Commit 95c0b2ef authored by Leonard Guetta's avatar Leonard Guetta
Browse files

changed a tiny detail

parent 1896ebf4
...@@ -238,8 +238,7 @@ From the previous proposition, we deduce the following very useful corollary. ...@@ -238,8 +238,7 @@ From the previous proposition, we deduce the following very useful corollary.
L(C) \ar[r,"L(\gamma)"]& L(D) L(C) \ar[r,"L(\gamma)"]& L(D)
\end{tikzcd} \end{tikzcd}
\] \]
is a \emph{homotopy} cocartesian square of $\Cat$ when equipped with the Thomason is a Thomason \emph{homotopy} cocartesian square of $\Cat$.
equivalences.
\end{corollary} \end{corollary}
\begin{proof} \begin{proof}
Since the nerve $N$ induces an equivalence of op-prederivators Since the nerve $N$ induces an equivalence of op-prederivators
...@@ -1629,7 +1628,7 @@ Now let $\sS_2$ be labelled as ...@@ -1629,7 +1628,7 @@ Now let $\sS_2$ be labelled as
In particular, square \eqref{eq:squarebouquethybridvertical} is again a In particular, square \eqref{eq:squarebouquethybridvertical} is again a
cocartesian square of identification of two objects of a free category, and cocartesian square of identification of two objects of a free category, and
thus, it is Thomason homotopy cocartesian. This implies that square thus, it is Thomason homotopy cocartesian. This implies that square
\eqref{eq:squarebouquethybrid} is homotopy cocartesian. Since $\sS_0$, $\sD_1$ and $\sS_2$ are \good{} and since the morphisms $\langle C, D \rangle : \sS_0 \to \sS_2$ and $i_1 : \sS_0 \to \sD_1$ are folk cofibrations, this proves that $P$ is \good{} \eqref{eq:squarebouquethybrid} is Thomason homotopy cocartesian. Since $\sS_0$, $\sD_1$ and $\sS_2$ are \good{} and since the morphisms $\langle C, D \rangle : \sS_0 \to \sS_2$ and $i_1 : \sS_0 \to \sD_1$ are folk cofibrations, this proves that $P$ is \good{}
and has the homotopy type of the bouquet of a $1$\nbd{}sphere with a $2$\nbd{}sphere. and has the homotopy type of the bouquet of a $1$\nbd{}sphere with a $2$\nbd{}sphere.
\end{paragr} \end{paragr}
......
...@@ -245,7 +245,7 @@ is a homotopy cocartesian square of simplicial sets. Since $N_{\oo}$ induces an ...@@ -245,7 +245,7 @@ is a homotopy cocartesian square of simplicial sets. Since $N_{\oo}$ induces an
\ar[from=1-1,to=2-2,phantom,"\ulcorner",very near end] \ar[from=1-1,to=2-2,phantom,"\ulcorner",very near end]
\end{tikzcd} \end{tikzcd}
\] \]
is homotopy cocartesian with respect to Thomason equivalences, then $C$ is \good{}. is Thomason homotopy cocartesian, then $C$ is \good{}.
\end{proposition} \end{proposition}
\begin{proof} \begin{proof}
Since the morphisms $i_k$ are folk cofibration and the Since the morphisms $i_k$ are folk cofibration and the
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