Commit efb96c27 authored by Leonard Guetta's avatar Leonard Guetta
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Edited typos + made a better looking title page

parent bc4fc08f
......@@ -154,7 +154,7 @@ In this section, we review some homotopical results on free
where $\eta$ is the unit of the adjunction $c \dashv N$.
\end{paragr}
\begin{lemma}[Dwyer-Kan]\label{lemma:dwyerkan}
For every $k\leq 1$, the canonical inclusion map
For every $k\geq 1$, the canonical inclusion map
\[
N^{k}(G) \to N^{k+1}(G)
\]
......
......@@ -586,7 +586,7 @@ For later reference, we put here the following trivial but important lemma, whos
An $\omega$-category is cofibrant for the folk model structure if and only if it is free.
\end{proposition}
\begin{proof}
The fact that every free $\omega$-category is cofibrant follows immediately from the fact that the $i_n : \sS_{n-1} \to \sD_n$ are cofibrations and that every $\omega$-category $C$ is the colimit of the canonical diagram (Lemma \ref{lemma:filtration})
The fact that every free $\omega$-category is cofibrant follows immediately from the fact that the $i_n : \sS_{n-1} \to \sD_n$ are cofibrations and that every $\oo$\nbd{}category $C$ is the colimit of the canonical diagram (Lemma \ref{lemma:filtration})
\[
\sk_{0}(C) \to \sk_{1}(C) \to \cdots \to \sk_n(C) \to \sk_{n+1}(C) \to \cdots
\]
......@@ -603,7 +603,7 @@ For later reference, we put here the following trivial but important lemma, whos
See \cite[Proposition 5.1.2.7]{lucas2017cubical} or \cite{ara2019folk}.
\end{proof}
\fi
\section{Equivalences of \texorpdfstring{$\oo$}{ω}-categories vs. Thomason equivalences}
\section{Equivalences of \texorpdfstring{$\oo$}{ω}-categories vs Thomason equivalences}
\begin{lemma}\label{lemma:nervehomotopical}
The nerve functor $N_{\omega} : \omega\Cat \to \Psh{\Delta}$ sends equivalences of $\omega$-categories to weak equivalences of simplicial sets.
\end{lemma}
......
......@@ -191,7 +191,7 @@ we shall use in the sequel.
% \todo{Gonzalez ne formule pas son théorème exactement de cette manière. Il
% faudrait vérifier que je n'ai pas dit de bêtises en le reformulant.}
\section{(op)Derivators and homotopy colimits}
\section{(op-)Derivators and homotopy colimits}
\begin{notation}We denote by $\CCat$ the $2$\nbd{}category of small categories and
$\CCAT$ the $2$\nbd{}category of big categories. For a $2$\nbd{}category
$\underline{A}$, the $2$\nbd{}category obtained from $\underline{A}$ by switching
......
......@@ -104,7 +104,7 @@ for the category of (strict) $\oo$\nbd{}categories.
enough, the general answer to the above question is \emph{no}. A
counterexample was found by Maltsiniotis and Ara. Let $B$ be the commutative
monoid $(\mathbb{N},+)$, seen as a $2$-category with only one $0$-cell and no
non-trivial $1$-cell. This $2$-category is free (as an $\oo$\nbd{}category)
non-trivial $1$-cells. This $2$-category is free (as an $\oo$\nbd{}category)
and a quick computation shows that:
\[
H_k^{\pol}(B)=\begin{cases} \mathbb{Z} &\text{ if } k=0,2 \\ 0 &\text{
......@@ -182,7 +182,7 @@ for the category of (strict) $\oo$\nbd{}categories.
unavailable to prove the left derivability of $\lambda$ and the difficulty was
to find a workaround solution.
From now on, we now set
From now on, we set
\[
\sH^{\sing}(C):=\LL \lambda^{\Th}(C).
\]
......@@ -268,7 +268,7 @@ for the category of (strict) $\oo$\nbd{}categories.
Every (small) category $C$ is homologically coherent.
\end{center}
In order for this result to make sense, one has to consider categories as
$\oo$\nbd{}categories with only unit cells in dimension above $1$. Beware that
$\oo$\nbd{}categories with only unit cells above dimension $1$. Beware that
this does not make the result trivial because given a polygraphic resolution
$P \to C$ of a small category $C$, the $\oo$\nbd{}category $P$ need \emph{not}
have only unit cells above dimension $1$.
......@@ -288,12 +288,12 @@ for the category of (strict) $\oo$\nbd{}categories.
$\oo$\nbd{}functor} when for every cell $x$ of $C$, if $f(x)$ can be written
as
\[
f(x)=y\comp_k y',
f(x)=y'\comp_k y'',
\]
then there exists a unique pair $(x,x')$ of cells of $C$ that are
then there exists a unique pair $(x',x'')$ of cells of $C$ that are
$k$\nbd{}composable and such that
\[
f(x)=y,\, f(x')=y \text{ and } x=x\comp_k x'.
f(x')=y',\, f(x'')=y'' \text{ and } x=x'\comp_k x''.
\]
The main result that we prove concerning discrete Conduché $\oo$\nbd{}functors
is that for a discrete $\oo$\nbd{}functor $f : C \to D$, if the
......@@ -316,9 +316,9 @@ for the category of (strict) $\oo$\nbd{}categories.
\end{tikzcd}
\]
where $P$ is a free $2$\nbd{}category, when is it \emph{homotopy cocartesian}
with respect to Thomason equivalences? As a consequence, a substantial part of
with respect to the Thomason equivalences? As a consequence, a substantial part of
the work presented here consists in developing tools to detect homotopy
cocartesian squares of $2$\nbd{}categories with respect to Thomason
cocartesian squares of $2$\nbd{}categories with respect to the Thomason
equivalences. While it appears that these tools do not allow to completely
answer the above question, they still make it possible to detect such homotopy
cocartesian squares in many concrete situations. In fact, a whole section of
......@@ -445,7 +445,7 @@ for the category of (strict) $\oo$\nbd{}categories.
A]{quillen1973higher} and its $\oo$\nbd{}categorical generalization by Ara and
Maltsiniotis \cite{ara2018theoreme,ara2020theoreme}.
In the fourth chapter is certainly we define the polygraphic and singular homologies of
In the fourth chapter, we define the polygraphic and singular homologies of
$\oo$\nbd{}categories and properly formulate the problem of their comparison.
Up to Section \ref{section:polygraphichmlgy} included, all the results were
known prior to this thesis (at least in the folklore), but starting from
......@@ -457,7 +457,7 @@ for the category of (strict) $\oo$\nbd{}categories.
\ref{prop:comphmlgylowdimension}, which states that low-dimensional singular
and polygraphic homology groups always coincide.
The fifth chapter is mainly geared towards Theorem
The fifth chapter is mainly geared towards the fundamental Theorem
\ref{thm:categoriesaregood}, which states that every category is \good{}. To
prove this theorem, we first focus on a particular class of
$\oo$\nbd{}categories, which we call \emph{contractible
......
No preview for this file type
......@@ -20,9 +20,42 @@
\title{Homology of strict $\omega$-categories}
\author{Léonard Guetta}
%\subtitle{test}
\begin{document}
\maketitle
\begin{titlepage}
\begin{center}
\vspace*{1cm}
\Huge
\textbf{Homology of strict $\omega$-categories}
% \vspace{0.5cm}
% \LARGE
% Thesis Subtitle
\vspace{1.5cm}
\large
\textbf{Léonard Guetta}
\vfill
PhD thesis in mathematics under the supervision of\\
François Métayer and Clemens Berger
\vspace{0.8cm}
\includegraphics[width=0.4\textwidth]{logo_UP.jpg}
\Large
IRIF\\
Université de Paris\\
France\\
\today
\end{center}
\end{titlepage}
\abstract{In this dissertation, we study the homology of strict
$\oo$\nbd{}categories. More precisely, we intend to compare the ``classical''
......
......@@ -5,7 +5,7 @@
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{showlabels}
%\usepackage{showlabels}
\usepackage{chngpage} %allows for temporary adjustment of margin
......
......@@ -1857,7 +1857,7 @@ $\oo$\nbd{}categories.
$\T[\Sigma]$.
\end{itemize}
Intuitively speaking, $\rho_{\Sigma}$ is to be understood as an ``evaluation
map'': for every $w$ well formed expression on units and formal cells of
map'': given $w$ a well formed expression on units and formal cells of
$\Sigma$, $\rho_{\Sigma}(w)$ is the evaluation of $w$ as an $(n+1)$\nbd{}cell
of $C$.
\end{paragr}
......@@ -1973,7 +1973,8 @@ with
\item[-] $\wt{F}(\,(\,)=($,
\item[-] $\wt{F}(\,)\,)=)$.
\end{itemize}
Notice that for every word $w \in \W[\Sigma^C]$, we have $|\wt{F}(w)|=|w|$ and $\mathcal{L}(\wt{F}(w))=\mathcal{L}(w)$.
Notice that for every word $w \in \W[\Sigma^C]$, we have
\[\vert \wt{F}(w) \vert=\vert w \vert\text{ and }\mathcal{L}(\wt{F}(w))=\mathcal{L}(w).\]
\end{paragr}
\begin{lemma}\label{lemmamapinducedonwords}
Let $F : C \to D$ be an $\omega$-functor, $\Sigma^C \subseteq C_{n}$ and $\Sigma^D \subseteq D_{n}$ such that $F_{n}(\Sigma^C)\subseteq \Sigma^D$. For every $u \in \W[\Sigma^C]$:
......
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