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.

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

\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

\[

N_{\omega} : \oo\Cat\to\Psh{\Delta}

\]

...

...

@@ -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 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 analoguous 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 $H^{\St}_k(C)$.

...

...

@@ -114,74 +114,6 @@ This way of understanding polygraphic homology as a left derived functor has bee

In order for this result to make sense, one has to consider categories as $\oo$\nbd-categories with only unit cells in dimension above $1$. Beware that this doesn't make the result trivial because given a polygraphic resolution $P \to C$ of a small category $C$, the $\oo$-category $P$ need \emph{not} have only unit cells above dimension $1$.

As such, this result is only a small generalization of Lafont and Métayer's result concerning monoids (although this new result, even restricted to monoids, is more precise because it means that the \emph{canonical comparison map} is an isomorphim). But the novelty lies in the proof which is more conceptual that the one of Lafont and Métayer and of which we now give a outline.

Let $C$ be a small category. Recall that we have a canonical isomorphim

\[

\colim_{c \in C}C/c \simeq C,

\]

where $C/c$ is the slice category over the object $c$ of $C$. Now the idea of the proof is to show that this colimit is in fact a homotopy colimit both in $\ho(\oo\Cat^{\Th})$ and in $\ho(\oo\Cat^{\Th})$. This means respectively that:

\begin{enumerate}[label=(\alph*)]

\item the canonical morphism of $\ho(\oo\Cat^{\Th})$

(The subscript of the homotopy colimits are there to make sure not to remind in which ``homotopy theory'' on $\oo\Cat$ the homotopy colimit is to be understood.)

Case (a) follows somewhat easily from Thomason's homotopy colimit theorem \cite{thomason1979homotopy}. Case (b) on the other hand is a new result and first appears in \cite{guetta2020homology}. The idea of the proof is as follows: let $f : P \to C$ a polygraphic resolution, i.e.\ $f$ is folk trivial fibration and $P$ is a free $\oo$-category, and consider the diagram

\begin{align*}

C &\to\oo\Cat\\

c &\mapsto P/c,

\end{align*}

where $P/c$ is the $\oo$-category defined as the following fibred product

\[

\begin{tikzcd}

P/c \ar[r]\ar[d]& P \ar[d,"f"]\\

C/c \ar[r]& C. \ar[from=1-1,to=2-2,phantom,very near start,"\lrcorner"]

\end{tikzcd}

\]

Using the fact that $f$ is a trivial fibration and some 2-out-of-3 property, the problem can be reduced to showing that the canonical map

is an isomorphism of $\ho(\oo\Cat^{\folk})$. Now, this follows from the ``miracle'' that the diagram $c \mapsto P/c$ is a cofibrant object of the projective model structure on the category $\underline{\Hom}(C,\oo\Cat)$ induced by the folk model structure on $\oo\Cat$. The proof of the miracle is rather technical and needed the introduction of what I named \emph{discrete Conduché $\oo$-functor} in reference to the existing notion of discrete Conduché functor (or fibration) between categories (see \cite{johnstone1999note} for example). The key point is the following result, to which \cite{guetta2020polygraphs} is dedicated:

\begin{center}

Let $f : C \to D$ be a discrete Conduché $\oo$-functor. If $D$ is free then so is $C$.

\end{center}

Let us now come back to the the proof that all categories are homologically coherent. The second key result is that categories with final objects are homologically coherent and the homology of such categories is (the homology of) $\mathbb{Z}$ seen as a chain complex concentrated in degree $0$. In fact, these categories belong to a larger class of homologically coherent $\oo$-categories, referred to as \emph{contractible $\oo$-categories} in this dissertation. Finally, a formal argument shows that both polygraphic homology and Street homology preserve homotopy colimits. Altoghether, we have

and a thorough analysis of naturality shows that the resulting isomorphim $\sH^{\pol}(C)\simeq\sH^{\Th}(C)$ is the canonical comparison map.

\iffalse It follows from Thomason's homotopy colimit theorem \cite{thomason1979homotopy} that this colimit is also homotopic when $\Cat$ is equipped with Thomason weak equivalences, which means that the canonical morphism

is an isomorphism in $\ho(\Cat^{\Th})$. A formal argument using Gagna's theorem shows that the canonical inclusion $\Cat\to\oo\Cat$ preserves homotopy colimits (when both categories are equipped with Thomason weak equivalences), and in particular the above morphism is also an isomorphim of $\ho(\oo\Cat^{\Th})$ when the colimit and homotopy colimit are understood in $\oo\Cat$. Then, an easy argument shows that Street homology preserves homotopy colimits and, combined with the fact that a category with terminal object has the homotopy type of a point \cite[Paragraph 1]{quillen1973higher}, we obtain that

where $\mathbb{Z}$ is seen as chain complex (up to quasi-isomorphism) concentrated in degree $0$. Note that this formula for the homology of the nerve of a small category at least goes back to Gabriel and Zisman \cite[Appendix II, Proposition 3.3]{gabriel1967calculus}.

On the other hand, it can also be shown that the colimit

\[

\colim_{c \in C}C/c \simeq C

\]

is a homotopy colimit in $\ho(\oo\Cat^{\folk})$. Contrary to the ``Thomason'' case, this result is new and first appears in \cite{guetta2020homology}. We shall come back to its proof shortly, but for now let us see how it implies that every category is homologically coherent. First, categories with terminal object belong to a class of what we refer to as \emph{contractible $\oo$-categories} and which are homologically coherent. In particular, we have \[\sH^{\pol}(C/c)\simeq\sH^{\St}(C/c)\simeq\mathbb{Z}\] for every object $c$ of $C$. Then, again by a formal argument, it can be shown that polygraphic homology preserves homotopy colimits and thus we have

(and a thorough analysis shows that the isomorphim is in fact the canonical comparison map).

Now, let us go back to the fact that the colimit $\colim_{c \in C}C/c$ is a homotopy colimit in $\ho(\oo\Cat^{\folk})$. Contrary to the ``Thomason'' case, we cannot reason in $\Cat$ because the inclusion $\Cat\to\oo\Cat$ does not

\fi

\end{named}

\begin{named}[The big picture]

Let us end this introduction with another point of view on the comparison of Street and polygraphic homologies. This point of view is not adressed at all in the rest of the dissertation because it is higly conjectural. It ought to be thought of as a guideline for future work.

...

...

@@ -207,33 +139,5 @@ This way of understanding polygraphic homology as a left derived functor has bee

Yet, as we have already seen, such property is not true: there are folk cofibrant objects that are \emph{not} enough cofibrant to compute (Street) homology. The archetypal example being the ``bubble'' of Ara and Maltsiniotis. However, even if false, the idea that folk cofibrant objects are sufficiently cofibrants for homology is seducing and I conjecturally believe that this defect is a mere consequence of working in a too narrow setting, as I shall now explain.

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 a variation of 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 notion of weak $\oo$-categories, which are conjectured to be all equivalent, at least in some higher categorical sense.)

\iffalse This means that $P$ is homologically coherent. On the other hand, it is not generally true that a folk cofibrant object $P'$ is Thomason cofibrant. However, \emph{if} the natural transformation $\pi$ in the above $2$-square was an isomorphism, then a quick 2-out-3 reasoning would show that the canonical map

\[

\LL\lambda^{\Th}(P')\to\lambda(P')

\]

is an isomorphism.

\fi

\end{named}

%This result might sound surprising to the reader familiar with the fact that strict $\oo$-groupoids, even with weak inverses, do \emph{not} model homotopy types (as explained, for example, in \cite{simpson1998homotopy}). However, Gagna's result

\iffalse This result is in fact a generalization of a well-known result concerning the homotopy theory of (small) categories. Indeed, the category of (small) categories $\Cat$ can be identified with the full subcategory of $\oo\Cat$ with only unit cells above dimension $1$ and the restriction of $N_{\omega}$ to $\Cat$ is nothing but the usual functor for categories

\[

N : \Cat\to\Psh{\Delta}

\]

\fi

%Note that since (small) categories can be seen as $\oo$-categories with only unit cells above dimension $1$, we can restrict the functor to $N_{\omega}$ to the category of small categories $\Cat$, seen as a full subcategory of $\oo\Cat$. In this case, $N_{\omega}\vert_{\Cat}$ coincide with the usual nerve of (small) categories and Thomason weak equivalences between small categories are exactly the weak equivalences of the model structure on $\Cat$ established

\iffalse

Before entering the heart of the subject and explaining precisely what the above means, let us quickly linger on a terminological detail concerning strict $\oo$-category. In this flavour of higher category theory, the axioms for compositions and units hold strictly, which means that they are witnessed by genuine equalities and not only up to higher dimensional cells. On the other hand, \emph{no} invertibility axioms is required on any dimension. In particular, strict $\oo$-categories are \emph{not} a particular case of $(\infty,1)$-categories in the sense of Lurie (improperly called $\infty$-categories).

\fi

\iffalse

In this vast generalization of category theory, the objects of study have objects and arrows, as for categories, but also arrows between arrows, called $2$-arrows or $2$-cells, arrows between arrows between arrows ($3$-cells), and so on. All these data come equipped with various composition laws between cells. Higher categories come in many different flavours which loosely depend on three main parameters:

\begin{itemize}[label=-]

\item The maximum dimension of the cells. One then speak of $1$-categories (which is a synonym for the usual categories), $2$-categories, $3$-catgegories and so on.

\item The requirement that units and associativity axioms for compositions either are witnessed by genuine equalities

\end{itemize}

\fi

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.)

@@ -579,3 +579,103 @@ Asking for an op-prederivator to have left Kan extensions is one of the axioms i

\end{enumerate}

\end{definition}

\fi

%%%%% Introduction

Let $C$ be a small category. Recall that we have a canonical isomorphim

\[

\colim_{c \in C}C/c \simeq C,

\]

where $C/c$ is the slice category over the object $c$ of $C$. Now the idea of the proof is to show that this colimit is in fact a homotopy colimit both in $\ho(\oo\Cat^{\Th})$ and in $\ho(\oo\Cat^{\Th})$. This means respectively that:

\begin{enumerate}[label=(\alph*)]

\item the canonical morphism of $\ho(\oo\Cat^{\Th})$

(The subscript of the homotopy colimits are there to make sure not to remind in which ``homotopy theory'' on $\oo\Cat$ the homotopy colimit is to be understood.)

Case (a) follows somewhat easily from Thomason's homotopy colimit theorem \cite{thomason1979homotopy}. Case (b) on the other hand is a new result and first appears in \cite{guetta2020homology}. The idea of the proof is as follows: let $f : P \to C$ a polygraphic resolution, i.e.\ $f$ is folk trivial fibration and $P$ is a free $\oo$-category, and consider the diagram

\begin{align*}

C &\to\oo\Cat\\

c &\mapsto P/c,

\end{align*}

where $P/c$ is the $\oo$-category defined as the following fibred product

\[

\begin{tikzcd}

P/c \ar[r]\ar[d]& P \ar[d,"f"]\\

C/c \ar[r]& C. \ar[from=1-1,to=2-2,phantom,very near start,"\lrcorner"]

\end{tikzcd}

\]

Using the fact that $f$ is a trivial fibration and some 2-out-of-3 property, the problem can be reduced to showing that the canonical map

is an isomorphism of $\ho(\oo\Cat^{\folk})$. Now, this follows from the ``miracle'' that the diagram $c \mapsto P/c$ is a cofibrant object of the projective model structure on the category $\underline{\Hom}(C,\oo\Cat)$ induced by the folk model structure on $\oo\Cat$. The proof of the miracle is rather technical and needed the introduction of what I named \emph{discrete Conduché $\oo$-functor} in reference to the existing notion of discrete Conduché functor (or fibration) between categories (see \cite{johnstone1999note} for example). The key point is the following result, to which \cite{guetta2020polygraphs} is dedicated:

\begin{center}

Let $f : C \to D$ be a discrete Conduché $\oo$-functor. If $D$ is free then so is $C$.

\end{center}

Let us now come back to the the proof that all categories are homologically coherent. The second key result is that categories with final objects are homologically coherent and the homology of such categories is (the homology of) $\mathbb{Z}$ seen as a chain complex concentrated in degree $0$. In fact, these categories belong to a larger class of homologically coherent $\oo$-categories, referred to as \emph{contractible $\oo$-categories} in this dissertation. Finally, a formal argument shows that both polygraphic homology and Street homology preserve homotopy colimits. Altoghether, we have

and a thorough analysis of naturality shows that the resulting isomorphim $\sH^{\pol}(C)\simeq\sH^{\Th}(C)$ is the canonical comparison map.

\iffalse It follows from Thomason's homotopy colimit theorem \cite{thomason1979homotopy} that this colimit is also homotopic when $\Cat$ is equipped with Thomason weak equivalences, which means that the canonical morphism

is an isomorphism in $\ho(\Cat^{\Th})$. A formal argument using Gagna's theorem shows that the canonical inclusion $\Cat\to\oo\Cat$ preserves homotopy colimits (when both categories are equipped with Thomason weak equivalences), and in particular the above morphism is also an isomorphim of $\ho(\oo\Cat^{\Th})$ when the colimit and homotopy colimit are understood in $\oo\Cat$. Then, an easy argument shows that Street homology preserves homotopy colimits and, combined with the fact that a category with terminal object has the homotopy type of a point \cite[Paragraph 1]{quillen1973higher}, we obtain that

where $\mathbb{Z}$ is seen as chain complex (up to quasi-isomorphism) concentrated in degree $0$. Note that this formula for the homology of the nerve of a small category at least goes back to Gabriel and Zisman \cite[Appendix II, Proposition 3.3]{gabriel1967calculus}.

On the other hand, it can also be shown that the colimit

\[

\colim_{c \in C}C/c \simeq C

\]

is a homotopy colimit in $\ho(\oo\Cat^{\folk})$. Contrary to the ``Thomason'' case, this result is new and first appears in \cite{guetta2020homology}. We shall come back to its proof shortly, but for now let us see how it implies that every category is homologically coherent. First, categories with terminal object belong to a class of what we refer to as \emph{contractible $\oo$-categories} and which are homologically coherent. In particular, we have \[\sH^{\pol}(C/c)\simeq\sH^{\St}(C/c)\simeq\mathbb{Z}\] for every object $c$ of $C$. Then, again by a formal argument, it can be shown that polygraphic homology preserves homotopy colimits and thus we have

(and a thorough analysis shows that the isomorphim is in fact the canonical comparison map).

Now, let us go back to the fact that the colimit $\colim_{c \in C}C/c$ is a homotopy colimit in $\ho(\oo\Cat^{\folk})$. Contrary to the ``Thomason'' case, we cannot reason in $\Cat$ because the inclusion $\Cat\to\oo\Cat$ does not

\fi

\iffalse This means that $P$ is homologically coherent. On the other hand, it is not generally true that a folk cofibrant object $P'$ is Thomason cofibrant. However, \emph{if} the natural transformation $\pi$ in the above $2$-square was an isomorphism, then a quick 2-out-3 reasoning would show that the canonical map

\[

\LL\lambda^{\Th}(P')\to\lambda(P')

\]

is an isomorphism.

\fi

%This result might sound surprising to the reader familiar with the fact that strict $\oo$-groupoids, even with weak inverses, do \emph{not} model homotopy types (as explained, for example, in \cite{simpson1998homotopy}). However, Gagna's result

\iffalse This result is in fact a generalization of a well-known result concerning the homotopy theory of (small) categories. Indeed, the category of (small) categories $\Cat$ can be identified with the full subcategory of $\oo\Cat$ with only unit cells above dimension $1$ and the restriction of $N_{\omega}$ to $\Cat$ is nothing but the usual functor for categories

\[

N : \Cat\to\Psh{\Delta}

\]

\fi

%Note that since (small) categories can be seen as $\oo$-categories with only unit cells above dimension $1$, we can restrict the functor to $N_{\omega}$ to the category of small categories $\Cat$, seen as a full subcategory of $\oo\Cat$. In this case, $N_{\omega}\vert_{\Cat}$ coincide with the usual nerve of (small) categories and Thomason weak equivalences between small categories are exactly the weak equivalences of the model structure on $\Cat$ established

\iffalse

Before entering the heart of the subject and explaining precisely what the above means, let us quickly linger on a terminological detail concerning strict $\oo$-category. In this flavour of higher category theory, the axioms for compositions and units hold strictly, which means that they are witnessed by genuine equalities and not only up to higher dimensional cells. On the other hand, \emph{no} invertibility axioms is required on any dimension. In particular, strict $\oo$-categories are \emph{not} a particular case of $(\infty,1)$-categories in the sense of Lurie (improperly called $\infty$-categories).

\fi

\iffalse

In this vast generalization of category theory, the objects of study have objects and arrows, as for categories, but also arrows between arrows, called $2$-arrows or $2$-cells, arrows between arrows between arrows ($3$-cells), and so on. All these data come equipped with various composition laws between cells. Higher categories come in many different flavours which loosely depend on three main parameters:

\begin{itemize}[label=-]

\item The maximum dimension of the cells. One then speak of $1$-categories (which is a synonym for the usual categories), $2$-categories, $3$-catgegories and so on.

\item The requirement that units and associativity axioms for compositions either are witnessed by genuine equalities