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Commit 11053638 authored by Leonard Guetta's avatar Leonard Guetta
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added gentle introduction slides

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\documentclass[handout]{beamer}
\documentclass{beamer}
%\usepackage[utf8]{inputenc}
\usepackage{mystyle}
\usepackage{graphicx}
\usepackage{tikz,tikz-3dplot}
\definecolor{cof}{RGB}{219,144,71}
\definecolor{pur}{RGB}{186,146,162}
\definecolor{greeo}{RGB}{91,173,69}
\definecolor{greet}{RGB}{52,111,72}
\tdplotsetmaincoords{70}{165}
% \usepackage{animate}
% \usepackage{xmpmulti}
......@@ -41,16 +50,193 @@
% \frametitle{Table of Contents}
% \tableofcontents
% \end{frame}
\section{}
\begin{frame}\frametitle{What is this PhD about ?}
The underlying theme of this thesis is the \emph{homotopy theory} of \emph{strict
$\oo$\nbd{}categories}.
\pause Very (very) roughly: Homotopy theory = study of geometric shapes up to deformation
\pause
% \animategraphics[loop,width=0.25\linewidth]{10}{Mug_and_Torus-}{57}{57}
% \transduration<0-57>{0}
% \multiinclude[<+->][format=png, graphics={width=0.25\textwidth}]{Mug_and_Torus}
% \begin{frame}\frametitle{What is this PhD about ?}
% The underlying theme of this thesis is the \alert{homotopy theory} of \alert{strict
% $\oo$\nbd{}categories}.
% \begin{itemize}[label=$\bullet$]
% \item Homotopy theory: study of geometric shapes up to deformation
% \begin{center}
% \begin{tikzpicture}
% \draw (0,0) rectangle (1,1);
% \draw (1.5,0.5) node{=};
% \draw (2.5,0.5) circle (0.5);
% \draw (3.5,0.5) node{=};
% \draw (4,0) -- (4.5,1) -- (5,0) -- cycle;
% \end{tikzpicture}
% \end{center}
% \end{itemize}
% \begin{itemize}
% \item Main tool in homotopy theory: \alert{homotopical invariants}. This means an
% invariant which does not change by a deformation.
% \item Example: number of connected components (= number of pieces).
% \item If two geometric
% shapes have different numbers of connected components, they cannot be deformed from
% one to the other.
% \end{itemize}
% \begin{center}
% \begin{tikzpicture}
% \fill[blue!40!white] (0,0) circle (0.25);
% \draw (0.5,0) node{$\neq$};
% \fill[blue!40!white] (1,0) circle (0.25);
% \fill[blue!40!white] (1.75,0) circle (0.25);
% \end{tikzpicture}
% \end{center}
% \end{frame}
% \begin{frame}\frametitle{What is this PhD about ?}
% \begin{itemize}[label=$\bullet$]
% \item Strict $\oo$\nbd{}categories are \alert{geometrico-algebraic} mathematical structures.
% \begin{itemize}[label=-]
% \item Geometric nature:
% \begin{center}
% \begin{tabular}{l | c }
% strict $\oo$\nbd{}category & polyhedron \\ \hline
% $0$\nbd{}cell & vertex \\
% $1$\nbd{}cell & edge \\
% $2$\nbd{}cell & face \\
% $\cdots$ & $\cdots$ \\
% \end{tabular}
% \begin{tikzpicture}[scale=0.75,tdplot_main_coords]
% \coordinate (O) at (0,0,0);
% % \draw[thick,->] (0,0,0) -- (1,0,0) node[anchor=north east]{$x$};
% % \draw[thick,->] (0,0,0) -- (0,1,0) node[anchor=north west]{$y$};
% % \draw[thick,->] (0,0,0) -- (0,0,1) node[anchor=south]{$z$};
% \tdplotsetcoord{A}{1}{90}{0} % cartesian (1,0,0)
% \tdplotsetcoord{B}{1}{90}{90} % cartesian (0,1,0)
% \tdplotsetcoord{C}{1}{90}{180} % cartesian (-1,0,0)
% \tdplotsetcoord{D}{1}{90}{270} % cartesian (0,-1,0)
% \tdplotsetcoord{E}{1}{0}{0} % cartesian (0,0,1)
% \tdplotsetcoord{F}{1}{180}{0} % cartesian (0,0,-1)
% \draw (A) -- (B) -- (C);
% \draw (E) -- (A) -- (F);
% \draw (E) -- (B) -- (F);
% \draw (E) -- (C) -- (F);
% \draw[dashed] (C) -- (D) -- (A);
% \draw[dashed](E) -- (D) -- (F);
% \fill[cof,opacity=0.6](A) -- (B) -- (E) -- cycle;
% \fill[pur,opacity=0.6](A) -- (B) -- (F) -- cycle;
% \fill[greeo,opacity=0.6](B) -- (C) -- (E) -- cycle;
% \fill[greet,opacity=0.6](B) -- (C) -- (F) -- cycle;
% \end{tikzpicture}
% \end{center}
% \item Algebraic nature: operations on the cells.
% \end{itemize}
% \end{itemize}
% Roughly :
% \begin{center}
% strict $\oo$\nbd{}categories $\simeq$ algebraic way of representing
% ``geometric shapes''.
% \end{center}
% \pause Hence, we can do homotopy theory of strict $\oo$\nbd{}categories.
% \pause In this thesis, we study and compare two invariants on
% $\oo$\nbd{}categories: one is called \alert{polygraphic homology} and the other
% \alert{singular homology}...
% \end{frame}
\begin{frame}\frametitle{De quoi parle cette thèse ?}
Le thème sous-jacent à cette thèse est la \alert{théorie de l'homotopie} des
\alert{$\oo$\nbd{}catégories strictes}.
\pause
\begin{itemize}[label=$\bullet$]
\item Théorie de l'homotopie: étude des formes géométriques à déformation
près
\pause
\begin{center}
\begin{tikzpicture}
\draw (0,0) rectangle (1,1);
\draw (1.5,0.5) node{=};
\draw (2.5,0.5) circle (0.5);
\draw (3.5,0.5) node{=};
\draw (4,0) -- (4.5,1) -- (5,0) -- cycle;
\end{tikzpicture}
\end{center}
\end{itemize}
\begin{itemize}
\item<4-> Outil principal en théorie de l'homotopie : les \alert{invariants
homotopiques}. C'est-à-dire des invariants qui ne changent pas par une
déformation.
\item<5-> Example: nombre de composantes connexes (= nombre de morceaux).
\item<6-> Si deux formes géométriques n'ont pas le même nombre de morceaux, on ne
peut pas déformer l'une en l'autre.
\end{itemize}
\pause\pause\pause
\begin{center}
\begin{tikzpicture}
\fill[blue!40!white] (0,0) circle (0.25);
\draw (0.5,0) node{$\neq$};
\fill[blue!40!white] (1,0) circle (0.25);
\fill[blue!40!white] (1.75,0) circle (0.25);
\end{tikzpicture}
\end{center}
\end{frame}
\begin{frame}\frametitle{De quoi parle cette thèse ?}
\begin{itemize}[label=$\bullet$]
\item Les $\oo$\nbd{}catégories strictes sont des objets
\alert{géométrico-algébriques}.
\pause
\begin{itemize}[label=-]
\item Nature géométrique:
\begin{center}
\begin{tabular}{l | c }
$\oo$\nbd{}catégorie stricte & polyèdre \\ \hline
$0$\nbd{}cellule & sommet \\
$1$\nbd{}cellule & arête \\
$2$\nbd{}cellule & face \\
$\cdots$ & $\cdots$ \\
\end{tabular}
\begin{tikzpicture}[scale=0.75,tdplot_main_coords]
\coordinate (O) at (0,0,0);
% \draw[thick,->] (0,0,0) -- (1,0,0) node[anchor=north east]{$x$};
% \draw[thick,->] (0,0,0) -- (0,1,0) node[anchor=north west]{$y$};
% \draw[thick,->] (0,0,0) -- (0,0,1) node[anchor=south]{$z$};
\tdplotsetcoord{A}{1}{90}{0} % cartesian (1,0,0)
\tdplotsetcoord{B}{1}{90}{90} % cartesian (0,1,0)
\tdplotsetcoord{C}{1}{90}{180} % cartesian (-1,0,0)
\tdplotsetcoord{D}{1}{90}{270} % cartesian (0,-1,0)
\tdplotsetcoord{E}{1}{0}{0} % cartesian (0,0,1)
\tdplotsetcoord{F}{1}{180}{0} % cartesian (0,0,-1)
\draw (A) -- (B) -- (C);
\draw (E) -- (A) -- (F);
\draw (E) -- (B) -- (F);
\draw (E) -- (C) -- (F);
\draw[dashed] (C) -- (D) -- (A);
\draw[dashed](E) -- (D) -- (F);
\fill[cof,opacity=0.6](A) -- (B) -- (E) -- cycle;
\fill[pur,opacity=0.6](A) -- (B) -- (F) -- cycle;
\fill[greeo,opacity=0.6](B) -- (C) -- (E) -- cycle;
\fill[greet,opacity=0.6](B) -- (C) -- (F) -- cycle;
\end{tikzpicture}
\end{center}
\pause
\item Nature algébrique : opérations sur les cellules.
\end{itemize}
\end{itemize}
\pause
En bref :
\begin{center}
$\oo$\nbd{}catégories strictes $\simeq$ façon algébrique de représenter des
``formes géométriques''.
\end{center}
\pause Ainsi, on peut faire la théorie de l'homotopie des $\oo$\nbd{}catégories.
\pause Dans cette thèse, on étudie et compare deux invariants sur les
$\oo$\nbd{}catégories strictes : l'un s'appelle l'\alert{homologie
polygraphique} et l'autre s'appelle l'\alert{homologie singulière}...
\end{frame}
\section{The setting}
\begin{frame}
\frametitle{Preliminary conventions}
......@@ -379,6 +565,9 @@
free on a polygraph.
\end{center}
\pause
Examples of free $\oo$\nbd{}categories: the orientals, the globes, the
spheres,...
\pause
\begin{exampleblock}{Important fact}
If $C$ is a free $\oo$\nbd{}category, then there is a \emph{unique} set
of generating cells possible.
......@@ -462,12 +651,23 @@
\end{frame}
\begin{frame}
\frametitle{Ara and Maltsiniotis' counter-example}
Let $B$ be the commutative monoid $(\mathbb{N},+)$ considered as a
$2$\nbd{}category with exactly one $0$\nbd{}cell and one $1$\nbd{}cell:
\[
B = \begin{tikzcd}[ampersand replacement=\&] \bullet \& \bullet
\ar[l,shift right] \ar[l,shift left] \& \mathbb{N} \ar[l,shift right] \ar[l,shift left].\end{tikzcd}
\]
Let $B$ be the $2$\nbd{}category freely generated by
\begin{itemize}[label=-]
\item one object: $\bullet$,
\item one $2$\nbd{}cell: $1_{\bullet} \Rightarrow 1_{\bullet}$.
\end{itemize}
\pause
\[B=
\begin{tikzcd}[ampersand replacement=\&]
\bullet \ar[loop,in=120,out=60,distance=1cm,Rightarrow]
\end{tikzcd}
\]
% the commutative monoid $(\mathbb{N},+)$ considered as a
% $2$\nbd{}category with exactly one $0$\nbd{}cell and one $1$\nbd{}cell:
% \[
% B = \begin{tikzcd}[ampersand replacement=\&] \bullet \& \bullet
% \ar[l,shift right] \ar[l,shift left] \& \mathbb{N} \ar[l,shift right] \ar[l,shift left].\end{tikzcd}
% \]
\pause $B$ is free as an $\oo$\nbd{}category and we have
\[
H_k^{\pol}(B)\simeq \begin{cases}\mathbb{Z} &\text{ if } k=0,2 \\ 0 &
......@@ -609,6 +809,9 @@
has an inverse ``up to an oplax transformation''.
\end{block}
\pause
Example: A $1$\nbd{}category with a terminal object is oplax
contractible.
\pause
\begin{exampleblock}{Lemma}
Every oplax contractible $\oo$\nbd{}category is homologically
coherent (and has the homotopy type of a point).
......
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