security commit

pres.tex

-\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}
     
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