slowly but surely

pres.tex

-\documentclass{beamer}
+\documentclass[handout]{beamer}
 
 %\usepackage[utf8]{inputenc}
 \usepackage{mystyle}
@@ -166,15 +166,15 @@
   \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
+      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, 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 $\kappa(N_{\oo}(C))$
       \[
         \begin{aligned}
@@ -233,7 +233,122 @@
         equivalence of categories.
 
       \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}
+    
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