security commit

2cat.tex

@@ -1165,51 +1165,90 @@ isomorphisms, which means by definition that $P$, $P'$ and $P''$ are \good{}.
     \item $G(\gamma)=\alpha\comp_1\beta$.
     \end{itemize}
     Notice that we have $F\circ G = \mathrm{id}_{P'}$, which means that $P'$ is a retract of $P$. In particular, $\sH^{\sing}(P)$ is a retract of $\sH^{\sing}(P')$ and since $P'$ has the homotopy type of $K(\mathbb{Z},2)$-space (\ref{paragr:bubble}), this proves that $P$ have non-trivial singular homology groups in all even dimension. But since it is a free $2$\nbd{}category, all its polygraphic homology groups are trivial above dimension $2$, which means that $P$ is \emph{not} \good{}.
-  \end{paragr}
-  \begin{paragr}
-    Let $P$ be the free $2$\nbd{}category defined as follows:
-    \begin{itemize}[label=-]
-    \item generating $0$\nbd{}cells: $A$ and $B$,
-    \item generating $1$\nbd{}cells: $f,g : A \to B$,
-      \item generating $2$\nbd{}cells: $\alpha,\beta,\gamma : f \to g$.
-      \end{itemize}
-      In picture, this gives
-      \[
-        \begin{tikzcd}[column sep=huge]
-          A \ar[r,bend left=75,"f",""{name=A,below,pos=8/20},""{name=C,below,pos=1/2},""{name=E,below,pos=12/20}] \ar[r,bend
-          right=75,"g"',""{name=B,above,pos=8/20},""{name=D,above,pos=1/2},""{name=F,above,pos=12/20}]& B.
-          \ar[from=A,to=B,Rightarrow,"\alpha"',bend right]
-          \ar[from=C,to=D,Rightarrow,"\beta"]
-          \ar[from=E,to=F,Rightarrow,"\gamma",bend left]
-        \end{tikzcd}
-      \]
-      Now let $P'$ be the sub-$2$\nbd{}category of $P$ spanned by $A$,$B$,
-      $\alpha$ and $beta$, and let $P''$ be the sub-$2$\nbd{}category of $P$
-      spanned by $A$,$B$,$\beta$ and $\gamma$. These $2$\nbd{}categories are
-      simply copies of $\sS_2$. Notice that we have a cocartesian
-      square
-      \begin{equation}\label{square:bouquet}
-      \begin{tikzcd}
-        \sD_1 \ar[r,"\langle \beta \rangle"] \ar[d,"\langle  \beta \rangle"] &
-        P' \ar[d] \\
-        P'' \ar[r] & P,
-        \ar[from=1-1,to=2-2,phantom,"\ulcorner",very near end]
-      \end{tikzcd}
-    \end{equation}
-    and by reasoning as in the proof of Lemma \ref{lemma:squarenerve}, one can
-    show that the square induced by the nerve
-    \[
-      \begin{tikzcd}
-      N_{\oo}(\sD_1) \ar[r,"\langle \beta \rangle"] \ar[d,"\langle  \beta \rangle"] &
-        N_{\oo}(P') \ar[d] \\
-        N_{\oo}(P'') \ar[r] & N_{\oo}(P)
-      \end{tikzcd}
-    \]
-    is also cocartesian. This proves that square \eqref{square:bouquet} is
-    Thomason homotopy cocartesian \todo{détailler?} and in particular that $P$ has the homotopy
-    type of a bouqet of two $2$\nbd{}spheres. Since $\sD_1$, $P'$ and $P''$ are
-    free and \good{} and since $\langle \beta \rangle : \sD_1 \to P'$ and
-    $\langle \beta \rangle : \sD_1 \to P'$, this proves that $P$ is \good{}.
+\end{paragr}
+\begin{paragr}
+  Let $P$ be the free $2$\nbd{}category defined as follows:
+  \begin{itemize}[label=-]
+  \item generating $0$\nbd{}cell: $A$,
+  \item generating $1$\nbd{}cell: $f,g: A \to A$,
+    \item generating $2$\nbd{}cell: $\alpha,\beta : f \to g$.
+  \end{itemize}
+  In picture, this gives:
+  %% \[
+  %% \begin{tikzcd}
+    
+  %%   \end{tikzcd}
+  %% \]
+\end{paragr}
+\begin{paragr}
+  Let $P$ be the free $2$\nbd{}category defined as follows:
+  \begin{itemize}[label=-]
+  \item generating $0$\nbd{}cells: $A$ and $B$,
+  \item generating $1$\nbd{}cells: $f,g : A \to B$,
+  \item generating $2$\nbd{}cells: $\alpha,\beta,\gamma : f \to g$.
+  \end{itemize}
+  In picture, this gives
+  \[
+  \begin{tikzcd}[column sep=huge]
+    A \ar[r,bend left=75,"f",""{name=A,below,pos=8/20},""{name=C,below,pos=1/2},""{name=E,below,pos=12/20}] \ar[r,bend
+      right=75,"g"',""{name=B,above,pos=8/20},""{name=D,above,pos=1/2},""{name=F,above,pos=12/20}]& B.
+    \ar[from=A,to=B,Rightarrow,"\alpha"',bend right]
+    \ar[from=C,to=D,Rightarrow,"\beta"]
+    \ar[from=E,to=F,Rightarrow,"\gamma",bend left]
+  \end{tikzcd}
+  \]
+  Now let $P'$ be the sub-$2$\nbd{}category of $P$ spanned by $A$,$B$,
+  $\alpha$ and $beta$, and let $P''$ be the sub-$2$\nbd{}category of $P$
+  spanned by $A$,$B$,$\beta$ and $\gamma$. These $2$\nbd{}categories are
+  simply copies of $\sS_2$. Notice that we have a cocartesian
+  square
+  \begin{equation}\label{square:bouquet}
+    \begin{tikzcd}
+      \sD_1 \ar[r,"\langle \beta \rangle"] \ar[d,"\langle  \beta \rangle"] &
+      P' \ar[d] \\
+      P'' \ar[r] & P,
+      \ar[from=1-1,to=2-2,phantom,"\ulcorner",very near end]
+    \end{tikzcd}
+  \end{equation}
+  and by reasoning as in the proof of Lemma \ref{lemma:squarenerve}, one can
+  show that the square induced by the nerve
+  \[
+  \begin{tikzcd}
+    N_{\oo}(\sD_1) \ar[r,"\langle \beta \rangle"] \ar[d,"\langle  \beta \rangle"] &
+    N_{\oo}(P') \ar[d] \\
+    N_{\oo}(P'') \ar[r] & N_{\oo}(P)
+  \end{tikzcd}
+  \]
+  is also cocartesian. Since $\langle \beta \rangle : \sD_1 \to P'$ and $\langle \beta \rangle : \sD_1 \to P''$ are monomorphisms and $N_{\oo}$ preserves monomorphisms, it follows from Lemma \ref{lemma:hmtpycocartesianreedy} that square \eqref{square:bouquet} is
+  Thomason homotopy cocartesian and in particular that $P$ has the homotopy
+  type of a bouquet of two $2$\nbd{}spheres. Since $\sD_1$, $P'$ and $P''$ are
+  free and \good{} and since $\langle \beta \rangle : \sD_1 \to P'$ and
+  $\langle \beta \rangle : \sD_1 \to P'$, this also proves that $P$ is \good{} (see \ref{paragr:criterion2cat}).
+\end{paragr}
+\begin{paragr}
+  Let $P$ be the free $2$\nbd{}category defined as follows:
+  \begin{itemize}[label=-]
+  \item generating $0$\nbd{}cells: $A$ and $B$,
+  \item generating $1$\nbd{}cells: $f,g,h : A \to B$,
+  \item generating $2$\nbd{}cells: $\alpha,\beta:f \to g$ and $\delta,\gamma:g \to h$.
+  \end{itemize}
+  In picture, this gives:
+  \[
+  \begin{tikzcd}[column sep=huge]
+    A \ar[r,bend left=75,"f",""{name=A,below,pos=8/20},""{name=E,below,pos=12/20}] \ar[r,"g",""{name=B,above,pos=8/20},""{name=C,below,pos=8/20},""{name=F,above,pos=12/20},""{name=G,below,pos=12/20}] \ar[r,bend right=75,"h"',""{name=D,above,pos=8/20},""{name=H,above,pos=12/20}] & B.
+    \ar[from=A,to=B,Rightarrow,"\alpha"',bend right]
+    \ar[from=C,to=D,Rightarrow,"\delta"',bend right]
+    \ar[from=E,to=F,Rightarrow,"\beta",bend left]
+    \ar[from=G,to=H,Rightarrow,"\gamma",bend left]
+  \end{tikzcd}
+  \]
+  Let us prove that this $2$\nbd{}category 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 $A$, $B$, $g$, $h$, $\gamma$ and $\delta$. These two $2$\nbd{}categories are copies of $\sS_2$ and we have a cocartesian square
+  \[
+  \begin{tikzcd}
+    
+  \end{tikzcd}
+  \]
+  using the second part of Corollary 
   \end{paragr}
 \section{The ``Bubble-free'' conjecture}
 \begin{definition}