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Leonard Guetta
PhD-presentation
Commits
11053638
Commit
11053638
authored
Jan 26, 2021
by
Leonard Guetta
Browse files
added gentle introduction slides
parent
9f16f3b1
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pres.pdf
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pres.tex
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11053638
\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
\[
B
=
\begin
{
tikzcd
}
[
ampersand replacement
=
\&
]
\bullet
\ar
[
loop,in
=
120
,out
=
60
,distance
=
1
cm,Rightarrow
]
\end
{
tikzcd
}
\]
% 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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