Commit b1657859 authored by Leonard Guetta's avatar Leonard Guetta
Browse files

Almost all that is left is the 2-category part

parent 77e33870
\documentclass[handout]{beamer} \documentclass{beamer}
%\usepackage[utf8]{inputenc} %\usepackage[utf8]{inputenc}
\usepackage{mystyle} \usepackage{mystyle}
...@@ -318,7 +318,7 @@ ...@@ -318,7 +318,7 @@
\begin{center} \begin{center}
etc. etc.
\end{center} \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{itemize}
\end{frame} \end{frame}
\begin{frame} \begin{frame}
...@@ -621,23 +621,94 @@ ...@@ -621,23 +621,94 @@
\begin{itemize}[label=$\bullet$] \begin{itemize}[label=$\bullet$]
\item<2-> each $A/a$ is oplax contractible, hence homologically coherent, \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 \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} \end{itemize}
\pause\pause \pause\pause
All that is left to show is that we also have \[\hocolim_{a \in All that is left to show is that we also have \[\hocolim_{a \in
A}^{\folk}A/a\simeq \colim_{a \in A}A/a\simeq A.\] 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 % \pause In order to do
% this, let $f : P \to A$ be a folk cofibrant replacement of $A$, and for % 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{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{frame}\frametitle{In practice}
\begin{exampleblock}{Corollary} \begin{exampleblock}{Corollary}
Let Let
......
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