Commit 5ad01d75 authored by Leonard Guetta's avatar Leonard Guetta
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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 $[n]=\{0<\cdots<n\}$ and whose morphisms are the non-decreasing maps. For $n > 0$ 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
%% \[
......
No preview for this file type
\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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