Commit 452b3570 authored by Leonard Guetta's avatar Leonard Guetta
Browse files

Slowly but surely

parent 492695a6
......@@ -4,6 +4,18 @@
\usepackage{mystyle}
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\usecolortheme{beaver}
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\setbeamertemplate{footline}[frame number]{}
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\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}
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