Commit e7cb9aa8 authored by Leonard Guetta's avatar Leonard Guetta
Browse files

security commit

parent 6253f148
\documentclass[handout]{beamer}
\documentclass{beamer}
%\usepackage[utf8]{inputenc}
\usepackage{mystyle}
......@@ -107,12 +107,14 @@
N_{\oo} : \oo\Cat &\to \Psh{\Delta} \\
C &\mapsto N_{\oo}(C).
\end{aligned}
\]
\end{frame}
\begin{frame}
\frametitle{$\oo$\nbd{}categories as spaces}
TODO : Exemple en basse dimension
\]
\pause
\begin{exampleblock}{Example}
When $C$ is a (1-)category, $N_{\oo}(C)$ is nothing but the usual nerve of
$C$.
\end{exampleblock}
\end{frame}
\begin{frame}
\frametitle{$\oo$\nbd{}categories as spaces}
\begin{block}{Definition}
......@@ -341,12 +343,72 @@
\end{frame}
\begin{frame}
\frametitle{Polygraphs}
\begin{block}{Definition}
An $\oo$\nbd{}category is free on a polygraph if it can be obtained
recursively from the empty category by freely
attaching cells.
\end{block}
\pause
Terminological convention:
\begin{center}
free $\oo$\nbd{}category = $\oo$\nbd{}category
free on a polygraph.
\end{center}
free on a polygraph.
\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 $C$.
\end{exampleblock}
TODO : Laisser le bloc ci-dessus ?
\end{frame}
\begin{frame}
\frametitle{Abelianization of $\oo$\nbd{}categories}
Recall that by a variation of the Dold--Kan equivalence, we have:
\[
\Ab(\oo\Cat) \simeq \Ch,
\]
\pause
hence, a forgetful functor
\[\Ch\simeq \Ab(\oo\Cat) \to \oo\Cat,
\]
\pause which
has a left adjoint
\[
\lambda : \oo\Cat \to \Ch,
\]
which we refer to as the \alert{abelianization functor}.
\end{frame}
\begin{frame}
\frametitle{Polygraphic homology}
\begin{alertblock}{Proposition (folklore ?)}
The functor $\lambda : \oo\Cat \to \Ch$ is left Quillen w.r.t the folk
model structure on $\oo\Cat$ and the projective model structure on $\Ch$.
\end{alertblock}
\pause
\begin{block}{Definition}
The \alert{polygraphic homology functor} is the left derived functor of
$\lambda$:
\[
\sH^{\pol}:=\LL \lambda \colon \Ho(\oo\Cat^{\folk})\to \Ho(\Ch),
\]
where $\Ho(\ooCat^{\folk})$ is the localization of $\oo\Cat$ w.r.t the
equivalences of $\oo$\nbd{}categories.
\end{block}
\end{frame}
\begin{frame}
\frametitle{Polygraphic homology practically}
TODO
\end{frame}
\begin{frame}
% \frametitle{}
A natural question:
\begin{center}
Let $C$ be an $\oo$\nbd{}category. Do we have $\sH^{\pol}(C) \simeq
\sH^{\sing}(C)$ ?
\end{center}
% \pause
% Answer : In general, \textbf{no} !
\end{frame}
\end{document}
%%% Local Variables:
......
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