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 $\Delta_{\leq1}$ the full subcategory of $\Delta$ spanned by $[0]$ and $[1]$. Then, the category $\Rgrph$ is nothing but $\Psh{\Delta_{\leq1}}$, the category of pre-sheaves on $\Delta_{\leq1}$.

\end{paragr}

There is another important description of the category $\Rgrph$. Consider $\Delta_{\leq1}$ the full subcategory of $\Delta$ spanned by $[0]$ and $[1]$. Then, the category $\Rgrph$ is nothing but $\Psh{\Delta_{\leq1}}$, the category of pre-sheaves on $\Delta_{\leq1}$. In particular, the canonical inclusion $i : \Delta_{\leq1}\arrow\Delta$ 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

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