Commit 598b64ca authored by Leonard Guetta's avatar Leonard Guetta
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idem

parent 02338bc4
\chapter{Homology and abelianization of \texorpdfstring{$\oo$}{ω}-categories} \chapter{Homology and abelianization of \texorpdfstring{$\oo$}{ω}-categories}
\chaptermark{Homology of $\omega$-categories}
\section{Homology via the nerve} \section{Homology via the nerve}
\begin{paragr} \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 cofibrantly generated model structure, known as the \emph{projective model structure on $\Ch$}, where: We denote by $\Ch$ the category of non-negatively graded chain complexes of abelian groups. Recall that $\Ch$ can be equipped with a cofibrantly generated model structure, known as the \emph{projective model structure on $\Ch$}, where:
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...@@ -526,7 +526,9 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with ...@@ -526,7 +526,9 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with
\begin{remark} \begin{remark}
All the results we have seen in this section are still true if we replace ``oplax'' by ``lax'' everywhere. All the results we have seen in this section are still true if we replace ``oplax'' by ``lax'' everywhere.
\end{remark} \end{remark}
\section{Equivalences of \texorpdfstring{$\oo$}{ω}-categories and the folk model structure} \section{Equivalences of \texorpdfstring{$\oo$}{ω}-categories and the folk model
structure}
\sectionmark{The folk model structure}
\begin{paragr}\label{paragr:ooequivalence} \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: 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} \begin{itemize}
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...@@ -542,6 +542,7 @@ We now turn to the most important way of obtaining op\nbd{}prederivators. ...@@ -542,6 +542,7 @@ We now turn to the most important way of obtaining op\nbd{}prederivators.
reader can refer to any of the references on derivators previously cited. reader can refer to any of the references on derivators previously cited.
\end{paragr} \end{paragr}
\section{Morphisms of op-derivators, preservation of homotopy colimits} \section{Morphisms of op-derivators, preservation of homotopy colimits}
\sectionmark{Morphisms of op-derivators}
We refer to \cite{leinster1998basic} for the precise definitions of We refer to \cite{leinster1998basic} for the precise definitions of
pseudo-natural transformation (called strong transformation there) and pseudo-natural transformation (called strong transformation there) and
modification. modification.
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...@@ -20,6 +20,11 @@ ...@@ -20,6 +20,11 @@
%%% %%%
\fi \fi
\usepackage{fancyhdr}
\pagestyle{fancy}
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\fancyfoot[C]{\thepage}
%%% For line numbering (used for proodreading purposes) %%% For line numbering (used for proodreading purposes)
%% \usepackage[pagewise,displaymath, mathlines]{lineno} %% \usepackage[pagewise,displaymath, mathlines]{lineno}
...@@ -166,6 +171,10 @@ homotopy theory, polygraphs. ...@@ -166,6 +171,10 @@ homotopy theory, polygraphs.
\newpage \newpage
\pagenumbering{arabic} \pagenumbering{arabic}
\fancyhf{}
\fancyfoot[C]{\thepage}
\fancyhead[RE]{\rightmark}
\fancyhead[LO]{\leftmark}
\include{introduction} \include{introduction}
\include{introduction_fr} \include{introduction_fr}
\include{omegacat} \include{omegacat}
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