Commit 492695a6 authored by Leonard Guetta's avatar Leonard Guetta
Browse files

doucement mais surement

parent 52ab118b
\documentclass{beamer}
\usepackage[utf8]{inputenc}
%\usepackage[utf8]{inputenc}
\usepackage{mystyle}
\usetheme{Madrid}
\usecolortheme{beaver}
\title{Homology of strict $\omega$-categories}
%\subtitle{PhD defense}
\author{Léonard Guetta}
\date{28 January 2021}
\date{PhD defense, 28 January 2021}
\institute{IRIF - Université de Paris}
\begin{document}
\frame{\titlepage}
% \begin{frame}
% \frametitle{Table of Contents}
% \tableofcontents
% \end{frame}
%%% oo-categories as spaces
\begin{frame}
\frametitle{Table of Contents}
\tableofcontents
\frametitle{$\oo$\nbd{}categories as spaces}
Starting point: Street's \emph{orientals}
\[
\Or \colon \Psh{\Delta} \to \oo\Cat.
\]
In pictures:
\[
\Or_0 = \bullet,
\]
\pause
\[
\Or_1=\begin{tikzcd}[ampersand replacement=\&]
\bullet \ar[r] \&\bullet,
\end{tikzcd}
\]
\pause
\[
\Or_2=
\begin{tikzcd}[ampersand replacement=\&]
\& \bullet \ar[rd]\& \\
\bullet \ar[ru]\ar[rr,""{name=A,above}]\&\&\bullet,
\ar[Rightarrow,from=A,to=1-2]
\end{tikzcd}
\]
\pause
\[
\Or_3=
\begin{tikzcd}[ampersand replacement=\&]
\& \bullet \ar[rd]\& \\
\bullet\ar[ru] \ar[rd,""{name=B,above}] \ar[rr,""{name=A,above}]\& \& \bullet \ar[ld]\\
\& \bullet \&
\ar[from=A,to=1-2,Rightarrow, shorten <= 0.25em, shorten >= 0.25em]
\ar[from=B,to=2-3,Rightarrow, near start, shorten <= 1.1em, shorten >= 1.5em]
\end{tikzcd}
\Rrightarrow
\begin{tikzcd}[ampersand replacement=\&]
\& \bullet \ar[rd] \ar[dd,""{name=B,right}] \& \\
\bullet\ar[ru] \ar[rd,""{name=A,above}] \& \& \bullet. \ar[ld]\\
\& \bullet \&
\ar[from=A,to=1-2,Rightarrow, near start, shorten <= 1em, shorten >= 1.5em]
\ar[from=B,to=2-3,Rightarrow, shorten <= 0.75em, shorten >=0.75em]
\end{tikzcd}
\]
\end{frame}
\begin{frame}
\frametitle{$\oo$\nbd{}categories as spaces}
\begin{block}{Definition}
The \alert{nerve} of an $\oo$\nbd{}category $C$ is the simplicial
set
\[
\begin{aligned}
N_{\omega}(C) : \Delta^{\op} &\to \Set\\
[n] &\mapsto \Hom_{\omega\Cat}(\Or_n,C).
\end{aligned}
\]
\end{block}
\end{frame}
\end{document}
%%% Local Variables:
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