### security commit

parent b159e7d1
 ... ... @@ -113,15 +113,15 @@ From the previous proposition, we deduce the following very useful corollary. Let $\begin{tikzcd} A \ar[d] \ar[r] B \ar[d] \\ C \ar[r] D A \ar[d,"\alpha"] \ar[r,"\beta"] B \ar[d,"\delta"] \\ C \ar[r,"\gamma"] D \end{tikzcd}$ be a cocartesian square in $\RGrph$. If either the arrow $A \to B$ or the arrow $A \to C$ is a monomorphism, then the induced square be a cocartesian square in $\RGrph$. If either $\alpha$ or $\beta$ is a monomorphism, then the induced square $\begin{tikzcd} L(A) \ar[d] \ar[r] L(B) \ar[d] \\ L(C) \ar[r] L(D) L(A) \ar[d,"L(\alpha)"] \ar[r,"L(\beta)"] L(B) \ar[d,"L(\delta)"] \\ L(C) \ar[r,"L(\gamma)"] L(D) \end{tikzcd}$ is a \emph{homotopy} cocartesian square of $\Cat$ equipped with the Thomason weak equivalences. ... ... @@ -134,19 +134,45 @@ From the previous proposition, we deduce the following very useful corollary. it suffices to prove that the induced square of simplicial sets $\begin{tikzcd} NL(A) \ar[d] \ar[r] NL(B) \ar[d] \\ NL(C) \ar[r] NL(D) NL(A) \ar[d,"NL(\alpha)"] \ar[r,"NL(\beta)"] NL(B) \ar[d,"NL(\delta)"] \\ NL(C) \ar[r,"NL(\gamma)"] NL(D) \end{tikzcd}$ is homotopy cocartesian. But, since $L \simeq c\circ i_!$, it follows from Lemma \ref{cor:hmtpysquaregraph} that this last square is weakly equivalent to the square of simplicial sets $\begin{tikzcd} i_!(A) \ar[d] \ar[r] i_!(B) \ar[d] \\ i_!(C) \ar[r] i_!(D). i_!(A) \ar[d,"i_!(\alpha)"] \ar[r,"i_!(\beta)"] i_!(B) \ar[d,"i_!(\delta)"] \\ i_!(C) \ar[r,"i_!(\gamma)"] i_!(D). \end{tikzcd}$ This square is cocartesian because $i_!$ is a left adjoint and since $i_!$ preserves monomorphism (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{proposition} Let $\begin{tikzcd} A \ar[d,"\alpha"] \ar[r,"\beta"] B \ar[d,"\delta"] \\ C \ar[r,"\gamma"] D \end{tikzcd}$ be a cocartesian square in $\RGrph$. Suppose that both following conditions are satisfied \begin{itemize}[label=\alph*)] \item Either $\alpha$ or $\beta$ is injective on objects. \item Either $\alpha$ or $\beta$ is injective on morphisms. \end{itemize} Then, the square $\begin{tikzcd} L(A) \ar[d,"L(\alpha)"] \ar[r,"L(\beta)"] L(B) \ar[d,"L(\delta)"] \\ L(C) \ar[r,"L(\gamma)"] L(D) \end{tikzcd}$ is a \emph{homotopy} cocartesian square of $\Cat$ equipped with the Thomason weak equivalences. \end{proposition} \begin{proof} The case where $\alpha$ or $\beta$ is injective both on objects and morphisms is Corollary \ref{cor:hmtpysquaregraph} hence we only have to treat the case when $\alpha$ is injective on objects and $\beta$ is injective on arrows. Let use denote by $E$ the set of objects of $C$ that lies in the image of $\alpha$. For each element $x$ of $E$, we denote by \end{proof} \begin{example}[Adding a generator] Let $C$ be a free category, $A$ and $B$ (possibly equal) two objects of $C$ and let $C'$ be the category obtained from $C$ by adding a generator $A \to B$, i.e. defined with the following cocartesian square: \[ ... ...
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