where $\eta$ is the unit of the adjunction $\tau^{i}_{\leq n}\dashv\iota_n$.

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@@ -937,7 +937,7 @@ This is \cite[Theorem 5]{lafont2010folk}. (Although the part concerning generati

\item Let $y$ be a $0$\nbd-cell of $\iota_n\tau^{i}_{\leq n}(D)$. The map $\eta_D$ being surjective on $0$\nbd-cells (even if $n=0$), there exists $y'$ such that $\eta_D(y')=y$. Since $f$ is a folk weak equivalence, there exists $x' \in C_0$ such that $x'\simeq_{\oo} y'$ and then $x:=\eta_C(x')\simeq_{\oo} y'$.

\item Let $x$ and $y$ be parallel

\end{enumerate}

\toto{À finir}

\todo{À finir}

\end{proof}

\begin{paragr}

Let $\Ch^{\leq n}$ be the category of chain complexes in degree comprised between $0$ and $n$. This means that an object $K$ of $\Ch^{\leq n}$ is a diagram of abelian groups of the form

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@@ -1183,3 +1183,8 @@ Finally, we obtain the result we were aiming for.

%% is commutative (up to an isomorphism).

%% \end{lemma}

%%\section{Homology and Homotopy of $\oo$-categories in low dimension}

The general framework in which the work to be presented in this dissertation takes place is the \emph{homotopy theory of strict $\oo$-categories}, and, as the title suggests, the focus is on homological aspects of this theory. The goal pursued is to study and compare two different homological invariants for strict $\oo$\nbd-categories; that is to say, two different functors \[\mathbf{Str}\oo\Cat\to\ho(\Ch)\] from the category of strict $\oo$-categories to the homotopy category of chain complexes in non-negative degre (i.e.\ the localization of the category of chain complexes in non-negative degree with respect to quasi-isomorphisms). Before entering the heart of the subject and explaining precisely what the above means, let us immediately mention that, with the only exception of the end of this introduction, all the $\oo$-categories considered will be strict. Hence, we drop the adjective ``strict'' and simply say \emph{$\oo$-category} instead of \emph{strict $\oo$-category}. Consequently, we write $\oo\Cat$ instead of $\mathbf{Str}\oo\Cat$ for the category of (strict) $\oo$-categories.

The general framework in which the work to be presented in this dissertation takes place is the \emph{homotopy theory of strict $\oo$-categories}, and, as the title suggests, the focus is on homological aspects of this theory. The goal pursued is to study and compare two different homological invariants for strict $\oo$\nbd-categories; that is to say, two different functors \[\mathbf{Str}\oo\Cat\to\ho(\Ch)\] from the category of strict $\oo$-categories to the homotopy category of chain complexes in non-negative degree (i.e.\ the localization of the category of chain complexes in non-negative degree with respect to quasi-isomorphisms). Before entering the heart of the subject and explaining precisely what the above means, let us immediately mention that, with the only exception of the end of this introduction, all the $\oo$-categories considered will be strict. Hence, we drop the adjective ``strict'' and simply say \emph{$\oo$-category} instead of \emph{strict $\oo$-category}. Consequently, we write $\oo\Cat$ instead of $\mathbf{Str}\oo\Cat$ for the category of (strict) $\oo$-categories.

\begin{named}[Background: $\oo$-categories as spaces] The homotopy theory of $\oo$-categories most certainly started with the introduction by Street \cite{street1987algebra} of a nerve functor

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@@ -10,7 +10,7 @@ that associates to any $\oo$-category $C$ a simplicial set $N_{\oo}(C)$ called t

where $\ho(\oo\Cat^{\Th})$ is the localization of $\oo\Cat$ with respect to Thomason weak equivalences and $\ho(\Psh{\Delta})$ is the localization of $\Psh{\Delta}$ with respect to (Quillen) weak equivalences. As it happens, the functor $\overline{N_{\omega}}$ is an equivalence of categories, as proved by Gagna in \cite{gagna2018strict}. In other words, the homotopy theory of $\oo$-categories induced by Thomason weak equivalences is the same as the homotopy theory of spaces. Gagna's result is in fact a generalization of the analoguous result for the usual nerve of small categories, which is attributed to Quillen in \cite{illusie1972complexe}. In the case of small categories, Thomason even showed the existence of a model structure whose weak equivalences are the ones induced by the nerve \cite{thomason1980cat}. The analoguous result for $\oo\Cat$ is conjectured but not yet established \cite{ara2014vers}.

where $\ho(\oo\Cat^{\Th})$ is the localization of $\oo\Cat$ with respect to Thomason weak equivalences and $\ho(\Psh{\Delta})$ is the localization of $\Psh{\Delta}$ with respect to (Quillen) weak equivalences. As it happens, the functor $\overline{N_{\omega}}$ is an equivalence of categories, as proved by Gagna in \cite{gagna2018strict}. In other words, the homotopy theory of $\oo$-categories induced by Thomason weak equivalences is the same as the homotopy theory of spaces. Gagna's result is in fact a generalization of the analoguous result for the usual nerve of small categories, which is attributed to Quillen in \cite{illusie1972complexe}. In the case of small categories, Thomason even showed the existence of a model structure whose weak equivalences are the ones induced by the nerve \cite{thomason1980cat}. The analogous result for $\oo\Cat$ is conjectured but not yet established \cite{ara2014vers}.

\end{named}

\begin{named}[Two homologies for $\oo$-categories]

Having in mind the nerve functor of Street, a most natural thing to do is to define the \emph{$k$-th homology group of an $\oo$-category $C$} as the $k$-th homology group of the nerve of $C$. In light of Gagna's result, these homology groups are just another way of looking at the homology groups of spaces. In order to explicitly avoid future confusion, we shall now use the name \emph{Street homology groups} of $C$ for these homology groups and use the notation $H^{\St}_k(C)$.

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@@ -142,3 +142,8 @@ This way of understanding polygraphic homology as a left derived functor has bee

%% In the same way that bicategories and tricategories are ``weak'' variations of the notions of (strict) $2$-categories and $3$-categories, there exists a general notion of \emph{weak $\oo$-categories}. These objects can be defined, for example, using the formalism of Grothendieck's coherators \cite{maltsiniotis2010grothendieck}, or of Batanin's globular operads \cite{batanin1998monoidal}. (In fact, each of these formalism give rise to many different possible notions of weak $\oo$-categories, which are conjectured to be all equivalent, at least in some higher categorical sense.)