Commit 11053638 by Leonard Guetta

 \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). ... ...