Commit 6253f148 authored by Leonard Guetta's avatar Leonard Guetta
Browse files

slowly but surely

parent e07cf015
\documentclass{beamer} \documentclass[handout]{beamer}
%\usepackage[utf8]{inputenc} %\usepackage[utf8]{inputenc}
\usepackage{mystyle} \usepackage{mystyle}
...@@ -166,15 +166,15 @@ ...@@ -166,15 +166,15 @@
\begin{frame} \begin{frame}
\frametitle{Singular homology of $\oo$\nbd{}categories} \frametitle{Singular homology of $\oo$\nbd{}categories}
\begin{block}{Definition} \begin{block}{Definition}
The \emph{singular homology functor} $\sH^{\sing} \colon \Ho(\oo\Cat^{\Th}) \to The \emph{singular homology functor} $\sH^{\sing} \colon
\Ho(\Ch)$ is defined as the composition \Ho(\oo\Cat^{\Th}) \to \Ho(\Ch)$ is defined as the composition
\[ \[
\Ho(\oo\Cat^{\Th}) \overset{\overline{N_{\oo}}}{\longrightarrow} \Ho(\oo\Cat^{\Th}) \overset{\overline{N_{\oo}}}{\longrightarrow}
\Ho(\Psh{\Delta}) \overset{\overline{\kappa}}{\longrightarrow} \Ho(\Ch). \Ho(\Psh{\Delta}) \overset{\overline{\kappa}}{\longrightarrow} \Ho(\Ch).
\] \]
\end{block} \end{block}
\pause \pause
In pratice, the $k$\nbd{}th singular homology group of an $\oo$\nbd{}category $C$ is In practice, the $k$\nbd{}th singular homology group of an $\oo$\nbd{}category $C$ is
the $k$\nbd{}th homology group of $\kappa(N_{\oo}(C))$ the $k$\nbd{}th homology group of $\kappa(N_{\oo}(C))$
\[ \[
\begin{aligned} \begin{aligned}
...@@ -233,7 +233,122 @@ ...@@ -233,7 +233,122 @@
equivalence of categories. equivalence of categories.
\end{frame} \end{frame}
\end{document} \begin{frame}
\frametitle{Equivalence of $\oo$\nbd{}categories and the folk model
structure}
For every $n \in \mathbb{N}$,
\begin{itemize}[label=-]
\item<2-> let $\sD_n$ be the ``$n$\nbd{}globe'' $\oo$\nbd{}category:
\begin{columns}
\column{0.5\textwidth}
\pause\pause \[
\sD_0=\bullet,
\]
\pause
\[
\begin{tikzcd}[ampersand replacement=\&]
\sD_1=\bullet \to \bullet,
\end{tikzcd}
\]
\column{0.5\textwidth}
\pause
\[
\sD_2=
\begin{tikzcd}[ampersand replacement=\&]
\bullet\ar[r,bend left=50,""{name=A,below}] \ar[r,bend
right=50,""{name=B,above}]\& \bullet,
\ar[from=A,to=B,Rightarrow]
\end{tikzcd}
\]
\pause
\[
\sD_3=
\begin{tikzcd}[ampersand replacement=\&]
\bullet \ar[r,bend left=50,""{name = U,below,near
start},""{name = V,below,near end}] \ar[r,bend
right=50,""{name=D,near start},""{name = E,near end}]\&\bullet,
\ar[Rightarrow, from=U,to=D, bend right,""{name=
L,above}]\ar[Rightarrow, from=V,to=E, bend left,""{name=
R,above}] \arrow[phantom,"\Rrightarrow",from=L,to=R]
\end{tikzcd}
\]
\end{columns}
\begin{center}
etc.
\end{center}
\item<7-> let $\sS_n$ be the ``$n$\nbd{}sphere'' $\oo$\nbd{}category:
\begin{columns}
\column{0.5\textwidth}
\pause\pause
\[
\sS_0=\emptyset,
\]
\pause
\[
\sS_1=
\begin{tikzcd}[ampersand replacement=\&]
\bullet \& \bullet
\end{tikzcd}
\]
\column{0.5\textwidth}
\pause
\[
\sS_2 =
\begin{tikzcd}[ampersand replacement=\&]
\bullet\ar[r,bend left=50] \ar[r,bend right=50]\& \bullet
\end{tikzcd}
\]
\pause
\[
\sS_3=
\begin{tikzcd}[ampersand replacement=\&]
\bullet \ar[r,bend left=50,""{name = U,below,near
start},""{name = V,below,near end}] \ar[r,bend
right=50,""{name=D,near start},""{name = E,near end}]\&\bullet.
\ar[Rightarrow, from=U,to=D, bend right,""{name=
L,above}]\ar[Rightarrow, from=V,to=E, bend left,""{name=
R,above}]
\end{tikzcd}
\]
\end{columns}
\begin{center}
etc.
\end{center}
\item<12-> let $i_n : \sS_n \to \sD_n$ be the ``boundary'' inclusion.
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Equivalence of $\oo$\nbd{}categories and the folk model
structure}
\begin{alertblock}{Theorem (Lafont,Métayer,Worytkiewicz - 2009)}
There exists a model structure on $\oo\Cat$ such that:
\begin{itemize}[label=$\bullet$]
\item the weak equivalences are the equivalences of
$\oo$\nbd{}categories,
\item the set $\{i_n : \sS_n \to \sD_n \vert n \in \mathbb{N}\}$ is
a set of generating cofibrations.
\end{itemize}
\end{alertblock}
\pause
It is known as the \alert{folk model structure} on $\oo\Cat$.
\pause
\begin{alertblock}{Theorem (Métayer - 2008)}
The cofibrant objects of the folk model structure are exactly the
$\oo$\nbd{}categories that are free on a polygraph.
\end{alertblock}
\end{frame}
\begin{frame}
\frametitle{Polygraphs}
Terminological convention:
\begin{center}
free $\oo$\nbd{}category = $\oo$\nbd{}category
free on a polygraph.
\end{center}
\end{frame}
\end{document}
%%% Local Variables: %%% Local Variables:
%%% mode: latex %%% mode: latex
%%% TeX-master: t %%% TeX-master: t
......
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