Commit 2a65308a authored by Leonard Guetta's avatar Leonard Guetta
Browse files

very very slowly but surely

parent 05c221bd
......@@ -499,6 +499,7 @@
\]
which is natural in $C$. We refer to it as the \alert{canonical
comparison map}.
\pause
\begin{block}{Definition}
An $\oo$\nbd{}category $C$ is \alert{homogically coherent} if the
map
......@@ -584,13 +585,38 @@
\begin{block}{Proposition}
Let $C$ be an $\oo$\nbd{}category. Suppose that there exists $d : I
\to \oo\Cat$ such that:
% \begin{enumerate}[label=($\roman$)]
% \item \[
% \hocolim^{\pol}_I(d)\simeq \hocolim^{\Th}_I(d) \simeq C
% \]
% \end{enumerate}
\end{block}
\end{frame}
\begin{enumerate}[label=(\roman*)]
\item<2-> $\displaystyle\hocolim^{\pol}_I(d)\simeq \hocolim^{\Th}_I(d)
\simeq C,$
\item<3-> for each $i \in \Ob(I)$, the $\oo$\nbd{}category $d(i)$ is
homologically coherent.
\end{enumerate}
\pause\pause Then $C$ is homologically coherent.
\end{block}
% \pause
% Often, we will use:
\end{frame}
\begin{frame}\frametitle{In practice}
\begin{exampleblock}{Corollary}
Let
\[
\begin{tikzcd}[ampersand replacement=\&]
A \ar[r,"u"] \ar[d,"v"] \& B \ar[d] \\
C \ar[r] \& D
\ar[from=1-1,to=2-2,"\ulcorner",phantom]
\end{tikzcd}
\]
be a cocartesian square in $\oo\Cat$. If
\begin{itemize}[label=$\bullet$]
\item<2-> $A$,$B$ and $C$ are homologically coherent,
\item<3-> $u$ or $v$ is a folk cofibration,
\item<4-> the square is homotopy cocartesian w.r.t Thomason equivalence,
\end{itemize}
then $D$ is homologically coherent.
\end{exampleblock}
\end{frame}
\end{document}
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