Commit 6253f148 by Leonard Guetta

slowly but surely

parent e07cf015
 \documentclass{beamer} \documentclass[handout]{beamer} %\usepackage[utf8]{inputenc} %\usepackage[utf8]{inputenc} \usepackage{mystyle} \usepackage{mystyle} ... @@ -166,15 +166,15 @@ ... @@ -166,15 +166,15 @@ \begin{frame} \begin{frame} \frametitle{Singular homology of $\oo$\nbd{}categories} \frametitle{Singular homology of $\oo$\nbd{}categories} \begin{block}{Definition} \begin{block}{Definition} The \emph{singular homology functor} $\sH^{\sing} \colon \Ho(\oo\Cat^{\Th}) \to The \emph{singular homology functor}$\sH^{\sing} \colon \Ho(\Ch)$is defined as the composition \Ho(\oo\Cat^{\Th}) \to \Ho(\Ch)$ is defined as the composition $\[ \Ho(\oo\Cat^{\Th}) \overset{\overline{N_{\oo}}}{\longrightarrow} \Ho(\oo\Cat^{\Th}) \overset{\overline{N_{\oo}}}{\longrightarrow} \Ho(\Psh{\Delta}) \overset{\overline{\kappa}}{\longrightarrow} \Ho(\Ch). \Ho(\Psh{\Delta}) \overset{\overline{\kappa}}{\longrightarrow} \Ho(\Ch).$ \] \end{block} \end{block} \pause \pause In pratice, the $k$\nbd{}th singular homology group of an $\oo$\nbd{}category $C$ is In practice, the $k$\nbd{}th singular homology group of an $\oo$\nbd{}category $C$ is the $k$\nbd{}th homology group of $\kappa(N_{\oo}(C))$ the $k$\nbd{}th homology group of $\kappa(N_{\oo}(C))$ \[ \begin{aligned} \begin{aligned} ... @@ -233,7 +233,122 @@ ... @@ -233,7 +233,122 @@ equivalence of categories. equivalence of categories. \end{frame} \end{frame} \end{document} \begin{frame} \frametitle{Equivalence of \oo\nbd{}categories and the folk model structure} For every n \in \mathbb{N}, \begin{itemize}[label=-] \item<2-> let \sD_n be the n\nbd{}globe'' \oo\nbd{}category: \begin{columns} \column{0.5\textwidth} \pause\pause \[ \sD_0=\bullet, \pause $\begin{tikzcd}[ampersand replacement=\&] \sD_1=\bullet \to \bullet, \end{tikzcd}$ \column{0.5\textwidth} \pause $\sD_2= \begin{tikzcd}[ampersand replacement=\&] \bullet\ar[r,bend left=50,""{name=A,below}] \ar[r,bend right=50,""{name=B,above}]\& \bullet, \ar[from=A,to=B,Rightarrow] \end{tikzcd}$ \pause $\sD_3= \begin{tikzcd}[ampersand replacement=\&] \bullet \ar[r,bend left=50,""{name = U,below,near start},""{name = V,below,near end}] \ar[r,bend right=50,""{name=D,near start},""{name = E,near end}]\&\bullet, \ar[Rightarrow, from=U,to=D, bend right,""{name= L,above}]\ar[Rightarrow, from=V,to=E, bend left,""{name= R,above}] \arrow[phantom,"\Rrightarrow",from=L,to=R] \end{tikzcd}$ \end{columns} \begin{center} etc. \end{center} \item<7-> let $\sS_n$ be the $n$\nbd{}sphere'' $\oo$\nbd{}category: \begin{columns} \column{0.5\textwidth} \pause\pause $\sS_0=\emptyset,$ \pause $\sS_1= \begin{tikzcd}[ampersand replacement=\&] \bullet \& \bullet \end{tikzcd}$ \column{0.5\textwidth} \pause $\sS_2 = \begin{tikzcd}[ampersand replacement=\&] \bullet\ar[r,bend left=50] \ar[r,bend right=50]\& \bullet \end{tikzcd}$ \pause $\sS_3= \begin{tikzcd}[ampersand replacement=\&] \bullet \ar[r,bend left=50,""{name = U,below,near start},""{name = V,below,near end}] \ar[r,bend right=50,""{name=D,near start},""{name = E,near end}]\&\bullet. \ar[Rightarrow, from=U,to=D, bend right,""{name= L,above}]\ar[Rightarrow, from=V,to=E, bend left,""{name= R,above}] \end{tikzcd}$ \end{columns} \begin{center} etc. \end{center} \item<12-> let $i_n : \sS_n \to \sD_n$ be the boundary'' inclusion. \end{itemize} \end{frame} \begin{frame} \frametitle{Equivalence of $\oo$\nbd{}categories and the folk model structure} \begin{alertblock}{Theorem (Lafont,Métayer,Worytkiewicz - 2009)} There exists a model structure on $\oo\Cat$ such that: \begin{itemize}[label=$\bullet$] \item the weak equivalences are the equivalences of $\oo$\nbd{}categories, \item the set $\{i_n : \sS_n \to \sD_n \vert n \in \mathbb{N}\}$ is a set of generating cofibrations. \end{itemize} \end{alertblock} \pause It is known as the \alert{folk model structure} on $\oo\Cat$. \pause \begin{alertblock}{Theorem (Métayer - 2008)} The cofibrant objects of the folk model structure are exactly the $\oo$\nbd{}categories that are free on a polygraph. \end{alertblock} \end{frame} \begin{frame} \frametitle{Polygraphs} Terminological convention: \begin{center} free $\oo$\nbd{}category = $\oo$\nbd{}category free on a polygraph. \end{center} \end{frame} \end{document} %%% Local Variables: %%% Local Variables: %%% mode: latex %%% mode: latex %%% TeX-master: t %%% TeX-master: t ... ...
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