@@ -157,7 +157,13 @@ From the previous proposition, we deduce the following very useful corollary.

...

@@ -157,7 +157,13 @@ From the previous proposition, we deduce the following very useful corollary.

\]

\]

This square is cocartesian because $i_!$ is a left adjoint. Since $i_!$ preserves monomorphisms (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}.

This square is cocartesian because $i_!$ is a left adjoint. Since $i_!$ preserves monomorphisms (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}

\end{proof}

Actually, by working a little more, we obtain the slightly more general result below.

\begin{paragr}

Actually, by working a little more, we obtain a more general result, which is stated in the propositon below. Let us say that a morphism of reflexive graphs, $\alpha : A \to B$, is \emph{quasi-injective on arrows} when for all $f$ and $g$ arrows of $A$, if

\[

\alpha(f)=\alpha(g),

\]

then either $f=g$ or $f$ and $g$ are both units. In other words, $\alpha$ never send a non-unit arrow to a unit arrow and $\alpha$ never identifies two non-unit arrows. It follows that if $\alpha$ is quasi-injective on arrows and injective on objects, then it is also injective on arrows and hence, a monomorphism of $\Rgrph$.

\end{paragr}

\begin{proposition}

\begin{proposition}

Let

Let

\[

\[

...

@@ -169,7 +175,7 @@ Actually, by working a little more, we obtain the slightly more general result b

...

@@ -169,7 +175,7 @@ Actually, by working a little more, we obtain the slightly more general result b

be a cocartesian square in $\Rgrph$. Suppose that the following two conditions are satisfied

be a cocartesian square in $\Rgrph$. Suppose that the following two conditions are satisfied

\begin{enumerate}[label=\alph*)]

\begin{enumerate}[label=\alph*)]

\item Either $\alpha$ or $\beta$ is injective on objects.

\item Either $\alpha$ or $\beta$ is injective on objects.

\item Either $\alpha$ or $\beta$ is injective on arrows.

\item Either $\alpha$ or $\beta$ is quasi-injective on arrows.