\chapter{Homology and abelianization of \texorpdfstring{$\oo$}{ω}-categories}

\chapter{Homology and abelianization of \texorpdfstring{$\oo$}{ω}-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 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}

\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 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.

...

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

\end{itemize}

\end{itemize}

\end{proposition}

\end{proposition}

\begin{proof}

\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}

\begin{tikzcd}

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

\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

...

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

\end{example}

\end{example}

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

\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}

\end{theorem}

\begin{proof}

\begin{proof}

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

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

...

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

%% is the identity on objects.

%% 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.

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

\end{paragr}

\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}

\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:

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

...

@@ -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 $A/b_0\to A$, .

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{enumerate}

\end{proof}

\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.

\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

...

@@ -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.

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)},

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)=\

...

@@ -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}

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

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

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

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

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

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

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

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