Commit a2823d7f by Leonard Guetta

### almost over

parent b1657859
 ... ... @@ -2,6 +2,7 @@ %\usepackage[utf8]{inputenc} \usepackage{mystyle} \usepackage{graphicx} \usetheme{Madrid} \usecolortheme{beaver} ... ... @@ -356,10 +357,9 @@ \end{center} \pause \begin{exampleblock}{Important fact} If $C$ is a free $\oo$\nbd{}category, the set of free generators of $C$ is uniquely determined from $C$. If $C$ is a free $\oo$\nbd{}category, then there is a \emph{unique} set of generating cells possible. \end{exampleblock} TODO : Laisser le bloc ci-dessus ? \end{frame} \begin{frame} \frametitle{Abelianization of $\oo$\nbd{}categories} ... ... @@ -703,14 +703,15 @@ \item<4-> Thus, for every $a \in A$, the $\oo$\nbd{}category $P/a$ is free. \item<5-> Besides, the set of generating cells of $P/a$ vary naturally in a''. \item<5-> Besides, the set of generating cells of $P/a$ is natural in a''. \item<6-> Worked out properly (cf. thesis), this means that $P/{-} : A \to \oo\Cat$ is cofibrant. \hfill CQFD \end{itemize} \end{frame} \begin{frame}\frametitle{In practice} \begin{exampleblock}{Corollary} \begin{frame}\frametitle{A criterion} A variation of the homotopy colimit criterion: \begin{exampleblock}{Proposition} Let $\begin{tikzcd}[ampersand replacement=\&] ... ... @@ -720,13 +721,14 @@ \end{tikzcd}$ be a cocartesian square in $\oo\Cat$. If \begin{itemize}[label=$\bullet$] \begin{enumerate}[label=(\roman*)] \item<2-> $A$,$B$ and $C$ are homologically coherent, \item<3-> $u$ or $v$ is a folk cofibration, \item<4-> the square is homotopy cocartesian w.r.t Thomason equivalences, \end{itemize} \end{enumerate} \pause\pause\pause\pause then $D$ is homologically coherent. \end{exampleblock} \end{exampleblock} \pause The third condition will usually be the hard one to prove. \end{frame} \begin{frame}\frametitle{Easy application: homology of globes and spheres} For every $n\geq 0$, $\sD_n$ is oplax contractible, hence ... ... @@ -747,8 +749,114 @@ \pause By an immediate induction, $\sS_n$ is homologically coherent (and has the homotopy type of an $n$\nbd{}sphere). \end{frame} \end{document} \begin{frame}\frametitle{2-categories} We would like to understand which 2-categories are homologically coherent. \begin{itemize} \item<2-> For simplification, we focus on \emph{free} 2-categories. \item<3-> This boils down to the following: given a cocartesian square $\begin{tikzcd}[ampersand replacement=\&] \sS_1 \ar[d,"i_1"'] \ar[r] \& P \ar[d] \\ \sD_2 \ar[r] \& P' \ar[from=1-1,to=2-2,"\ulcorner",very near end,phantom] \end{tikzcd}$ with $P$ and $P'$ free $2$\nbd{}categories, when is it homotopy cocartesian w.r.t the Thomason equivalences ? \item<4-> I do not have a general answer to this question... \item<5-> However, using tools that I don't have time to explain, I know how to answer this question in many concrete situations. \end{itemize} \end{frame} \begin{frame}\frametitle{Zoology of 2-categories: basic examples} For $n, m \geq 0$, let $A_{(m,n)}$ be the free $2$\nbd{}category, with one generating $2$\nbd{}cell whose source is a chain of length $m$ and target a chain of length $n$: \pause $A_{(m,n)} = \qquad \underbrace{\overbrace{\begin{tikzcd}[column sep=small, ampersand replacement=\&] \&\bullet \ar[r,description,"\cdots",phantom,""{name=A,below}] \& \bullet \ar[rd] \& \\ \bullet \ar[ru] \ar[rd] \& \& \&\bullet \\ \&\bullet \ar[r,description,"\cdots",phantom,""{name=B,above}] \& \bullet. \ar[ru]\ar[from=A,to=B,shorten <= 2em, shorten >= 2em,Rightarrow]\end{tikzcd}}^{m}}_{n}$ \pause Examples: \begin{itemize}[label=-] \item<4-> $A_{(1,1)}$ is $\sD_2$. \item<5-> $A_{(0,0)}$ is the $2$\nbd{}category $B$ from Ara and Maltsiniotis' counter-example. \end{itemize} \end{frame} \begin{frame}\frametitle{Zoology of $2$\nbd{}categories: basic examples} \begin{block}{Proposition} If $n+m>0$, the $2$\nbd{}category $A_{(m,n)}$ has the homotopy type of a point and is homologically coherent. Else, $A_{(0,0)}$ has the homotopy type of a $K(\mathbb{Z},2)$. \end{block} \pause Note: for $m+n=1$, the result was not \emph{a priori} clear. \pause For example: $A_{(1,0)}= \begin{tikzcd} \bullet \ar[loop,in=50,out=130,distance=1.5cm,""{name=A,below}] \ar[from=A,to=1-1,Rightarrow] \end{tikzcd}$ has many non-trivial $2$\nbd{}cells. \end{frame} \begin{frame}\frametitle{Zoology of $2$\nbd{}categories: variation of spheres} % \small \begin{center} \scalebox{0.85}{ \begin{tabular}{ l || c | c } \hline $2$\nbd{}category & \good{}? & homotopy type \\ \hline \hline \pause { $\begin{tikzcd}[ampersand replacement=\&] \bullet \ar[r,bend left=75,""{name=A,below,pos=9/20},""{name=C,below,pos=11/20}] \ar[r,bend right=75,""{name=B,above,pos=9/20},""{name=D,above,pos=11/20}] \& \bullet \ar[from=C,to=D,bend left,Rightarrow] \ar[from=A,to=B,bend right,Rightarrow] \end{tikzcd}$ } & yes & $\sS_2$\\ \hline \pause { $\begin{tikzcd}[ampersand replacement=\&] \bullet \ar[r,bend left=75,""{name=A,below}] \ar[r,bend right=75,""{name=B,above}] \& \bullet \ar[from=A,to=B,bend right,Rightarrow] \ar[from=B,to=A,bend right,Rightarrow] \end{tikzcd}$} & yes & $\sS_2$ \\ \hline \pause {$\begin{tikzcd}[ampersand replacement=\&] \bullet \ar[r,""{name=A,above}] \& \bullet \ar[from=A,to=A,loop, in=130, out=50,distance=1cm, Rightarrow] \end{tikzcd}$} & yes &$\sS_2$ \\ \hline \pause { $\begin{tikzcd} \bullet \ar[loop,in=30,out=150,distance=2cm,""{name=A,below}] \ar[from=A,to=1-1,bend right,Rightarrow] \ar[from=A,to=1-1,bend left,Rightarrow] \end{tikzcd}$ } & yes & $\sS_2$ \\ \hline \pause { $\begin{tikzcd} \bullet \ar[loop,in=30,out=150,distance=2cm,""{name=A,below}] \ar[from=A,to=1-1,bend right,Rightarrow] \ar[from=1-1,to=A,bend right,Rightarrow] \end{tikzcd}$ } & no & $K(\mathbb{Z},2)$ \\ \hline \pause {$\begin{tikzcd} \bullet \ar[loop,in=120,out=60,distance=1.2cm,Rightarrow] \end{tikzcd}$} & no & $K(\mathbb{Z},2)$ \\ \hline \end{tabular} } \end{center} \end{frame} \end{document} %%% Local Variables: %%% mode: latex ... ...
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