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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}
In this section, we review some homotopical results concerning free ($1$-)categories that will be of great help in the sequel.
\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
  \begin{itemize}
  \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 $x \in G_0$,
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  \[
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  \src(1_{x}) = \trgt (1_{x}) = x.
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  \]
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  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. There is a ``underlying reflexive graph'' functor
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  \[
  U : \Cat \to \Rgrph,
  \]
  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 of $L(G)$ is a chain
  \[
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  \begin{tikzcd}
  X_0 \ar[r,"f_1"]& X_1\ar[r,"f_2"]& X_2 \ar[r]& \cdots \ar[r] &X_{n-1} \ar[r,"f_n"]& X_{n} 
  \end{tikzcd}
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  \]
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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 $f$. Composition is given by concatenation of chains. 
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  \end{paragr}
\begin{lemma}
  A category $C$ is free in the sense of \todo{ref} 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 non unital $1$-cells of $G$.
\end{proof}
\begin{remark}
  Note that for a morphism of reflexive graphs $ f : G \to G'$, the functor $L(f)$ is not necessarily rigid in the sense of \todo{ref} because generating $1$-cells may be sent to units. 
\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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  \[
  i^* : \Psh{\Delta} \to \Rgrph,
  \]
  which, by the usual technique of Kan extensions, has a left adjoint
  \[
  i_! : \Rgrph \to \Psh{\Delta}.
  \]
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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 degenerated for $k>2$. 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}
  
\end{proof}
\begin{paragr}
  Let us denote by $N : \Psh{\Delta} \to \Cat$ (instead of $N_1$ as in Paragraph \todo{ref}) 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
  \[
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  \begin{tikzcd}
  X_0 \ar[r,"f_1"]& X_1\ar[r,"f_2"]& X_2 \ar[r]& \cdots \ar[r] &X_{n-1} \ar[r,"f_n"]& X_{n} 
  \end{tikzcd}
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  \]
  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
  \[
  \Cat \overset{N}{\rightarrow} \Psh{\Delta} \overset{i^*}{\rightarrow} \Rgrph
  \]
  is nothing but the forgetful functor $U : \Cat \to \Rgrph$. Thus, the functor $L : \Rgrph \to \Cat$ is (isomorphic to) the composite of
  \[
  \Rgrph \overset{i_!}{\rightarrow} \Psh{\Delta} \overset{c}{\rightarrow} \Cat.
  \]
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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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  \[
  \begin{tikzcd}
  X_0 \ar[r,"f_1"]& X_1\ar[r,"f_2"]& X_2 \ar[r]& \cdots \ar[r] &X_{n-1} \ar[r,"f_n"]& X_{n} 
  \end{tikzcd}
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  \]
  of arrows of $L(G)$ such that the length of each $f_i$ is \emph{at most} $k$. In particular, we have
  \[
  N^1(G)=i_!(G)
  \]
  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
  \[
  \eta_{i_!(G)} : i_!(G) \to Nci_!(G),
  \]
  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
  \[
  N^{k}(G) \to N^{k+1}(G)
  \]
  is a weak equivalence of simplicial sets.
\end{lemma}
\begin{proof}
  
\end{proof}
From this lemma, we deduce the following propositon.
\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 weak equivalence of simplicial sets.
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\end{proposition}
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\begin{proof}
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  It is an immediate consequence of Lemma \ref{lemma:dwyerkan} and the fact that filtered colimits are homotopic in a model category whose objects are all cofibrants \todo{ref}.
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\end{proof}
From the previous proposition, we deduce the following very useful corollary.
\begin{corollary}\label{cor:hmtpysquaregraph}
  Let
  \[
  \begin{tikzcd}
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    A \ar[d,"\alpha"] \ar[r,"\beta"] &B \ar[d,"\delta"] \\
    C \ar[r,"\gamma"]& D
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  \end{tikzcd}
  \]
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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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  \[
  \begin{tikzcd}
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    L(A) \ar[d,"L(\alpha)"] \ar[r,"L(\beta)"]& L(B) \ar[d,"L(\delta)"] \\
    L(C) \ar[r,"L(\gamma)"]& L(D)
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  \end{tikzcd}
  \]
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  is a \emph{homotopy} cocartesian square of $\Cat$ when equipped with the Thomason weak equivalences.
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\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
  \[
  \begin{tikzcd}
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    NL(A) \ar[d,"NL(\alpha)"] \ar[r,"NL(\beta)"]& NL(B) \ar[d,"NL(\delta)"] \\
    NL(C) \ar[r,"NL(\gamma)"]& NL(D)
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  \end{tikzcd}
  \]
  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
    \[
  \begin{tikzcd}
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    i_!(A) \ar[d,"i_!(\alpha)"] \ar[r,"i_!(\beta)"] &i_!(B) \ar[d,"i_!(\delta)"] \\
    i_!(C) \ar[r,"i_!(\gamma)"]& i_!(D).
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  \end{tikzcd}
  \]
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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 \todo{ref}.
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\end{proof}
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Actually, by working a little more, we obtain the slightly more general result below.
\begin{proposition}
  Let
  \[
  \begin{tikzcd}
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    A \ar[d,"\alpha"] \ar[r,"\beta"] &B \ar[d,"\delta"] \\
    C \ar[r,"\gamma"]& D
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  \end{tikzcd}
  \]
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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 injective on arrows.
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  \end{enumerate}
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  Then, the square
  \[
  \begin{tikzcd}
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    L(A) \ar[d,"L(\alpha)"] \ar[r,"L(\beta)"] &L(B) \ar[d,"L(\delta)"] \\
    L(C) \ar[r,"L(\gamma)"] &L(D)
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  \end{tikzcd}
  \]
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  is a \emph{homotopy} cocartesian square of $\Cat$ when equipped with the Thomason weak equivalences.
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\end{proposition}
\begin{proof}
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  The case where $\alpha$ or $\beta$ is injective both on objects and arrows is Corollary \ref{cor:hmtpysquaregraph}. Hence, we only have to treat the case when $\alpha$ is injective on objects and $\beta$ is injective on arrows. The other case being symmetric.

  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
  \[
  \begin{tikzcd}
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    \displaystyle\coprod_{x \in E}F_x \ar[r] \ar[d] & E \ar[d]\\
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    A \ar[r] & G,
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    \ar[from=1-1,to=2-2,phantom,very near end,"\ulcorner"]
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  \end{tikzcd}
  \]
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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 $\beta$.
  Notice importantly that the morphism \[ \coprod_{x \in E}F_x \to A\] is a monomorphism, i.e. injective on objects and arrows, and the morphism \[A \to G\] is injective on arrows. 
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\end{proof}
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\begin{example}[Adding a generator]
  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 $A \to B$, i.e. defined with the following cocartesian square:
  \[
  \begin{tikzcd}
        \partial\sD_1 \ar[d,"i_1"] \ar[r,"{\langle A, B \rangle}"] & C \ar[d] \\
    \sD_1 \ar[r] & C'.
  \end{tikzcd}
  \]
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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 $\Rgrph$ by the functor $L$ and the morphism $i_1 : \partial\sD_1 \to \sD_1$ comes from a monomorphism of $\Rgrph$. 
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\end{example}
\begin{remark}
  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 free (1-)categories are \good{} (which we already knew since we have seen that all (1-)categories are \good{}). 
\end{remark}
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\begin{example}[Identyfing two generators]
  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
  \[
  \begin{tikzcd}
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    \sS_1\ar[d] \ar[r,"{\langle f, g \rangle}"] &C \ar[d] \\
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    \sD_1  \ar[r] & C',  
  \end{tikzcd}
  \]
  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 $\sD_1$. Let us prove that the previous square is homotopy cocartesian in $\Cat$ (when equipped with Thomason weak equivalences). Notice that this square is the image of a cocartesian square of $\Rgrph$ by the functor $L$ as in the previous example. We now distinguish two cases. First, if $A \neq B$, then the map $\sS_1 \to C$ comes from a monomorphism of reflexive graphs and we conclude as in the previous example. Now if $A=B$, notice that the map $\sS_1 \to C$ factorizes as
  \[
  \sS_1 \to F_2 \to C
  \]
  where $F_2$ is the free monoid with two generators, seen as a category. In particular, it is free and notice that the map on the left comes from a monomorphism of reflexive graphs. Now, this factorization yields a factorization of our cocartesian square into two cocartesian squares
  \[
  \begin{tikzcd}
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    a
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  \end{tikzcd}
  \]
\end{example}
\begin{example}[Killing a generator]
  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:
  \[
  \begin{tikzcd}
    \sD_1 \ar[d] \ar[r,"\langle f \rangle"] & C \ar[d] \\
    \sD_0 \ar[r] & C'.
  \end{tikzcd}
  \]
  Then, this above square is homotopy cocartesion in $\Cat$ (equipped with the Thomason weak 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 $\Rgrph$.
\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. \todo{À mieux dire ?}
  \end{remark}
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