Commit 9f16f3b1 authored by Leonard Guetta's avatar Leonard Guetta
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qsdf

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\documentclass{beamer}
\documentclass[handout]{beamer}
%\usepackage[utf8]{inputenc}
\usepackage{mystyle}
......@@ -154,10 +154,10 @@
\]
where:
\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 $\oo\Cat$ w.r.t
the Thomason equivalences,
\item $\Ho(\Psh{\Delta})$ is the localization of $\Psh{\Delta}$ with
respect to weak equivalences of simplicial sets.
\item $\Ho(\Psh{\Delta})$ is the localization of $\Psh{\Delta}$ w.r.t
the weak equivalences of simplicial sets.
\end{itemize}
% Where $\Ho(-)$ stands for the localized category (or better
% the localized pre-derivator or even weak $(\oo,1)$\nbd{}category).
......@@ -523,7 +523,7 @@
\pause
Hence, both $\sH^{\pol}$ and $\sH^{\sing}$ are obtained as left derived
functors of $\lambda$ but not w.r.t the same class of weak equivalences.
\begin{exampleblock}{Corollary}
\begin{exampleblock}{Corollary (abstract non-sense)}
There is a canonical natural transformation
\[
\begin{tikzcd}[ampersand replacement=\&]
......@@ -544,7 +544,7 @@
comparison map}.
\pause
\begin{block}{Definition}
An $\oo$\nbd{}category $C$ is \alert{homogically coherent} if the
An $\oo$\nbd{}category $C$ is \alert{homologically coherent} if the
map
\[
\pi_C : \sH^{\sing}(C) \to \sH^{\folk}(C)
......@@ -552,7 +552,7 @@
is an isomorphism.
\end{block}
\pause
Goal: Understand which $\oo$\nbd{}categories are homogically coherent.
Goal: Understand which $\oo$\nbd{}categories are homologically coherent.
\end{frame}
\begin{frame}
\frametitle{Polygraphic homology is not homotopical}
......@@ -572,7 +572,7 @@
\pause
\begin{exampleblock}{New slogan}
The polygraphic homology is a
way of computing the singular homology of homogically coherent
way of computing the singular homology of homologically coherent
$\oo$\nbd{}categories.
\end{exampleblock}
\end{frame}
......@@ -597,7 +597,7 @@
for $k=2,3$, for any $\oo$\nbd{}category $C$ ?
\end{exampleblock}
\end{frame}
\section{Detecting homologically coherent $\oo$-categories I}
\section{Detecting homologically coherent $\oo$-categories}
\begin{frame}\frametitle{Preliminaries: oplax contractile
$\oo$\nbd{}categories}
\begin{block}{Definition}
......@@ -633,7 +633,7 @@
In other words, for a diagram $d : I \to \oo\Cat$, the
canonical map
\[
\hocolim_{I}^{\folk}(d) \to \hocolim_{I}^{\Th}(d)
\hocolim_{I}^{\Th}(d) \to \hocolim_{I}^{\folk}(d)
\]
is not an isomorphism in general.
\pause
......@@ -710,7 +710,7 @@
A}^{\folk}A/a\simeq \colim_{a \in A}A/a\simeq A.\]
\pause
Let $f : P \longrightarrow A$ be a folk cofibrant
resolution of $A$. \pause (Note that $P$ is free but need not be a
replacement of $A$. \pause (Note that $P$ is free but need not be a
$1$\nbd{}category).
% \pause How do we prove that ? Let us take a detour.
% \pause In order to do
......@@ -791,7 +791,7 @@
cofibrant. \hfill CQFD
\end{itemize}
\end{frame}
\section{Detecting homologically coherent $\oo$-categories II}
\section{The case of $2$-categories}
% \begin{frame}\frametitle{A criterion}
% A variation of the homotopy colimit criterion:
% \begin{exampleblock}{Proposition}
......
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