Commit 77e33870 authored by Leonard Guetta's avatar Leonard Guetta
Browse files

Slowly but surely

parent 2a65308a
\documentclass{beamer} \documentclass[handout]{beamer}
%\usepackage[utf8]{inputenc} %\usepackage[utf8]{inputenc}
\usepackage{mystyle} \usepackage{mystyle}
...@@ -470,7 +470,7 @@ ...@@ -470,7 +470,7 @@
\begin{frame} \begin{frame}
\frametitle{Singular homology as a derived functor} \frametitle{Singular homology as a derived functor}
\begin{alertblock}{Theorem (Guetta - 2020)} \begin{alertblock}{Theorem (G. - 2020)}
The functor $\lambda : \oo\Cat \to \Ch$ is left derivable w.r.t the The functor $\lambda : \oo\Cat \to \Ch$ is left derivable w.r.t the
\emph{Thomason equivalences} on $\oo\Cat$ and we have \emph{Thomason equivalences} on $\oo\Cat$ and we have
\[ \[
...@@ -586,7 +586,7 @@ ...@@ -586,7 +586,7 @@
Let $C$ be an $\oo$\nbd{}category. Suppose that there exists $d : I Let $C$ be an $\oo$\nbd{}category. Suppose that there exists $d : I
\to \oo\Cat$ such that: \to \oo\Cat$ such that:
\begin{enumerate}[label=(\roman*)] \begin{enumerate}[label=(\roman*)]
\item<2-> $\displaystyle\hocolim^{\pol}_I(d)\simeq \hocolim^{\Th}_I(d) \item<2-> $\displaystyle\hocolim^{\folk}_I(d)\simeq \hocolim^{\Th}_I(d)
\simeq C,$ \simeq C,$
\item<3-> for each $i \in \Ob(I)$, the $\oo$\nbd{}category $d(i)$ is \item<3-> for each $i \in \Ob(I)$, the $\oo$\nbd{}category $d(i)$ is
homologically coherent. homologically coherent.
...@@ -596,6 +596,47 @@ ...@@ -596,6 +596,47 @@
% \pause % \pause
% Often, we will use: % Often, we will use:
\end{frame}
\begin{frame}\frametitle{The case of 1-categories}
\begin{alertblock}{Theorem (G. - 2019)}
Every (small) category is homologically coherent.
\end{alertblock}
\pause
\underline{Remark 1:} The homology (polygraphic or singular)
of a category need not be trivial above dimension $1$.
%Hence the previous result is \emph{not} trivial.
\pause
\underline{Remark 2:} Extension of Lafont and Métayer's result on the
homology of monoids, but more precise and completely new proof.
\end{frame}
\begin{frame}\frametitle{The case of 1-categories}
\emph{Sketch of proof:}
Let $A$ be a small category. Recall that
\[
\colim_{a \in A}A/a \simeq A.
\]
\pause Moreover:
\begin{itemize}[label=$\bullet$]
\item<2-> each $A/a$ is oplax contractible, hence homologically coherent,
\item<3-> $\displaystyle\hocolim_{a \in A}^{\Th}A/a \simeq \colim_{a \in A}A/a
\simeq A$ (Thomason's result from 1980).
\end{itemize}
\pause\pause
All that is left to show is that we also have \[\hocolim_{a \in
A}^{\folk}A/a\simeq \colim_{a \in A}A/a\simeq A.\]
\pause How do we prove that ? Let us take a detour.
% \pause In order to do
% this, let $f : P \to A$ be a folk cofibrant replacement of $A$, and for
% each $a \in A$, the $\oo$\nbd{}category $P/a$ defined as
% \[
% \begin{tikzcd}[ampersand replacement=\&]
% P/a \ar[r] \ar[d] \& P \ar[d,"f"] \\
% A/a \ar[r] \& A.
% \ar[from=1-1,to=2-2,"\lrcorner",phantom,very near start]
% \end{tikzcd}
% \]
\end{frame} \end{frame}
\begin{frame}\frametitle{In practice} \begin{frame}\frametitle{In practice}
\begin{exampleblock}{Corollary} \begin{exampleblock}{Corollary}
...@@ -604,19 +645,38 @@ ...@@ -604,19 +645,38 @@
\begin{tikzcd}[ampersand replacement=\&] \begin{tikzcd}[ampersand replacement=\&]
A \ar[r,"u"] \ar[d,"v"] \& B \ar[d] \\ A \ar[r,"u"] \ar[d,"v"] \& B \ar[d] \\
C \ar[r] \& D C \ar[r] \& D
\ar[from=1-1,to=2-2,"\ulcorner",phantom] \ar[from=1-1,to=2-2,"\ulcorner",very near end,phantom]
\end{tikzcd} \end{tikzcd}
\] \]
be a cocartesian square in $\oo\Cat$. If be a cocartesian square in $\oo\Cat$. If
\begin{itemize}[label=$\bullet$] \begin{itemize}[label=$\bullet$]
\item<2-> $A$,$B$ and $C$ are homologically coherent, \item<2-> $A$,$B$ and $C$ are homologically coherent,
\item<3-> $u$ or $v$ is a folk cofibration, \item<3-> $u$ or $v$ is a folk cofibration,
\item<4-> the square is homotopy cocartesian w.r.t Thomason equivalence, \item<4-> the square is homotopy cocartesian w.r.t Thomason equivalences,
\end{itemize} \end{itemize}
then $D$ is homologically coherent. \pause\pause\pause\pause then $D$ is homologically coherent.
\end{exampleblock} \end{exampleblock}
\end{frame} \end{frame}
\begin{frame}\frametitle{Easy application: homology of globes and spheres}
For every $n\geq 0$, $\sD_n$ is oplax contractible, hence
homologically coherent.\pause Moreover, we have
\[
\begin{tikzcd}[ampersand replacement=\&]
\sS_{n-1} \ar[r,"i_n"] \ar[d,"i_n"] \& \sD_n \ar[d] \\
\sD_n \ar[r] \& \sS_n,
\ar[from=1-1,to=2-2,"\ulcorner",very near end, phantom]
\end{tikzcd}
\]
(with $\sS_{-1}=\emptyset$).
\pause
\begin{exampleblock}{Perfect situation:}
The image by $N_{\oo}$ of the previous square is a cocartesian square
of monos, hence homotopy cocartesian.
\end{exampleblock}
\pause By an immediate induction, $\sS_n$ is homologically coherent
(and has the homotopy type of an $n$\nbd{}sphere).
\end{frame}
\end{document} \end{document}
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