Commit 36687578 authored by Leonard Guetta's avatar Leonard Guetta
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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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