Commit e7cb9aa8 by Leonard Guetta

### 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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