\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 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:

\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 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,

...

...

@@ -1086,7 +1086,7 @@ We now turn to truncations of chain complexes.

\end{itemize}

\end{proposition}

\begin{proof}

This is a typical example of a transfer of a cofibrantely generated model structure along a right adjoint as in \cite[Proposition 2.3]{beke2001sheafifiableII}. The only \emph{a priori} non-trivial hypothesis to check is that there exists a set $J$ of generating trivial cofibrations of the projective model structure on $\Ch$ such that for every $j : A \to B$ in $J$ and every cocartesian square

This is a typical example of a transfer of a cofibrantly generated model structure along a right adjoint as in \cite[Proposition 2.3]{beke2001sheafifiableII}. The only \emph{a priori} non-trivial hypothesis to check is that there exists a set $J$ of generating trivial cofibrations of the projective model structure on $\Ch$ such that for every $j : A \to B$ in $J$ and every cocartesian square

\[

\begin{tikzcd}

\tau^{i}_{\leq n}(A)\ar[r]\ar[d,"\tau^{i}_{\leq n}(j)"']& X \ar[d,"g"]\\

@@ -548,7 +548,7 @@ For later reference, we put here the following trivial but important lemma, whos

\end{example}

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

There exists a cofibrantely generated model structure on $\omega\Cat$ such that the weak equivalences are the equivalences of $\omega$-categories, and the set $\{i_n : \sS_{n-1}\to\sD_n \vert n \in\mathbb{N}\}$ (see \ref{paragr:defglobe}) is a set of generating cofibrations.

There exists a cofibrantly generated model structure on $\omega\Cat$ such that the weak equivalences are the equivalences of $\oo$\nbd{}categories, and the set $\{i_n : \sS_{n-1}\to\sD_n \vert n \in\mathbb{N}\}$ (see \ref{paragr:defglobe}) is a set of generating cofibrations.

\end{theorem}

\begin{proof}

This is the main result of \cite{lafont2010folk}.

...

...

@@ -655,7 +655,7 @@ The nerve functor $N_{\omega} : \omega\Cat \to \Psh{\Delta}$ sends the equivalen

%% is the identity on objects.

This functor cannot be an equivalence since this would imply that every Thomason equivalence is an equivalence of $\oo$\nbd{}categories.

\end{paragr}

\section{Slices of\texorpdfstring{$\oo$}{ω}-category and a folk Theorem A}

\section{Slice \texorpdfstring{$\oo$}{ω}-categories and a folk Theorem A}

\begin{paragr}\label{paragr:slices}

Let $A$ be an $\oo$\nbd{}category and $a_0$ an object of $A$. We define the slice $\oo$\nbd{}category $A/a_0$ as the following fibred product:

\[

...

...

@@ -897,7 +897,6 @@ The nerve functor $N_{\omega} : \omega\Cat \to \Psh{\Delta}$ sends the equivalen

whose image by $u/c_0$ is equivalent for the relation $\sim_{\oo}$ to the above $(n+1)$\nbd{}cell of $B/c_0$. In particular, the source and target of $\alpha$ are respectively $f$ and $f'$. Finally, we obtain that $\alpha\sim_{\oo}\beta$ by applying the canonical $\oo$\nbd{}functor $A/b_0\to A$, .

\end{enumerate}

\end{proof}

%\todo{Il faudrait vérifier que je n'ai pas écrit de bêtises dans la preuve précédente.}

\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.

@@ -118,7 +118,7 @@ x\comp_n y \comp_m z = (x \comp_n y) \comp_m z \text{ and } x \comp_m y \comp_n

\]

whenever these equations make sense.

A \emph{morphism of $\oo$-magmas}$f : X \to Y$ is a morphism of underlying $\oo$\nbd{}graphs that is compatible with units and compositions, i.e.\ for every $n$\nbd{}cell $x$, we have

A \emph{morphism of $\oo$\nbd{}magmas}$f : X \to Y$ is a morphism of underlying $\oo$\nbd{}graphs that is compatible with units and compositions, i.e.\ for every $n$\nbd{}cell $x$, we have

\[

f(1_x)=1_{f(x)},

\]

...

...

@@ -1623,7 +1623,7 @@ with $w_1$ and $w_2$ well formed words and $0\leq k \leq n$. Then $\src_k(w_1)=\

\section{Proof of Theorem \ref{thm:conduche}: part II}

Recall that we have seen in Proposition \ref{prop:smallestcategoricalcongruence}

that there exists a smallest categorical congruence on every $n$\nbd{}magma $X$

with $n \geq1$. However, the description of this congruence that we we used for

with $n \geq1$. However, the description of this congruence that we used for

proving its existence is rather abstract. The main goal of this section is to

give a more concrete description of the smallest categorical congruence in the

case that $X=\E^+$ for an $n$\nbd{}cellular extension $\E$. This description