Commit aa388927 by Leonard Guetta

### security commit

parent 18dfc912
 ... ... @@ -883,7 +883,7 @@ For any $n \geq 0$, consider the following cocartesian square Similarly, if $m >0$ and $n\geq 0$, then $A_{(m,n)}$ has the homotopy type of a point and is \good{}. \end{paragr} Combined with the result of Paragraph \ref{paragr:bubble}, we have proved the following proposition. \begin{proposition} \begin{proposition}\label{prop:classificationAmn} Let $m,n \geq 0$ and consider the $2$-category $A_{(m,n)}$. If $m\neq 0$ or $n\neq 0$, then $A_{(m,n)}$ is \good{} and has the homotopy type of a point. If $n=m=0$, then $A_{(0,0)}$ is not \good{} and has the homotopy type of a $K(\mathbb{Z},2)$\nbd{}space. \end{proposition} \section{Zoology of $2$-categories: more examples} ... ... @@ -897,9 +897,15 @@ For any $n \geq 0$, consider the following cocartesian square In picture, this gives $\begin{tikzcd}[column sep=huge] A \ar[r,bend left=75,"f",""{name=A,below}]\ar[r,bend right=75,"g"',""{name=B,above}] \ar[r,"1_A" pos=1/3,""{name=C,above},""{name=D,below}]& A. A \ar[r,bend left=75,"f",""{name=A,below}]\ar[r,bend right=75,"g"',""{name=B,above}] \ar[r,"1_A" pos=1/3,""{name=C,above},""{name=D,below}]& A \ar[from=A,to=C,Rightarrow,"\alpha"] \ar[from=B,to=D,"\beta" pos=9/20,Rightarrow] \end{tikzcd} \text{ or } \begin{tikzcd} A. \ar[loop,in=130,out=50,distance=1.5cm,"f"',""{name=A,below}] \ar[loop,in=-130,out=-50,distance=1.5cm,"g",""{name=B,above}] \ar[from=A,to=1-1,Rightarrow,"\alpha"] \ar[from=B,to=1-1,Rightarrow,"\beta"] \end{tikzcd}$ Notice that this category has many non-trivial $2$\nbd{}cells and it is not \emph{a priori} clear what its homotopy type is and whether or not it is \good{}. ... ... @@ -1022,7 +1028,50 @@ For similar reasons as for $F$, the $2$\nbd{}functor $G$ is a Thomason equivalen Since $F$ and $G$ are Thomason equivalences, the two bottom horizontal morphisms are isomorphisms. Besides, a simple computation using Proposition \ref{prop:abelianizationfreeoocat}, which we left to the reader, shows that the two top horizontal morphisms are isomorphisms. Since $P''$ is \good{} (Proposition \ref{prop:spheresaregood}), the morphism is $\pi_{P''}$ is an isomorphism. This implies that $\pi_{P'}$ and $\pi_{P'}$ are isomorphisms, which means by definition that $P$ and $P'$ are \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: $f : A \to A$, \item generating $2$\nbd{}cells: $\alpha : f \Rightarrow 1_A$ and $\beta: 1_A \Rightarrow f$. \end{itemize} In picture, this gives $\begin{tikzcd}[column sep=huge] A \ar[r,bend left=75,"f",""{name=A,below}]\ar[r,bend right=75,"f"',""{name=B,above}] \ar[r,"1_A" pos=1/3,""{name=C,above},""{name=D,below}]& A \ar[from=A,to=C,Rightarrow,"\alpha"] \ar[from=D,to=B,"\beta" pos=9/20,Rightarrow] \end{tikzcd} \text{ or } \begin{tikzcd} A. \ar[loop,in=30,out=150,distance=3cm,"f",""{name=A,below}] \ar[from=A,to=1-1,bend right,Rightarrow,"\alpha"'] \ar[from=1-1,to=A,bend right,Rightarrow,"\beta"'] \end{tikzcd}$ Now, let $P'$ be a copy of $B^2\mathbb{N}$ labelled as follows $\begin{tikzcd} A' \ar[r,bend left=75,"1_{A'}",""{name=A,below}] \ar[r,bend right=75,"1_{A'}"',""{name=B,above}] & A', \ar[from=A,to=B,"\gamma",Rightarrow] \end{tikzcd}$ which can be also pictured as $\begin{tikzcd} A'. \ar[loop,in=120,out=60,distance=1.5cm,"\gamma"',Rightarrow] \end{tikzcd}$ Let $F : P \to P'$ be the unique $2$\nbd{}functor such that: \begin{itemize}[label=-] \item $F(A)=A'$, \item $F(f)=1_{A'}$ \item $F(\alpha)=\1^2_{A'}$ and $F(\beta)=\gamma$, \end{itemize} and $G : P' \to P$ be the unique $2$\nbd{}functor such that: \begin{itemize}[label=-] \item $G(A')=A$, \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} \section{The Bubble-free'' conjecture} \begin{definition} ... ...
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