Commit 5094e675 by Leonard Guetta

security

parent 79808a66
 ... ... @@ -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}. \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} Let \[ ... ... @@ -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 \begin{enumerate}[label=\alph*)] \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. \end{enumerate} Then, the square \[ ... ...
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