Commit 9f16f3b1 authored by Leonard Guetta's avatar Leonard Guetta
Browse files

qsdf

parent f6abe0b6
No preview for this file type
\documentclass{beamer} \documentclass[handout]{beamer}
%\usepackage[utf8]{inputenc} %\usepackage[utf8]{inputenc}
\usepackage{mystyle} \usepackage{mystyle}
...@@ -154,10 +154,10 @@ ...@@ -154,10 +154,10 @@
\] \]
where: where:
\begin{itemize}[label=$\bullet$] \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, the Thomason equivalences,
\item $\Ho(\Psh{\Delta})$ is the localization of $\Psh{\Delta}$ with \item $\Ho(\Psh{\Delta})$ is the localization of $\Psh{\Delta}$ w.r.t
respect to weak equivalences of simplicial sets. the weak equivalences of simplicial sets.
\end{itemize} \end{itemize}
% Where $\Ho(-)$ stands for the localized category (or better % Where $\Ho(-)$ stands for the localized category (or better
% the localized pre-derivator or even weak $(\oo,1)$\nbd{}category). % the localized pre-derivator or even weak $(\oo,1)$\nbd{}category).
...@@ -523,7 +523,7 @@ ...@@ -523,7 +523,7 @@
\pause \pause
Hence, both $\sH^{\pol}$ and $\sH^{\sing}$ are obtained as left derived 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. 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 There is a canonical natural transformation
\[ \[
\begin{tikzcd}[ampersand replacement=\&] \begin{tikzcd}[ampersand replacement=\&]
...@@ -544,7 +544,7 @@ ...@@ -544,7 +544,7 @@
comparison map}. comparison map}.
\pause \pause
\begin{block}{Definition} \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 map
\[ \[
\pi_C : \sH^{\sing}(C) \to \sH^{\folk}(C) \pi_C : \sH^{\sing}(C) \to \sH^{\folk}(C)
...@@ -552,7 +552,7 @@ ...@@ -552,7 +552,7 @@
is an isomorphism. is an isomorphism.
\end{block} \end{block}
\pause \pause
Goal: Understand which $\oo$\nbd{}categories are homogically coherent. Goal: Understand which $\oo$\nbd{}categories are homologically coherent.
\end{frame} \end{frame}
\begin{frame} \begin{frame}
\frametitle{Polygraphic homology is not homotopical} \frametitle{Polygraphic homology is not homotopical}
...@@ -572,7 +572,7 @@ ...@@ -572,7 +572,7 @@
\pause \pause
\begin{exampleblock}{New slogan} \begin{exampleblock}{New slogan}
The polygraphic homology is a 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. $\oo$\nbd{}categories.
\end{exampleblock} \end{exampleblock}
\end{frame} \end{frame}
...@@ -597,7 +597,7 @@ ...@@ -597,7 +597,7 @@
for $k=2,3$, for any $\oo$\nbd{}category $C$ ? for $k=2,3$, for any $\oo$\nbd{}category $C$ ?
\end{exampleblock} \end{exampleblock}
\end{frame} \end{frame}
\section{Detecting homologically coherent $\oo$-categories I} \section{Detecting homologically coherent $\oo$-categories}
\begin{frame}\frametitle{Preliminaries: oplax contractile \begin{frame}\frametitle{Preliminaries: oplax contractile
$\oo$\nbd{}categories} $\oo$\nbd{}categories}
\begin{block}{Definition} \begin{block}{Definition}
...@@ -633,7 +633,7 @@ ...@@ -633,7 +633,7 @@
In other words, for a diagram $d : I \to \oo\Cat$, the In other words, for a diagram $d : I \to \oo\Cat$, the
canonical map 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. is not an isomorphism in general.
\pause \pause
...@@ -710,7 +710,7 @@ ...@@ -710,7 +710,7 @@
A}^{\folk}A/a\simeq \colim_{a \in A}A/a\simeq A.\] A}^{\folk}A/a\simeq \colim_{a \in A}A/a\simeq A.\]
\pause \pause
Let $f : P \longrightarrow A$ be a folk cofibrant 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). $1$\nbd{}category).
% \pause How do we prove that ? Let us take a detour. % \pause How do we prove that ? Let us take a detour.
% \pause In order to do % \pause In order to do
...@@ -791,7 +791,7 @@ ...@@ -791,7 +791,7 @@
cofibrant. \hfill CQFD cofibrant. \hfill CQFD
\end{itemize} \end{itemize}
\end{frame} \end{frame}
\section{Detecting homologically coherent $\oo$-categories II} \section{The case of $2$-categories}
% \begin{frame}\frametitle{A criterion} % \begin{frame}\frametitle{A criterion}
% A variation of the homotopy colimit criterion: % A variation of the homotopy colimit criterion:
% \begin{exampleblock}{Proposition} % \begin{exampleblock}{Proposition}
......
Markdown is supported
0% or .
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!
Please register or to comment