Commit 77e33870 by Leonard Guetta

### 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} %%% Local Variables: %%% Local Variables: ... ...
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