@@ -47,7 +47,7 @@ In this section, we review some homotopical results concerning free ($1$-)catego

\]

For a graph $G$, the simplicial set $i_l(G)$ has $G_0$ as its set of $0$-simplices, $G_1$ as its set of $1$-simplices and all $k$-simplices degenerated for $k>2$. For future reference we put here the following lemma.

\end{paragr}

\begin{lemma}

\begin{lemma}\label{lemma:monopreserved}

The functor $i_! : \Rgrph\to\Psh{\Delta}$ preserves monomorphism.

\end{lemma}

\begin{proof}

...

...

@@ -66,5 +66,111 @@ In this section, we review some homotopical results concerning free ($1$-)catego

We now review a construction of Dwyer and Kan from \cite{dwyer1980simplicial}.

Let $G$ be a reflexive graph. For every $i\leq1$, we define the simplicial set $N^k(G)$ as the sub-simplicial set of $N(L(G))$ whose $n$-simplices are chains

where $\eta$ is the unit of the adjunction $c \dashv N$.

\end{paragr}

\begin{lemma}[Dwyer-Kan]\label{lemma:dwyerkan}

For every $k\leq1$, 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.

\end{propostion}

\begin{proof}

It is an immediate consequence of Lemma \ref{lemma:dwyerkan} and the fact that filtered colimits are homotopic in a model category such that all objects are cofibrants \todo{ref}.

\end{proof}

From the previous proposition, we deduce the following very useful corollary.

\begin{corollary}\label{cor:hmtpysquaregraph}

Let

\[

\begin{tikzcd}

A \ar[d]\ar[r] B \ar[d]\\

C \ar[r] D

\end{tikzcd}

\]

be a cocartesian square in $\RGrph$. If either the arrow $A \to B$ or the arrow $A \to C$ is a monomorphism, then the induced square

\[

\begin{tikzcd}

L(A)\ar[d]\ar[r] L(B)\ar[d]\\

L(C)\ar[r] L(D)

\end{tikzcd}

\]

is a \emph{homotopy} cocartesian square of $\Cat$ equipped with the Thomason weak equivalences.

\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]\ar[r] NL(B)\ar[d]\\

NL(C)\ar[r] 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]\ar[r] i_!(B)\ar[d]\\

i_!(C)\ar[r] i_!(D).

\end{tikzcd}

\]

This square is cocartesian because $i_!$ is a left adjoint and since $i_!$ preserves monomorphism (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}.

\end{proof}

\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}

\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}

\]

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 $\Rgrh$.

\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{}).