\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:

We denote by $\Ch$ the category of non-negatively graded chain complexes of abelian groups. Recall that $\Ch$ can be equipped with a cofibrantely generated model structure where:

\begin{itemize}

\item[-] the weak equivalences are the quasi-isomorphisms, i.e. morphisms of chain complexes that induce an isomorphism on homology groups,

\item[-] the weak equivalences are the quasi-isomorphisms, i.e.\morphisms of chain complexes that induce an isomorphism on homology groups,

\item[-] the cofibrations are the morphisms of chain complexes $f: X\to Y$ such that for every $n\geq0$, $f_n : X_n \to Y_n$ is a monomorphism with projective cokernel,

\item[-] the fibrations are the morphisms of chain complexes $f : X \to Y$ such that for every $n>0$, $f_n : X_n \to Y_n$ is an epimorphism.

\end{itemize}

...

...

@@ -527,7 +527,7 @@ The following proposition is an immediate consequence of Theorem \ref{thm:cisins

Let $\begin{tikzcd} f : y \ar[r,shift left]&z :g\ar[l,shift left]\end{tikzcd}$ be an adjunction and $h : x \to y$ an equivalence with quasi-inverse $k : y \to x$. Then $fh$ is left adjoint to $kg$.

\end{lemma}

We can now state and prove the promised result.

\begin{proposition}\label{prop:hmlgyderived}

\begin{theorem}\label{thm:hmlgyderived}

Consider that $\oo\Cat$ is equipped with the Thomason equivalences. The abelianization functor $\lambda : \oo\Cat\to\Ch$ is strongly left derivable and the left derived morphism

\[

\LL\lambda^{\Th} : \Ho(\oo\Cat^{\Th})\to\Ho(\Ch)

...

...

@@ -536,7 +536,7 @@ The following proposition is an immediate consequence of Theorem \ref{thm:cisins

\[

\sH^{\sing} : \Ho(\oo\Cat^{\Th})\to\Ho(\Ch).

\]

\end{proposition}

\end{theorem}

\begin{proof}

Let $\nu$ be the right adjoint of the abelianization functor (see Lemma \ref{lemma:adjlambda}) and consider the following adjunctions

\[

...

...

@@ -589,7 +589,7 @@ The following proposition is an immediate consequence of Theorem \ref{thm:cisins

A thorough reading of the proofs of Proposition \ref{prop:gonzalezcritder} and Proposition\ref{prop:hmlgyderived} enables us to give the following description of $\alpha^{\sing}$. By post-composing the co-unit of the adjunction $c_{\oo}\dashv N_{\oo}$ with the abelianization functor, we obtain $2$-morphism

A thorough reading of the proofs of Proposition \ref{prop:gonzalezcritder} and Theorem\ref{thm:hmlgyderived} enables us to give the following description of $\alpha^{\sing}$. By post-composing the co-unit of the adjunction $c_{\oo}\dashv N_{\oo}$ with the abelianization functor, we obtain $2$-morphism

@@ -177,9 +177,7 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with

\[

\gamma^{\Th} : n\Cat\to\Ho(n\Cat^{\Th})

\]

for the localization morphism.

The reason is to avoid confusion with other weak equivalences on $n\Cat$ we will later introduce.

for the localization morphism. The reason is to avoid confusion with other weak equivalences on $n\Cat$ which we will introduce later.

\end{paragr}

\begin{paragr}

By definition, the nerve functor induces a morphism of localizers ${N_n : (n\Cat,\W_n^{\Th})\to(\Psh{\Delta},\W_{\Delta})}$ and hence a morphism of op-prederivators

is an equivalence of $\oo$-categories, then so is $u$.

\end{proposition}

\end{theorem}

\begin{proof}

Before anything else, let us note the following trivial but important fact: for any $\oo$\nbd{}functor $F : X \to Y$ and any $n$-cells $x$ and $y$ of $X$, if $x \sim_{\oo} y$, then $F(x)\sim_{\oo} F(y)$.

\begin{paragr} The name ``folk Theorem A'' is an explicit reference of Quillen's Theorem A \cite[Theorem A]{quillen1973higher} and its $\oo$\nbd{}categorical generalization by Ara and Maltsiniotis \cite{ara2018theoreme,ara2020theoreme}. For the sake of comparison we recall below the latter one.

\end{paragr}

\begin{proposition}[Ara and Maltsiniotis' Theorem A] Let

\begin{theorem}[Ara and Maltsiniotis' Theorem A] Let

The present chapter sticks out from the others as it contains no original results. Its goal is simply to introduce the language and tools of homotopical algebra that we shall need in the rest of the dissertation. In consequence, most of the results are simply asserted and the reader will find references to literature for the proofs. The main notion of homotopical algebra we are aiming for is the one of \emph{homotopy colimits} and the language we chose to express this notion is the one given by the theory of Grothendieck's \emph{derivators}\cite{grothendieckderivators}. We do not assume that the reader is familiar with this theory and will quickly recall the basics. If needed, gentle introductions can be found in \cite{maltsiniotis2001introduction} and in a letter from Grothendieck to Thomason \cite{grothendieck1991letter}; more detailed introductions can be found in \cite{groth2013derivators} and in the first section of \cite{cisinski2003images}; finally, a rather complete (yet unfinished and unpublished) textbook on the subject is \cite{groth2013book}.

The present chapter stands out from the others as it contains no original results. Its goal is simply to introduce the language and tools of homotopical algebra that we shall need in the rest of the dissertation. In consequence, most of the results are simply asserted and the reader will find references to literature for the proofs. The main notion of homotopical algebra we are aiming for is the one of \emph{homotopy colimits} and the language we chose to express this notion is the one given by the theory of Grothendieck's \emph{derivators}\cite{grothendieckderivators}. We do not assume that the reader is familiar with this theory and will quickly recall the basics. If needed, gentle introductions can be found in \cite{maltsiniotis2001introduction} and in a letter from Grothendieck to Thomason \cite{grothendieck1991letter}; more detailed introductions can be found in \cite{groth2013derivators} and in the first section of \cite{cisinski2003images}; finally, a rather complete (yet unfinished and unpublished) textbook on the subject is \cite{groth2013book}.

\iffalse Let us quickly motive this choice for the reader unfamiliar with this theory.