Commit 9f16f3b1 by Leonard Guetta

### qsdf

parent f6abe0b6
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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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