Commit 040e2e4b authored by Leonard Guetta's avatar Leonard Guetta
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Premier jet fini. J'ai ajouté le pdf au git également.

parent 12cfb713
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...@@ -15,7 +15,7 @@ ...@@ -15,7 +15,7 @@
%gets rid of footer %gets rid of footer
%will override 'frame number' instruction above %will override 'frame number' instruction above
%comment out to revert to previous/default definitions %comment out to revert to previous/default definitions
\setbeamertemplate{footline}{} %\setbeamertemplate{footline}{}
\title{Homology of strict $\omega$-categories} \title{Homology of strict $\omega$-categories}
%\subtitle{PhD defense} %\subtitle{PhD defense}
...@@ -23,15 +23,23 @@ ...@@ -23,15 +23,23 @@
\date{PhD defense, 28 January 2021} \date{PhD defense, 28 January 2021}
\institute{IRIF - Université de Paris} \institute{IRIF - Université de Paris}
\AtBeginSection[]
{
\begin{frame}[noframenumbering,plain]
\frametitle{Table of Contents}
\tableofcontents[currentsection]
\end{frame}
}
\begin{document} \begin{document}
\frame{\titlepage} \frame[noframenumbering,plain]{\titlepage}
% \begin{frame} % \begin{frame}
% \frametitle{Table of Contents} % \frametitle{Table of Contents}
% \tableofcontents % \tableofcontents
% \end{frame} % \end{frame}
\section{The setting}
\begin{frame} \begin{frame}
\frametitle{Preliminary conventions} \frametitle{Preliminary conventions}
In this talk: In this talk:
...@@ -39,9 +47,12 @@ ...@@ -39,9 +47,12 @@
\item<2-> $\oo$\nbd{}category = strict $\omega$\nbd{}category \item<2-> $\oo$\nbd{}category = strict $\omega$\nbd{}category
\item<3-> $n$\nbd{}category = $\oo$\nbd{}category with only unit cells above \item<3-> $n$\nbd{}category = $\oo$\nbd{}category with only unit cells above
dimension $n$ dimension $n$
\item<4-> the functor $n\Cat \to \oo\Cat$ is an inclusion \item<4-> $1$\nbd{}category = (small) category
\item<5-> the functor $n\Cat \to \oo\Cat$ is an inclusion
\end{itemize} \end{itemize}
\end{frame} \end{frame}
%%% oo-categories as spaces %%% oo-categories as spaces
\begin{frame} \begin{frame}
...@@ -50,7 +61,7 @@ ...@@ -50,7 +61,7 @@
\[ \[
\Or \colon \Psh{\Delta} \to \oo\Cat. \Or \colon \Psh{\Delta} \to \oo\Cat.
\] \]
In pictures: \pause In pictures:
\[ \[
\Or_0 = \bullet, \Or_0 = \bullet,
\] \]
...@@ -159,7 +170,7 @@ ...@@ -159,7 +170,7 @@
where $\Ch$ is the category of non-negatively graded chain complexes.\pause where $\Ch$ is the category of non-negatively graded chain complexes.\pause
This functor sends weak equivalences of simplicial sets to quasi-isomorphisms. This functor sends weak equivalences of simplicial sets to quasi-isomorphisms.\pause
Hence, Hence,
\[ \[
\overline{\kappa} \colon \Ho(\Psh{\Delta}) \to \Ho(\Ch), \overline{\kappa} \colon \Ho(\Psh{\Delta}) \to \Ho(\Ch),
...@@ -396,8 +407,27 @@ ...@@ -396,8 +407,27 @@
\end{block} \end{block}
\end{frame} \end{frame}
\begin{frame} \begin{frame}
\frametitle{Polygraphic homology practically} \frametitle{Polygraphic homology in practice}
TODO Let $C$ be a free $\oo$\nbd{}category and write $\Sigma_k$ for its set
of generating $k$\nbd{}cells.
\pause The polygraphic homology of $C$ is the homology of the chain
complex
\[
\mathbb{Z}\Sigma_0 \overset{\partial}{\longleftarrow}
\mathbb{Z}\Sigma_1 \overset{\partial}{\longleftarrow}
\mathbb{Z}\Sigma_2 \overset{\partial}{\longleftarrow} \cdots,
\]
\pause where for $x \in \Sigma_n$, we have
\[
\partial(x)=\text{``generators in the target of x''
}-\text{``generators in the source of x''}.
\]
\pause
\begin{exampleblock}{Motivation}
For a \emph{free} $\oo$\nbd{}category, the polygraphic homology is \emph{a
priori} simpler to compute than the singular homology.
\end{exampleblock}
\end{frame} \end{frame}
\begin{frame} \begin{frame}
\frametitle{Polygraphic homology vs singular homology} \frametitle{Polygraphic homology vs singular homology}
...@@ -448,6 +478,7 @@ ...@@ -448,6 +478,7 @@
\pause \pause
This is what I tried to answer in my PhD. This is what I tried to answer in my PhD.
\end{frame} \end{frame}
\section{Abstract reformulation}
\begin{frame} \begin{frame}
\frametitle{Equivalence of $\oo$\nbd{}categories vs Thomason \frametitle{Equivalence of $\oo$\nbd{}categories vs Thomason
equivalences} equivalences}
...@@ -554,8 +585,25 @@ ...@@ -554,8 +585,25 @@
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}
\begin{frame}\frametitle{An abstract criterion} \section{Detecting homologically coherent $\oo$-categories I}
Back on the triangle: \begin{frame}\frametitle{Preliminaries: oplax contractile
$\oo$\nbd{}categories}
\begin{block}{Definition}
An $\oo$\nbd{}category $C$ is \alert{oplax contractible} if the
canonical morphism
\[
C \to \sD_0
\]
has an inverse ``up to an oplax transformation''.
\end{block}
\pause
\begin{exampleblock}{Lemma}
Every oplax contractible $\oo$\nbd{}category is homologically
coherent (and has the homotopy type of a point).
\end{exampleblock}
\end{frame}
\begin{frame}\frametitle{An abstract criterion to detect homological coherence}
Back to the triangle:
\[ \[
\begin{tikzcd}[ampersand replacement=\&] \begin{tikzcd}[ampersand replacement=\&]
\Ho(\oo\Cat^{\folk})\ar[d,"\mathcal{J}"'] \ar[dr,"\sH^{\pol}",""{name=A,below}]\& \\ \Ho(\oo\Cat^{\folk})\ar[d,"\mathcal{J}"'] \ar[dr,"\sH^{\pol}",""{name=A,below}]\& \\
...@@ -581,7 +629,7 @@ ...@@ -581,7 +629,7 @@
Idea: exploit that sometimes it \emph{is} an isomorphism. Idea: exploit that sometimes it \emph{is} an isomorphism.
\end{frame} \end{frame}
\begin{frame}\frametitle{An abstract criterion} \begin{frame}\frametitle{An abstract criterion to detect homological coherence}
\begin{block}{Proposition} \begin{block}{Proposition}
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:
...@@ -591,11 +639,33 @@ ...@@ -591,11 +639,33 @@
\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.
\end{enumerate} \end{enumerate}
\pause\pause Then $C$ is homologically coherent. \pause\pause\pause Then $C$ is homologically coherent.
\end{block} \end{block}
%\pause It is our main strategy to detect homologically coherent $\oo$\nbd{}categories
% \pause % \pause
% Often, we will use: % Often, we will use:
\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{equation}\label{squaresphere}\tag{$\ast$}
\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}
\end{equation}
(with $\sS_{-1}=\emptyset$). \pause This square is ``folk homotopy cocartesian''
because $i_n$ is a cofibration.
\pause
\begin{exampleblock}{Exceptional situation:}
The image by $N_{\oo}$ of \eqref{squaresphere} in $\Psh{\Delta}$ is a \emph{cocartesian} square
of monos, hence homotopy cocartesian. \pause It follows that square
\eqref{squaresphere} is ``Thomason 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{frame}
\begin{frame}\frametitle{The case of 1-categories} \begin{frame}\frametitle{The case of 1-categories}
\begin{alertblock}{Theorem (G. - 2019)} \begin{alertblock}{Theorem (G. - 2019)}
...@@ -708,47 +778,30 @@ ...@@ -708,47 +778,30 @@
\[P/{-} : A \to \oo\Cat\] is \[P/{-} : A \to \oo\Cat\] is
cofibrant. \hfill CQFD cofibrant. \hfill CQFD
\end{itemize} \end{itemize}
\end{frame} \end{frame}
\begin{frame}\frametitle{A criterion} \section{Detecting homologically coherent $\oo$-categories II}
A variation of the homotopy colimit criterion: % \begin{frame}\frametitle{A criterion}
\begin{exampleblock}{Proposition} % A variation of the homotopy colimit criterion:
Let % \begin{exampleblock}{Proposition}
\[ % Let
\begin{tikzcd}[ampersand replacement=\&] % \[
A \ar[r,"u"] \ar[d,"v"] \& B \ar[d] \\ % \begin{tikzcd}[ampersand replacement=\&]
C \ar[r] \& D % A \ar[r,"u"] \ar[d,"v"] \& B \ar[d] \\
\ar[from=1-1,to=2-2,"\ulcorner",very near end,phantom] % C \ar[r] \& D
\end{tikzcd} % \ar[from=1-1,to=2-2,"\ulcorner",very near end,phantom]
\] % \end{tikzcd}
be a cocartesian square in $\oo\Cat$. If % \]
\begin{enumerate}[label=(\roman*)] % be a cocartesian square in $\oo\Cat$. If
\item<2-> $A$,$B$ and $C$ are homologically coherent, % \begin{enumerate}[label=(\roman*)]
\item<3-> $u$ or $v$ is a folk cofibration, % \item<2-> $A$,$B$ and $C$ are homologically coherent,
\item<4-> the square is homotopy cocartesian w.r.t Thomason equivalences, % \item<3-> $u$ or $v$ is a folk cofibration,
\end{enumerate} % \item<4-> the square is homotopy cocartesian w.r.t Thomason equivalences,
\pause\pause\pause\pause then $D$ is homologically coherent. % \end{enumerate}
\end{exampleblock} % \pause\pause\pause\pause then $D$ is homologically coherent.
\pause The third condition will usually be the hard one to prove. % \end{exampleblock}
\end{frame} % \pause The third condition will usually be the hard one to prove.
\begin{frame}\frametitle{Easy application: homology of globes and spheres} % \end{frame}
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}
\begin{frame}\frametitle{2-categories} \begin{frame}\frametitle{2-categories}
We would like to understand which 2-categories are homologically We would like to understand which 2-categories are homologically
coherent. coherent.
...@@ -756,7 +809,7 @@ ...@@ -756,7 +809,7 @@
\item<2-> For simplification, we focus on \emph{free} 2-categories. \item<2-> For simplification, we focus on \emph{free} 2-categories.
\item<3-> This boils down to the following: given a cocartesian square \item<3-> Archetypal situation to understand: given a cocartesian square
\[ \[
\begin{tikzcd}[ampersand replacement=\&] \begin{tikzcd}[ampersand replacement=\&]
\sS_1 \ar[d,"i_1"'] \ar[r] \& P \ar[d] \\ \sS_1 \ar[d,"i_1"'] \ar[r] \& P \ar[d] \\
...@@ -857,7 +910,7 @@ ...@@ -857,7 +910,7 @@
\end{frame} \end{frame}
\begin{frame} \begin{frame}
\frametitle{Zoology of 2-categories: Bouquet of spheres} \frametitle{Zoology of 2-categories: Bouquets of spheres}
\begin{center} \begin{center}
\scalebox{0.85}{ \scalebox{0.85}{
\begin{tabular}{ l || c | c } \begin{tabular}{ l || c | c }
...@@ -919,9 +972,54 @@ ...@@ -919,9 +972,54 @@
\pause This $2$\nbd{}category has the homotopy type of the \alert{torus} and \pause This $2$\nbd{}category has the homotopy type of the \alert{torus} and
is homologically coherent. is homologically coherent.
\pause Not so easy to show ! Idea: show that it is Thomason equivalent to the \pause Idea of proof: show that $C$ is Thomason equivalent to the
monoid $(\mathbb{N}\times \mathbb{N},+)$ (which is not free) monoid $(\mathbb{N}\times \mathbb{N},+)$ (which is not free) and use the
equivalence of polygraphic and singular homologies for monoids.
\pause Not that easy to carry out properly !
\end{frame}
\begin{frame}\frametitle{Bubbles}
\begin{block}{Definition}
A \alert{bubble} in a $2$\nbd{}category is a non unit
$2$\nbd{}cell $\alpha$ whose source and target are units on a $0$\nbd{}cell.
\end{block}
\pause
In pictures:
\[
\begin{tikzcd}[ampersand replacement=\&]
A \ar[r,bend left=75,"1_A",""{name=A,below}] \ar[r,bend
right=75,"1_A"',pos=21/40,""{name=B,above}] \&A
\ar[from=A,to=B,"\alpha",Rightarrow]
\end{tikzcd}
\text{ or }
\begin{tikzcd}[ampersand replacement=\&]
A. \ar[loop,in=120,out=60,distance=1cm,"\alpha"',Rightarrow]
\end{tikzcd}
\]
\pause
\begin{block}{Definition}
A $2$\nbd{}category is \alert{bubble-free} if it has no bubbles.
\end{block}
\end{frame}
\begin{frame}\frametitle{The bubble-free conjecture}
The archetypal example of \emph{non} bubble-free $2$\nbd{}category is
the $2$\nbd{}category $B$ from Ara and Maltsiniotis' counter-example.
\pause
In all the examples, the free $2$\nbd{}categories that are homologically
coherent are exactly the bubble-free ones.
\pause
\begin{exampleblock}{Conjecture}
Let $C$ be a free $2$\nbd{}category. It is homologically coherent if and
only if it is bubble-free.
\end{exampleblock}
\end{frame} \end{frame}
\begin{frame}[noframenumbering,plain]
\begin{center}
Merci pour votre attention !
\end{center}
\end{frame}
\end{document} \end{document}
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