! \dashv \delta^* \dashv \delta_*.
  \]
  We say that a morphism $f : X \to Y$ of bisimplicial sets is a \emph{diagonal
    weak equivalence} (resp.\ \emph{diagonal fibration}) when $\delta^*(f)$ is a
  weak equivalence (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. We shall refer to this model
  structure as the \emph{diagonal model structure}.
\end{paragr}
\begin{proposition}\label{prop:diageqderivator}
  Consider that $\Psh{\Delta\times\Delta}$ is equipped with the diagonal model
  structure. Then, the adjunction
  \[
    \begin{tikzcd}
      \delta_! : \Psh{\Delta} \ar[r,shift left] & \Psh{\Delta\times\Delta}
      \ar[l,shift left]: \delta^*,
    \end{tikzcd}
  \]
  is a Quillen equivalence.
\end{proposition}
\begin{proof}
  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}.
\end{proof}
\begin{paragr}
  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.
\end{paragr}
Diagonal weak equivalences are not the only interesting weak equivalences for
bisimplicial sets.
\begin{paragr}
  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
  \[
    f_{\bullet,n} : X_{\bullet,n} \to Y_{\bullet,n}
  \]
  (resp.
  \[
    f_{n,\bullet} : X_{n,\bullet} \to Y_{n,\bullet})
  \]
  is a weak equivalence of simplicial sets. Recall now a very useful lemma.
\end{paragr}
\begin{lemma}\label{bisimpliciallemma}
  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.
\end{lemma}
\begin{proof}
  See for example \cite[Chapter XII,4.3]{bousfield1972homotopy} or
  \cite[Proposition 2.1.7]{cisinski2004localisateur}.
\end{proof}
\begin{paragr}
  In particular, the identity functor of the category of bisimplicial sets
  induces the morphisms of op-prederivators:
  \[
    \Ho(\Psh{\Delta\times\Delta}^{\mathrm{vert}}) \to
    \Ho(\Psh{\Delta\times\Delta}^{\mathrm{diag}})
  \]
  and
  \[
    \Ho(\Psh{\Delta\times\Delta}^{\mathrm{hor}}) \to
    \Ho(\Psh{\Delta\times\Delta}^{\mathrm{diag}}).
  \]
\end{paragr}
\begin{proposition}\label{prop:bisimplicialcocontinuous}
  The morphisms of op-prederivators
  \[
    \Ho(\Psh{\Delta\times\Delta}^{\mathrm{vert}}) \to
    \Ho(\Psh{\Delta\times\Delta}^{\mathrm{diag}})
  \]
  and
  \[
    \Ho(\Psh{\Delta\times\Delta}^{\mathrm{hor}}) \to
    \Ho(\Psh{\Delta\times\Delta}^{\mathrm{diag}})
  \]
  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}).
  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
  \[
    \begin{tikzcd}
      \delta^* : \Psh{\Delta\times\Delta} \ar[r,shift left] & \ar[l,shift left]
      \Psh{\Delta} : \delta_*
    \end{tikzcd}
  \]
  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. Now, the obvious identity
  $\delta^*=\delta^* \circ \mathrm{id}_{\Psh{\Delta\times\Delta}}$ implies that
  we have commutative triangles
  \[
    \begin{tikzcd}
      \Ho(\Psh{\Delta\times\Delta}^{\mathrm{vert}}) \ar[r]
      \ar[rd,"\overline{\delta^*}"']&
      \Ho(\Psh{\Delta\times\Delta}^{\mathrm{diag}})
      \ar[d,"\overline{\delta^*}"] \\
      &\Ho(\Psh{\Delta})
    \end{tikzcd}
  \]
  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 commutative square in the category of bisimplicial sets satisfying at least one of the two
  following conditions:
  \begin{enumerate}[label=(\alph*)]
  \item For every $n\geq 0$, the square of simplicial sets
    \[
      \begin{tikzcd}
        A_{\bullet,n} \ar[r,"{u_{\bullet,n}}"]\ar[d,"{f_{\bullet,n}}"] & B_{\bullet,n} \ar[d,"{g_{\bullet,n}}"] \\
        C_{\bullet,n} \ar[r,"{v_{\bullet,n}}"] & D_{\bullet,n}
      \end{tikzcd}
    \]
    is homotopy cocartesian.
  \item For every $n\geq 0$, the square of simplicial sets
    \[
      \begin{tikzcd}
        A_{n,\bullet} \ar[r,"{u_{n,\bullet}}"]\ar[d,"{f_{n,\bullet}}"] & B_{n,\bullet} \ar[d,"{g_{n,\bullet}}"] \\
        C_{n,\bullet} \ar[r,"{v_{n,\bullet}}"] & D_{n,\bullet}
      \end{tikzcd}
    \]
    is homotopy cocartesian.
  \end{enumerate}
  Then, the square
  \[
    \begin{tikzcd}
      \delta^*(A) \ar[r,"\delta^*(u)"]\ar[d,"\delta^*(f)"] & \delta^*(B) \ar[d,"\delta^*(g)"] \\
      \delta^*(C) \ar[r,"\delta^*(v)"] & \delta^*(D)
    \end{tikzcd}
  \]
  is a homotopy cocartesian square of simplicial sets.
\end{corollary}
\begin{proof}
  From \cite[Corollary 10.3.10(i)]{groth2013book} we know that the square of
  bisimplicial sets
  \[
    \begin{tikzcd}
      A \ar[r,"u"]\ar[d,"f"] & B \ar[d,"g"] \\
      C \ar[r,"v"] & D
    \end{tikzcd}
  \]
  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}.
\end{proof}
\section{Bisimplicial nerve for 2-categories}\label{section:bisimplicialnerve}
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}.
\begin{notation}
  \begin{itemize}
  \item[-] Once again, we write $N : \Cat \to \Psh{\Delta}$ instead of $N_1$ for
    the usual nerve of categories. Moreover, using the usual notation for the
    set of $k$-simplices of a simplicial set, if $C$ is a (small) category, then
    \[
      N(C)_k
    \]
    is the set of $k$-simplices of the nerve of $C$.
  \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}.
  \item[-] For a $2$-category $C$ and $x$ and $y$ objects of $C$, we denote by
    \[
      C(x,y)
    \]
    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 $C$.
  \end{itemize}
\end{notation}
\begin{paragr}
  Every $2$-category $C$ defines a simplicial object in $\Cat$,
  \[S(C): \Delta^{\op} \to \Cat,\] where, for each $n \geq 0$,
  \[
    S_n(C):= \coprod_{(x_0,\cdots,x_n)\in \Ob(C)^{\times (n+1)}}C(x_0,x_1)
    \times \cdots \times C(x_{n-1},x_n).
  \]
  Note that for $n=0$, the above formula reads $S_0(C)=C_0$. For $n>0$, the face operators  $\partial_i : S_{n}(C) \to S_{n-1}(C)$ are induced by horizontal
  composition for $0 < i <n$ and by projection for $i=0$ or $n$. The degeneracy operators $s_i \colon S_{n}(C) \to S_{n+1}(C)$ are
  induced by the units for the horizontal composition.

  Post-composing $S(C)$ with the nerve functor $N : \Cat \to \Psh{\Delta}$, we
  obtain a functor
  \[
    NS(C) : \Delta^{\op} \to \Psh{\Delta}.
  \]
\end{paragr}
\begin{remark}
  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
  category.
\end{remark}
\begin{definition}
  The \emph{bisimplicial nerve} of a $2$-category $C$ is the bisimplicial set
  $\binerve(C)$ defined as
  \[
    \binerve(C)_{n,m}:=N(S_n(C))_m,
  \]
  for all $n,m \geq 0$.
\end{definition}
\begin{paragr}\label{paragr:formulabisimplicialnerve}
  In other words, the bisimplicial nerve of $C$ is obtained by ``un-currying''
  the functor $NS(C) : \Delta^{op} \to \Psh{\Delta}$.
  
  Since the nerve $N$ commutes with products and sums, we obtain the formula
  \begin{equation}\label{fomulabinerve}
    \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
  \[
    m \underbrace{\left\{\begin{tikzcd}[column sep=huge,ampersand
          replacement=\&] \bullet \ar[r,"\vdots"]\ar[r,bend
          left=90,looseness=1.4,""{name=A,below}] \ar[r,bend
          left=35,""{name=B,above}] \ar[r,bend
          right=35,"\vdots",""{name=G,below}]\ar[r,bend
          right=90,looseness=1.4,""{name=H,above}] \&
          \bullet\ar[r,"\vdots"]\ar[r,bend
          left=90,looseness=1.4,""{name=C,below}] \ar[r,bend
          left=35,""{name=D,above}] \ar[r,bend
          right=35,"\vdots",""{name=I,below}]\ar[r,bend
          right=90,looseness=1.4,""{name=J,above}]
          \&\bullet\ar[r,phantom,description,"\cdots"]\&\bullet\ar[r,"\vdots"]\ar[r,bend
          left=90,looseness=1.4,""{name=E,below}] \ar[r,bend
          left=35,""{name=F,above}] \ar[r,bend
          right=35,"\vdots",""{name=K,below}]\ar[r,bend
          right=90,looseness=1.4,""{name=L,above}] \&\bullet
          \ar[from=A,to=B,Rightarrow] \ar[from=C,to=D,Rightarrow]
          \ar[from=E,to=F,Rightarrow] \ar[from=G,to=H,Rightarrow]
          \ar[from=I,to=J,Rightarrow] \ar[from=K,to=L,Rightarrow]
        \end{tikzcd}\right.}_{ n }.
  \]
\end{paragr}
In the definition of the bisimplicial nerve of a $2$\nbd{}category we gave, one
direction of the bisimplicial set is privileged over the other. We now give
an equivalent definition of the bisimplicial nerve which uses the other direction.
\begin{paragr}
  Let $C$ be a $2$\nbd{}category. For every $k \geq 1$, we define a
  $1$\nbd{}category $V_k(C)$ in the following fashion:
  \begin{itemize}[label=-]
  \item The objects of $V_k(C)$ are the objects of $C$.
  \item A morphism $\alpha$ is a sequence
    \[
      \alpha=(\alpha_1,\alpha_2,\cdots,\alpha_k)
    \]
    of vertically composable $2$-cells of $C$, i.e.\ such that for
    every $1 \leq i \leq k-1$, we have
    \[
      \src(\alpha_i)=\trgt(\alpha_{i+1}).
    \]
    The source and target of $\alpha$ are given by
    \[
      \src(\alpha):=\src_0(\alpha_1)\text{ and
      }\trgt(\alpha):=\trgt_0(\alpha_1).
    \]
    (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.)
  \item Composition is given by
    \[
      (\alpha_1,\alpha_2,\cdots,\alpha_k)\circ(\beta_1,\beta_2,\cdots,\beta_k):=(\alpha_1\comp_0\beta_1,\alpha_2\comp_0\beta_2,\cdots,\alpha_k\comp_0\beta_k)
    \]
    and the unit on an object $x$ is the sequence
    \[
      (1^2_x,\cdots, 1^2_x).