### security commit

parent 4e621240
 ... ... @@ -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 ?} \end{remark}
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