Commit b1657859 by Leonard Guetta

Almost all that is left is the 2-category part

parent 77e33870
 \documentclass[handout]{beamer} \documentclass{beamer} %\usepackage[utf8]{inputenc} \usepackage{mystyle} ... ... @@ -318,7 +318,7 @@ \begin{center} etc. \end{center} \item<12-> let $i_n : \sS_n \to \sD_n$ be the boundary'' inclusion. \item<12-> let $i_n : \sS_{n-1} \to \sD_n$ be the boundary'' inclusion. \end{itemize} \end{frame} \begin{frame} ... ... @@ -621,23 +621,94 @@ \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). \simeq A$(From Thomason's homotopy colimit theorem). \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 Let$f : P \longrightarrow A$be a folk cofibrant resolution 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 % 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} \begin{frame}\frametitle{The case of$1$\nbd{}categories (sequel)} For each$a \in A$, we define$P/a$as: $\begin{tikzcd}[ampersand replacement=\&] P/a \ar[r] \ar[d] \& P \ar[d,"f"] \\ A/a \ar[r] \& A. \ar[from=1-1,to=2-2,"\lrcorner",phantom,very near start] \end{tikzcd}$ %We have$\displaystyle\colim_{a \in A}P/a \simeq P. \pause \begin{exampleblock}{Crucial lemma} The functor \begin{aligned} P/{-} : A &\to \oo\Cat\\ a &\mapsto P/a \end{aligned} is a cofibrant object for the projective model structure on\underline{\Hom}(A,\oo\Cat)$induced by the folk model structure. \end{exampleblock} \pause Then,$\displaystyle\hocolim^{\folk}_{a \in A}A/a\simeq \hocolim^{\folk}_{a \in A}P/a \simeq \displaystyle\colim_{a \in A}P/a \simeq P \simeq A.$\hfill CQFD \pause But how do we prove the crucial lemma ? Let us take a detour. \end{frame} \begin{frame}\frametitle{Interlude: Conduché discrete$\oo$\nbd{}functor} \begin{block}{Definition} An$\oo$\nbd{}functor$f : C \to D$is \alert{discrete Conduché} if for every$n$\nbd{}cell$x$of$C$that decomposes as $f(x)=y'\comp_k y'',$ there exists a \emph{unique} pair$(x',x'')$of$k$\nbd{}composable$n$\nbd{}cells of$C$such that: \begin{itemize}[label=$\bullet$] \item$x=x'\comp_k x''$, \item$f(x')=y'$and$f(x'')=y''$. \end{itemize} \end{block} \pause \begin{alertblock}{Theorem (G. 2018)} Let$f : C \to D$be a discrete Conduché$\oo$\nbd{}functor. If$D$is free then so is$C$.\pause Moreover the set of generating cells of$C$is the inverse image of those of$D$by$f$. \end{alertblock} \pause \underline{Proof}: Long and tedious but not so hard conceptually. \end{frame} \begin{frame}\frametitle{Sketched proof of the crucial lemma} Back to the square: $\begin{tikzcd}[ampersand replacement=\&] P/a \ar[r] \ar[d] \& P \ar[d,"f"] \\ A/a \ar[r] \& A. \ar[from=1-1,to=2-2,"\lrcorner",phantom,very near start] \end{tikzcd}$ \pause \begin{itemize} \item<2-> It is easy to check that$A/a \to A$is discrete Conduché. \item<3-> Hence, so is$P/a \to P$(stability by pullback of Conduché). \item<4-> Thus, for every$a \in A$, the$\oo$\nbd{}category$P/a$is free. \item<5-> Besides, the set of generating cells of$P/a\$ vary naturally in a''. \item<6-> Worked out properly (cf. thesis), this means that $P/{-} : A \to \oo\Cat$ is cofibrant. \hfill CQFD \end{itemize} \end{frame} \begin{frame}\frametitle{In practice} \begin{exampleblock}{Corollary} Let ... ...
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