Commit 6b618234 authored by Leonard Guetta's avatar Leonard Guetta
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parent 7bd14d8e
......@@ -1574,7 +1574,7 @@ following proposition.
\end{equation}
Let us prove that this square is Thomason homotopy cocartesian using the
second part of Corollary \ref{prop:critverthorThomhmtpysquare}. This means
that we have to show that for every $k \geq 0$, the induced square
that we have to show that for every $k \geq 0$, the induced square of $\Cat$
\begin{equation}\label{squarebouquethorizontal}
\begin{tikzcd}
H_k(P_0) \ar[d] \ar[r] & H_k(P_2) \ar[d] \\\
......@@ -1623,6 +1623,61 @@ following proposition.
square \eqref{squarebouquet} is homotopy cocartesian. Hence, $P$ is \good{} and
has the homotopy type of a bouquet of two $2$\nbd{}spheres.
\end{paragr}
\begin{paragr}
Let $P$ be the free $2$\nbd{}category defined as follows:
\begin{itemize}[label=-]
\item generating $0$\nbd{}cells: $A$, $B$ and $C$,
\item generating $1$\nbd{}cells: $f,g : A \to B$ and $h,i : B \to C$,
\item generating $2$\nbd{}cells: $\alpha,\beta : f \Rightarrow g$ and $\gamma,\delta : h \Rightarrow i$.
\end{itemize}
In pictures, this gives:
\[
\begin{tikzcd}[column sep=huge]
A \ar[r,bend left=75,"f",""{name=A,below,pos=8/20},""{name=C,below,pos=12/20}] \ar[r,bend right=75,"g"',""{name=B,above,pos=8/20},""{name=D,above,pos=12/20}] & B \ar[r,bend left=75,"h",""{name=E,below,pos=8/20},""{name=G,below,pos=12/20}] \ar[r,bend right=75,"i"',""{name=F,above,pos=8/20},""{name=H,above,pos=12/20}] & C.
\ar[from=A,to=B,bend right,"\alpha",Rightarrow]
\ar[from=C,to=D,bend left,"\beta",Rightarrow]
\ar[from=E,to=F,bend right,"\gamma",Rightarrow]
\ar[from=G,to=H,bend left,"\delta",Rightarrow]
\end{tikzcd}
\]
Let us prove that $P$ is \good{}. Let $P'$ be the sub-$2$\nbd{}category of $P$ spanned by $A$, $B$, $f$, $g$, $\alpha$ and $\beta$ and let $P''$ be the sub-$2$\nbd{}category of $P$ spanned by $B$, $C$, $h$, $i$, $\beta$ and $\gamma$. These two $2$\nbd{}categories are copies of $\sS_2$ and we have a cocartesian square
\[
\begin{tikzcd}
\sD_0 \ar[r,"\langle B \rangle"] \ar[d,"\langle B \rangle"] & P' \ar[d,hook]\\
P'' \ar[r,hook] & P,
\ar[from=1-1,to=2-2,phantom,very near end]
\end{tikzcd}
\]
where the anonymous arrows are the canonical inclusions. Let us prove that this cocartesian
square is Thomason homotopy cocartesian using the first part of Proposition \ref{prop:critverthorThomhmtpysquare}. This means that we need to prove that for every $k\geq 0$, the induced square of $\Cat$
\begin{equation}\label{squarebouquetbis}
\begin{tikzcd}
V_k(\sD_0) \ar[r] \ar[d] & V_k(P') \ar[d]\\
V_k(P'') \ar[r] & V_k(P)
\end{tikzcd}
\end{equation}
is Thomason homotopy cocartesian. For $k=0$, we have the following isomorphisms
\[
V_0(\sD_0) \simeq \sD_0,
\]
\[
V_0(P') \simeq \sD_0 \amalg \sD_0,
\]
\[
V_0(P'') \simeq \sD_0 \amalg \sD_0,
\]
\[
V_0(P) \sime \sD_0 \amalg \sD_0 \amalg \sD_0
\]
and the square
\[
\begin{tikzcd}
\sD_0 \ar[r] \ar[d] & \sD_0\amald \sD_0 \ar[d] \\
\sD_0\amalg \sD_0 \ar[r] & \sD_0 \amalg \sD_0 \amalg \sD_0
\end{tikzcd}
\]
is obviously Thomason homotopy cocartesian.
\end{paragr}
\begin{paragr}
Let $P$ be the free $2$\nbd{}category defined as follows:
\begin{itemize}[label=-]
......
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