qsdf

pres.tex

@@ -120,15 +120,16 @@
     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
+  %\pause $\W^{\Th}$:=class of Thomason equivalences.
+  \pause By definition, the
     nerve functor induces 
     \[
-      \overline{N_{\oo}} : \Ho(\oo\Cat^{\Th}) \to \Ho(\Psh{\Delta}).
+      \overline{N_{\oo}} : \Ho(\oo\Cat^{\Th}) \to \Ho(\Psh{\Delta}),
     \]
-    Where:
+    where:
     \begin{itemize}[label=$\bullet$]
       \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
           respect to weak equivalences of simplicial sets.
       \end{itemize}
@@ -143,7 +144,7 @@
   \end{alertblock}
   \pause In other words:
   \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{frame}
   \begin{frame}
@@ -152,7 +153,7 @@
     \[
       \kappa \colon \Psh{\Delta} \to \Ch,
     \]
-    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.
@@ -173,12 +174,65 @@
       \]
     \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 $N_{\oo}(C)$,
+    In pratice, the $k$\nbd{}th singular homology group of an $\oo$\nbd{}category $C$ is
+      the $k$\nbd{}th homology group of $\kappa(N_{\oo}(C))$
       \[
-        H_k^{\sing}(C):=H_k(N_{\oo}(C)).
+        \begin{aligned}
+        H_k^{\sing}(C)&:=H_k(\sH^{\sing}(C))\\
+        &=H_k(\kappa(N_{\oo}(C))).
+        \end{aligned}
       \]
-  \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}
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