Slowly but surely

pres.tex

@@ -4,6 +4,18 @@
 \usepackage{mystyle}
 \usetheme{Madrid}
 \usecolortheme{beaver}
+
+%gets rid of bottom navigation bars
+\setbeamertemplate{footline}[frame number]{}
+
+%gets rid of bottom navigation symbols
+\setbeamertemplate{navigation symbols}{}
+
+%gets rid of footer
+%will override 'frame number' instruction above
+%comment out to revert to previous/default definitions
+\setbeamertemplate{footline}{}
+
 \title{Homology of strict $\omega$-categories}
 %\subtitle{PhD defense}
 \author{Léonard Guetta}
@@ -19,6 +31,16 @@
 %   \tableofcontents
 % \end{frame}
 
+\begin{frame}
+  \frametitle{Preliminary conventions}
+  In this talk:
+  \begin{itemize}
+  \item<2-> $\oo$\nbd{}category = strict $\omega$\nbd{}category
+    \item<3-> $n$\nbd{}category = $\oo$\nbd{}category with only unit cells above
+      dimension $n$
+      \item<4-> the functor $n\Cat \to \oo\Cat$ is an inclusion 
+  \end{itemize}
+  \end{frame}
 %%% oo-categories as spaces
 
 \begin{frame}
@@ -78,6 +100,84 @@
     \end{aligned}
     \]
   \end{block}
+  \pause
+  This yields the \alert{nerve functor} for $\oo$\nbd{}categories
+  \[
+    \begin{aligned}
+      N_{\oo} : \oo\Cat &\to \Psh{\Delta} \\
+      C &\mapsto N_{\oo}(C).
+      \end{aligned}
+  \]
+\end{frame}
+\begin{frame}
+  \frametitle{$\oo$\nbd{}categories as spaces}
+  TODO : Exemple en basse dimension
+\end{frame}
+\begin{frame}
+  \frametitle{$\oo$\nbd{}categories as spaces}
+  \begin{block}{Definition}
+    A morphism $f\colon C \to D$ of $\oo\Cat$ is a \emph{Thomason equivalence}
+    if $N_{\oo}(f)\colon N_{\oo}(C) \to N_{\oo}(D)$ is a weak equivalence of
+    simplicial sets. 
+  \end{block}
+    \pause $\W^{\Th}$:=class of Thomason equivalences. \pause By definition, the
+    nerve functor induces 
+    \[
+      \overline{N_{\oo}} : \Ho(\oo\Cat^{\Th}) \to \Ho(\Psh{\Delta}).
+    \]
+    Where:
+    \begin{itemize}[label=$\bullet$]
+      \item $\Ho(\oo\Cat^{\Th})$ is the localization of $\oo\Cat$ with respect to
+        $\W^{\Th}$,
+        \item $\Ho(\Psh{\Delta})$ is the localization of $\Psh{\Delta}$ with
+          respect to weak equivalences of simplicial sets.
+      \end{itemize}
+    % Where $\Ho(-)$ stands for the localized category (or better
+    % the localized pre-derivator or even weak $(\oo,1)$\nbd{}category).
+  \end{frame}
+\begin{frame}
+  \frametitle{$\oo$\nbd{}categories as spaces}
+  \begin{alertblock}{Theorem (Gagna, 2018)}
+    $\overline{N_{\oo}} : \Ho(\oo\Cat^{\Th}) \to \Ho(\Psh{\Delta})$ is an
+    equivalence of categories (or better an equivalence of derivators, or of weak $(\infty,1)$\nbd{}categories).
+  \end{alertblock}
+  \pause In other words:
+  \begin{center}
+    Homotopy theory of $\oo$\nbd{}categories induced by Thomason equivalences \\$\cong$\\ Homotopy theory of spaces
+    \end{center}
+  \end{frame}
+  \begin{frame}
+    \frametitle{Singular homology of $\oo$\nbd{}categories}
+    Recall that we have the (normalized) chain complex functor
+    \[
+      \kappa \colon \Psh{\Delta} \to \Ch,
+    \]
+    Where $\Ch$ is the category of non-negatively graded chain complexes.\pause
+
+    
+    This functor sends weak equivalences of simplicial sets to quasi-isomorphisms.
+    Hence,
+    \[
+      \overline{\kappa} \colon \Ho(\Psh{\Delta}) \to \Ho(\Ch),
+    \]
+    where $\Ho(\Ch)$ is the localization of $\Ch$ with respect to quasi-isomorphisms.
+  \end{frame}
+  \begin{frame}
+    \frametitle{Singular homology of $\oo$\nbd{}categories}
+    \begin{block}{Definition}
+      The \emph{singular homology functor} $\sH^{\sing} \colon \Ho(\oo\Cat^{\Th}) \to
+      \Ho(\Ch)$ is defined as the composition
+      \[
+        \Ho(\oo\Cat^{\Th}) \overset{\overline{N_{\oo}}}{\longrightarrow}
+        \Ho(\Psh{\Delta}) \overset{\overline{\kappa}}{\longrightarrow} \Ho(\Ch).
+      \]
+    \end{block}
+    \pause
+    In pratice, this means that the $k$\nbd{}th singular homology group of an $\oo$\nbd{}category $C$ is
+      the $k$\nbd{}th homology group of $N_{\oo}(C)$,
+      \[
+        H_k^{\sing}(C):=H_k(N_{\oo}(C)).
+      \]
   \end{frame}
 \end{document}
 %%% Local Variables: