@@ -147,19 +147,6 @@ From the previous proposition, we deduce the following very useful corollary.

\]

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:

\[

...

...

@@ -173,4 +160,36 @@ From the previous proposition, we deduce the following very useful corollary.

\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}

\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}

\sS_1\ar[d]\ar[r,"{\langle f, g \rangle}"] C \ar[d]\\

\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}

\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 ?}