### blabla

parent cacef22c
 ... ... @@ -20,7 +20,7 @@ In this section, we review some homotopical results concerning free ($1$-)catego $X_0 \overset{\rightarrow}{f_1} X_1 \overset{\rightarrow}{f_2} X_2 \rightarrow \cdots \rightarrow X_{n-1} \overset{\rightarrow}{f_n} X_{n}$ of composable 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. 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. \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 ... ... @@ -37,5 +37,34 @@ In this section, we review some homotopical results concerning free ($1$-)catego 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} There is another important description of the category $\Rgrph$. Consider $\Delta_{\leq 1}$ the full subcategory of $\Delta$ spanned by $$ and $$. Then, the category $\Rgrph$ is nothing but $\Psh{\Delta_{\leq 1}}$, the category of pre-sheaves on $\Delta_{\leq 1}$. \end{paragr} There is another important description of the category $\Rgrph$. Consider $\Delta_{\leq 1}$ the full subcategory of $\Delta$ spanned by $$ and $$. 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 $i : \Delta_{\leq 1} \arrow \Delta$ 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 $i_l(G)$ 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} 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 $X_0 \overset{\rightarrow}{f_1} X_1 \overset{\rightarrow}{f_2} X_2 \rightarrow \cdots \rightarrow X_{n-1} \overset{\rightarrow}{f_n} X_{n}$ 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 $\Rgrph \overset{i_!}{\rightarrow} \Psh{\Delta} \overset{c}{\rightarrow} \Cat.$ \end{paragr}
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