Commit 6b618234 by Leonard Guetta

### security commit

parent 7bd14d8e
 ... ... @@ -1574,7 +1574,7 @@ following proposition. 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 $\Cat$ \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 $\Cat$ \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} 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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