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 \chapter{Homotopy and homology type of free $2$-categories} \section{Preliminaries: the case of free $1$-categories}\label{section:prelimfreecat} \chapter{Homotopy and homology type of free 2-categories} \section{Preliminaries: the case of free 1-categories}\label{section:prelimfreecat} In this section, we review some homotopical results on free ($1$-)categories that will be of great help in the sequel. \begin{paragr} ... ... @@ -1021,7 +1021,7 @@ of $2$-categories. \ref{prop:streetvsbisimplicial} and Corollary \ref{cor:bisimplicialsquare}. \end{proof} \section{Zoology of $2$-categories: basic examples} \section{Zoology of 2-categories: basic examples} \begin{paragr}\label{paragr:criterion2cat} Before embarking on computations of homology and homotopy types of $2$\nbd{}categories, let us recall the following particular case of Corollary ... ... @@ -1176,7 +1176,7 @@ following proposition. If $n=m=0$, then $A_{(0,0)}$ is not \good{} and has the homotopy type of a $K(\mathbb{Z},2)$. \end{proposition} \section{Zoology of $2$-categories: more examples} \section{Zoology of 2-categories: more examples} As a warm-up, let us begin with an example which is direct consequence of the results at the end of the previous section. \begin{paragr} ... ...
 \chapter{Homology of contractible $\omega$-categories its consequences} \section{Contractible $\oo$-categories} \chapter{Homology of contractible \texorpdfstring{$\oo$}{ω}-categories its consequences} \section{Contractible \texorpdfstring{$\oo$}{ω}-categories} Recall that for any $\oo$\nbd{}category $C$, we write $p_C : C \to \sD_0$ the canonical morphism to the terminal object of $\sD_0$. \begin{definition}\label{def:contractible} An $\oo$\nbd{}category $C$ is \emph{oplax contractible} when the canonical morphism $p_C : C \to \sD_0$ is an oplax homotopy equivalence (Definition \ref{def:oplaxhmtpyequiv}). ... ...
 \chapter{Homology and abelianization of $\oo$-categories} \chapter{Homology and abelianization of \texorpdfstring{$\oo$}{ω}-categories} \section{Homology via the nerve} \begin{paragr} We denote by $\Ch$ the category of non-negatively graded chain complexes of abelian groups. Recall that $\Ch$ can be equipped with a cofibrantely model structure where: ... ...
 \chapter{Homotopy theory of $\oo$-categories} \chapter{Homotopy theory of \texorpdfstring{$\oo$}{ω}-categories} \section{Nerve} \begin{paragr}\label{paragr:simpset} We denote by $\Delta$ the category whose objects are the finite non-empty totally ordered sets $[n]=\{0<\cdots 0$ and $0\leq i\leq n$, we denote by ... ... @@ -500,7 +500,7 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with \begin{remark} All the results we have seen in this section are still true if we replace oplax'' by lax'' everywhere. \end{remark} \section{Equivalences of $\omega$-categories and the folk model structure} \section{Equivalences of \texorpdfstring{$\oo$}{ω}-categories and the folk model structure} \begin{paragr}\label{paragr:ooequivalence} Let $C$ be an $\omega$-category. We define the equivalence relation $\sim_{\omega}$ on the set $C_n$ by co-induction on $n \in \mathbb{N}$. For $x, y \in C_n$, we have $x \sim_{\omega} y$ when: \begin{itemize} ... ... @@ -605,7 +605,7 @@ 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 $\omega$-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} ... ... @@ -649,7 +649,7 @@ The nerve functor $N_{\omega} : \omega\Cat \to \Psh{\Delta}$ sends equivalences %% is the identity on objects. This functor cannot be an equivalence since this would imply that every Thomason equivalence is an equivalence of $\oo$-categories. \end{paragr} \section{Slices of $\oo$-category and a folk Theorem $A$} \section{Slices of \texorpdfstring{$\oo$}{ω}-category and a folk Theorem A} \begin{paragr}\label{paragr:slices} Let $A$ be an $\oo$-category and $a_0$ an object of $A$. We define the slice $\oo$-category $A/a_0$ as the following fibred product: \[ ... ...
 ... ... @@ -322,7 +322,7 @@ We now turn to the most important way of obtaining op-prederivators. where we use the notation $X\vert_{A/b}$ for $k^*(X)$ and $u_!(F)_b$ for $b^*(u_!(X))$. Note that this morphism is reminiscent of the formula that computes pointwise left Kan extensions in the classical'' sense (see for example \cite[chapter X, section 3]{mac2013categories}). %This formula is to be compare with formula \eqref{lknxtfrmla}. \end{paragr} \begin{definition}[Grothendieck] A \emph{right op-derivator} is an op-prederivator $\sD$ such that the following axioms are satisfied: A \emph{right op-derivator} is an op\nbd{}prederivator $\sD$ such that the following axioms are satisfied: \begin{description} \item[Der 1)] For any finite family $(A_i)_{i \in I}$ of small categories, the canonical functor \[ ... ...
 ... ... @@ -395,17 +395,19 @@ for the category of (strict) $\oo$\nbd{}categories. We can go even further and conjecture the same thing for weak $\oo$\nbd{}categories. In order to do so, we need a definition of singular homology for weak $\oo$\nbd{}categories. This is conjecturally done as follows. To every weak $\oo$\nbd{}category $C$, one can associate a weak $\oo$\nbd{}groupoid $L(C)$ by formally inverting all the cells of $C$. Then, if we believe in Grothendieck's conjecture (see \cite[Section 2]{maltsiniotis2010grothendieck}), the category of weak $\oo$\nbd{}groupoids equipped with weak equivalences of weak $\oo$\nbd{}groupoids (see Paragraph 2.2 of op.\ cit.) is a model for the homotopy theory of spaces. In particular, every weak $\oo$\nbd{}groupoid has homology groups and we can define the singular homology groups of a weak $\oo$\nbd{}category $C$ as the homology groups of $L(C)$. $\oo$\nbd{}categories. In order to do so, we need a definition of singular homology for weak $\oo$\nbd{}categories. This is conjecturally done as follows. To every weak $\oo$\nbd{}category $C$, one can associate a weak $\oo$\nbd{}groupoid $L(C)$ by formally inverting all the cells of $C$. Then, if we believe in Grothendieck's conjecture (see \cite{grothendieck1983pursuing} and \cite[Section 2]{maltsiniotis2010grothendieck}), the category of weak $\oo$\nbd{}groupoids equipped with weak equivalences of weak $\oo$\nbd{}groupoids (see Paragraph 2.2 of \cite{maltsiniotis2010grothendieck}) is a model for the homotopy theory of spaces. In particular, every weak $\oo$\nbd{}groupoid has homology groups and we can define the singular homology groups of a weak $\oo$\nbd{}category $C$ as the homology groups of $L(C)$. %% This defines a functor %% \[ ... ...
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 \documentclass[12pt,a4paper,draft]{report} \usepackage[final]{hyperref} \usepackage[unicode,psdextra,final]{hyperref} \usepackage{mystyle} ... ... @@ -23,8 +23,6 @@ \maketitle \tableofcontents \abstract{In this dissertation, we study the homology of strict $\oo$\nbd{}categories. More precisely, we intend to compare the classical'' homology of an $\oo$\nbd{}category (defined as the homology of its Street ... ... @@ -32,6 +30,8 @@ important results concerning free strict $\oo$\nbd{}categories on polygraphs (also known as computads) and concerning the homotopy theory of strict $\oo$\nbd{}categories. } \tableofcontents \include{introduction} \include{omegacat} \include{homtheo} ... ...
 ... ... @@ -263,6 +263,12 @@ author={Grothendieck, Alexandre}, year={~1990}, note={Available at \url{http://webusers.imj-prg.fr/~georges.maltsiniotis/groth/Derivateurs}} } @unpublished{grothendieck1983pursuing, title={Pursuing {S}tacks}, author={Grothendieck, Alexandre}, year={1983}, note={To appear in {D}ocuments {M}ath{'e}matiques, {S}oci{'e}t{'e} {M}ath{'e}matiques de {F}rance} } @unpublished{grothendieck1991letter, title={Lettre à {T}homason}, author={Grothendieck, Alexandre}, ... ...
 ... ... @@ -86,7 +86,7 @@ %Large categories \newcommand{\C}{\ensuremath{\mathcal{C}}} %A random category \renewcommand{\C}{\ensuremath{\mathcal{C}}} %A random category \newcommand{\D}{\ensuremath{\mathcal{D}}} %Another random category \newcommand{\A}{\ensuremath{\mathcal{A}}} %Idem \newcommand{\E}{\ensuremath{\mathcal{E}}} %Idem ... ... @@ -95,7 +95,7 @@ \newcommand{\W}{\ensuremath{\mathcal{W}}} %Idem \newcommand{\R}{\ensuremath{\mathcal{R}}} %Idem \newcommand{\T}{\ensuremath{\mathcal{T}}} %Idem \newcommand{\G}{\ensuremath{\mathcal{G}}} %Idem \renewcommand{\G}{\ensuremath{\mathcal{G}}} %Idem \newcommand{\J}{\ensuremath{\mathcal{J}}} %Idem \newcommand{\M}{\ensuremath{\mathcal{M}}} %Idem ... ...
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