Commit e07cf015 authored by Leonard Guetta's avatar Leonard Guetta
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qsdf

parent 452b3570
...@@ -120,15 +120,16 @@ ...@@ -120,15 +120,16 @@
if $N_{\oo}(f)\colon N_{\oo}(C) \to N_{\oo}(D)$ is a weak equivalence of if $N_{\oo}(f)\colon N_{\oo}(C) \to N_{\oo}(D)$ is a weak equivalence of
simplicial sets. simplicial sets.
\end{block} \end{block}
\pause $\W^{\Th}$:=class of Thomason equivalences. \pause By definition, the %\pause $\W^{\Th}$:=class of Thomason equivalences.
\pause By definition, the
nerve functor induces nerve functor induces
\[ \[
\overline{N_{\oo}} : \Ho(\oo\Cat^{\Th}) \to \Ho(\Psh{\Delta}). \overline{N_{\oo}} : \Ho(\oo\Cat^{\Th}) \to \Ho(\Psh{\Delta}),
\] \]
Where: where:
\begin{itemize}[label=$\bullet$] \begin{itemize}[label=$\bullet$]
\item $\Ho(\oo\Cat^{\Th})$ is the localization of $\oo\Cat$ with respect to \item $\Ho(\oo\Cat^{\Th})$ is the localization of $\oo\Cat$ with respect to
$\W^{\Th}$, the Thomason equivalences,
\item $\Ho(\Psh{\Delta})$ is the localization of $\Psh{\Delta}$ with \item $\Ho(\Psh{\Delta})$ is the localization of $\Psh{\Delta}$ with
respect to weak equivalences of simplicial sets. respect to weak equivalences of simplicial sets.
\end{itemize} \end{itemize}
...@@ -143,7 +144,7 @@ ...@@ -143,7 +144,7 @@
\end{alertblock} \end{alertblock}
\pause In other words: \pause In other words:
\begin{center} \begin{center}
Homotopy theory of $\oo$\nbd{}categories induced by Thomason equivalences \\$\cong$\\ Homotopy theory of spaces Homotopy theory of $\oo$\nbd{}categories induced by Thomason equivalences \\$\cong$\\ Homotopy theory of spaces.
\end{center} \end{center}
\end{frame} \end{frame}
\begin{frame} \begin{frame}
...@@ -152,7 +153,7 @@ ...@@ -152,7 +153,7 @@
\[ \[
\kappa \colon \Psh{\Delta} \to \Ch, \kappa \colon \Psh{\Delta} \to \Ch,
\] \]
Where $\Ch$ is the category of non-negatively graded chain complexes.\pause where $\Ch$ is the category of non-negatively graded chain complexes.\pause
This functor sends weak equivalences of simplicial sets to quasi-isomorphisms. This functor sends weak equivalences of simplicial sets to quasi-isomorphisms.
...@@ -173,12 +174,65 @@ ...@@ -173,12 +174,65 @@
\] \]
\end{block} \end{block}
\pause \pause
In pratice, this means that the $k$\nbd{}th singular homology group of an $\oo$\nbd{}category $C$ is In pratice, the $k$\nbd{}th singular homology group of an $\oo$\nbd{}category $C$ is
the $k$\nbd{}th homology group of $N_{\oo}(C)$, the $k$\nbd{}th homology group of $\kappa(N_{\oo}(C))$
\[ \[
H_k^{\sing}(C):=H_k(N_{\oo}(C)). \begin{aligned}
H_k^{\sing}(C)&:=H_k(\sH^{\sing}(C))\\
&=H_k(\kappa(N_{\oo}(C))).
\end{aligned}
\] \]
\end{frame} \end{frame}
\begin{frame}
\frametitle{Equivalence of $\oo$\nbd{}categories and the folk model
structure}
\begin{block}{Definition}
Let $C$ be an $\oo$\nbd{}category and $x,y$ two $n$\nbd{}cells of $C$.
We say that \alert{$x \sim_{\oo} y $} if there exist $(n+1)$\nbd{}cells $r : x \to y $ and $\overline{r} : y
\to x$ such that
\[
r \comp_n \overline{r} \sim_{\oo} 1_y \text{ and } \overline{r} \comp_n
r \sim_{\oo }1_x.
\]
(This definition is co-inductive.)
\end{block}
\pause
\begin{exampleblock}{Example 1}
Let $x$ and $y$ be two objects of a 1-category. We have $x \sim_{\oo} y $
if and only if $x$ and $y$ are \alert{isomorphic}.
\end{exampleblock}
\pause
\begin{exampleblock}{Example 2}
Let $x$ and $y$ be two objects of a $2$\nbd{}category. We have
$x\sim_{\oo} y$ if and only if $x$ and $y$ are \alert{equivalent}.
\end{exampleblock}
\end{frame}
\begin{frame}
\frametitle{Equivalence of $\oo$\nbd{}categories and the folk model
structure}
\begin{block}{Definition}
A morphism $f : C \to D$ of $\oo\Cat$ is an \alert{equivalence of
$\oo$\nbd{}categories} if:
\begin{itemize}[label=$\bullet$]
\item<2-> for every $0$\nbd{}cell $y$ of $D$, there exists a $0$\nbd{}cell $x$
of $C$ such that
\[
f(x)\sim_{\oo} y,
\]
\item<3-> for every parallel $n$\nbd{}cells $x$ and $x'$ of $C$ and for
every $(n+1)$\nbd{}cell $\beta : f(x) \to f(x')$ of $D$, there
exists an $(n+1)$\nbd{}cell $\alpha : x \to x'$ of $C$ such that
\[
f(\alpha) \sim_{\oo} \beta.
\]
\end{itemize}
\end{block}
\pause\pause\pause
When $C$ and $D$ are (1-)categories, we recover the usual notion of
equivalence of categories.
\end{frame}
\end{document} \end{document}
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