Commit 12cfb713 authored by Leonard Guetta's avatar Leonard Guetta
Browse files

almost there

parent a2823d7f
...@@ -856,6 +856,72 @@ ...@@ -856,6 +856,72 @@
\end{center} \end{center}
\end{frame} \end{frame}
\begin{frame}
\frametitle{Zoology of 2-categories: Bouquet of spheres}
\begin{center}
\scalebox{0.85}{
\begin{tabular}{ l || c | c }
\hline
$2$\nbd{}category & \good{}? & homotopy type \\ \hline \hline \pause
$\begin{tikzcd}[column sep=huge,ampersand replacement=\&]
\bullet \ar[r,bend
left=75,""{name=A,below,pos=8/20},""{name=C,below,pos=1/2},""{name=E,below,pos=12/20}]
\ar[r,bend
right=75,""{name=B,above,pos=8/20},""{name=D,above,pos=1/2},""{name=F,above,pos=12/20}]
\& \bullet \ar[from=A,to=B,Rightarrow,bend right]
\ar[from=C,to=D,Rightarrow]
\ar[from=E,to=F,Rightarrow,bend left]
\end{tikzcd}$ & yes & $\sS_2\vee \sS_2$ \\ \hline \pause
$\begin{tikzcd}[column sep=huge,ampersand replacement =\&]
\bullet \ar[r,bend
left=75,""{name=A,below,pos=8/20},""{name=E,below,pos=12/20}]
\ar[r,""{name=B,above,pos=8/20},""{name=C,below,pos=8/20},""{name=F,above,pos=12/20},""{name=G,below,pos=12/20}]
\ar[r,bend
right=75,""{name=D,above,pos=8/20},""{name=H,above,pos=12/20}] \& \bullet
\ar[from=A,to=B,Rightarrow,bend right]
\ar[from=C,to=D,Rightarrow,bend right]
\ar[from=E,to=F,Rightarrow,bend left]
\ar[from=G,to=H,Rightarrow,bend left]
\end{tikzcd}$ & yes & $\sS_2\vee \sS_2$ \\ \hline \pause
$\begin{tikzcd}[column sep=huge,ampersand replacement=\&]
\bullet \ar[r,bend
left=75,""{name=A,below,pos=8/20},""{name=C,below,pos=12/20}]
\ar[r,bend
right=75,""{name=B,above,pos=8/20},""{name=D,above,pos=12/20}] \& \bullet
\ar[r,bend
left=75,""{name=E,below,pos=8/20},""{name=G,below,pos=12/20}]
\ar[r,bend
right=75,""{name=F,above,pos=8/20},""{name=H,above,pos=12/20}] \& \bullet
\ar[from=A,to=B,bend right,Rightarrow] \ar[from=C,to=D,bend
left,Rightarrow] \ar[from=E,to=F,bend right,Rightarrow]
\ar[from=G,to=H,bend left,Rightarrow]
\end{tikzcd}$ & yes & $\sS_2\vee \sS_2$ \\ \hline \pause
$\begin{tikzcd}[column sep=huge,ampersand replacement=\&]
A \ar[r,bend
left=75,""{name=A,below,pos=9/20},""{name=C,below,pos=11/20}]
\ar[r,bend
right=75,""{name=B,above,pos=9/20},""{name=D,above,pos=11/20}] \& A
\ar[from=C,to=D,bend left,Rightarrow] \ar[from=A,to=B,bend
right,Rightarrow]
\end{tikzcd}$ & yes & $\sS_2 \vee \sS_1$ \\ \hline
\end{tabular}
}
\end{center}
\end{frame}
\begin{frame}\frametitle{Zoology of 2-categories: Torus}
Let $C$ be the free $2$\nbd{}category pictured as
\[
\begin{tikzcd}[ampersand replacement=\&]
A \ar[r,"f"] \ar[d,"g"'] \& A \ar[d,"g"] \\
A \ar[r,"f"'] \& A. \ar[from=2-1,to=1-2,Rightarrow]
\end{tikzcd}
\]
\pause This $2$\nbd{}category has the homotopy type of the \alert{torus} and
is homologically coherent.
\pause Not so easy to show ! Idea: show that it is Thomason equivalent to the
monoid $(\mathbb{N}\times \mathbb{N},+)$ (which is not free)
\end{frame}
\end{document} \end{document}
%%% Local Variables: %%% Local Variables:
......
Markdown is supported
0% or .
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!
Please register or to comment