Commit da42df59 authored by Leonard Guetta's avatar Leonard Guetta
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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 $[0]$ and $[1]$. 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 $[0]$ 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 $i : \Delta_{\leq 1} \arrow \Delta$ 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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