Since $F$ is a Thomason equivalence, the bottom horizontal arrow is an isomorphism and since $B^1(\mathbb{N}\times\mathbb{N})$ is a $1$\nbd{}category, it is \good{} (Theorem \ref{thm:categoriesaregood}), which means that the right vertical arrow is an isomorphism. The $1$\nbd{}category $B^1(\mathbb{N}\times\mathbb{N})$ is not free but since it has the homotopy type of the torus, we have $H^{\sing}_k(B^1(\mathbb{N}\times\mathbb{N}))=0$ for $k\geq0$ and it follows then from Corollary \ref{cor:polhmlgycofibrant} and Paragraph \ref{paragr:polhmlgylowdimension} that the map $\sH^{\pol}(F)$ may be identified with the image in $\ho(\Ch)$ of the map

It is straightforward to check that this last map is a quasi-isomorphism, which implies by a 2-out-of-3 property that $\pi_P : \sH^{\pol}(P)\to\sH^{\sing}(P)$ is an isomorphism. This means by definition that $P$ is \good{}.

which is canonically isomorphic to $\sk_{0}(X/-)\to X/-$ (see Lemma \ref{lemma:filtration}), is also a cofibration for the projective model structure. It is easy to check that $\sk_{0}(X/-)$ is cofibrant (as is any diagram $A \to\oo\Cat$ that factorizes through $0\Cat\to\oo\Cat$),

which implies that $X/- : A \to\oo\Cat$ is cofibrant.

\end{proof}

\begin{corollary}\label{cor:folkhmtpycol}

Let $A$ be a $1$\nbd{}category and $f : X \to A$ an $\oo$\nbd{}functor. The canonical arrow of $\ho(\oo\Cat^{\folk})$

...

...

@@ -589,7 +600,7 @@ We now recall an important Theorem due to Thomason.

It is possible to extend the previous corollary to prove that for any functor $f : X \to A$ ($X$ and $A$ being $1$-categories), we have \[\hocolim^{\Th}_{a \in A}(X/a)\simeq X.\] However, to prove that it is also the case when $X$ is an $\oo$\nbd{}category and $f$ an $\oo$\nbd{}functor, as in Corollary \ref{cor:folkhmtpycol}, one would need to extend the Grothendieck construction to functors with value in $\oo\Cat$ and to prove an $\oo$\nbd{}categorical analogous of Theorem \ref{thm:Thomason}. Such results, while being highly plausible, go way beyond the scope of this dissertation.

\end{remark}

Putting all the pieces together, we are now able to prove the awaited Theorem.

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

\section{Homology via the nerve}

\begin{paragr}

We denote by $\Ch$ the category of chain complexes of abelian groups in non-negative degree. 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 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 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,

...

...

@@ -41,14 +41,23 @@ This defines a chain complex $\kappa(X)$, which we call the \emph{normalized cha

The functor $\kappa : \Psh{\Delta}\to\Ch$ is left Quillen and sends weak equivalences of simplicial sets to quasi-isomorphisms.

\end{lemma}

\begin{proof}

From the Dold-Kan equivalence, we know that $\Ch$ is equivalent to the category $\Ab(\Delta)$ of simplicial abelian groups (see for example \cite[Chapter III, section 2]{goerss2009simplicial}). With this identification the functor $\kappa : \Psh{\Delta}\to\Ch$ is left adjoint of the canonical forgetful functor

Recall that the Quillen model structure on simplicial sets admits the set of inclusions

\[

\{\partial\Delta_n \hookrightarrow\Delta_n \vert n \in\mathbb{N}\}

\]

as generating cofibrations and the set of inclusions

\[

\{\Lambda^i_n \hookrightarrow\Delta_n \vert n \in\mathbb{N}, 0\leq i \leq n\}

\]

as a generating trivial cofibrations (see for example \cite[Section I.1]{goerss2009simplicial} for the notations). A quick computation, which we leave to the reader, shows that the image by $\kappa$ of $\partial\Delta_n \hookrightarrow\Delta_n$ is a monomorphism with projective cokernel and the image by $\kappa$ of $\Lambda^i_n \hookrightarrow\Delta_n$ is a quasi-isomorphism. This proves that $\kappa$ is left Quillen. Since all simplicial sets are cofibrant, it follows from Ken Brown's Lemma \cite[Lemma 1.1.12]{hovey2007model} that $\kappa$ also preserves weak equivalences.

\end{proof}

\begin{remark}

The previous lemma admits also as more conceptual proof as follows. From the Dold-Kan equivalence, we know that $\Ch$ is equivalent to the category $\Ab(\Delta)$ of simplicial abelian groups and with this identification the functor $\kappa : \Psh{\Delta}\to\Ch$ is left adjoint of the canonical forgetful functor

\[

U : \Ch\simeq\Ab(\Delta)\to\Psh{\Delta}

\]

induced by the forgetful functor from abelian groups to sets. It follows from \cite[Lemma 2.9 and Corollary 2.10]{goerss2009simplicial} that $U$ is right Quillen, hence $\kappa$ is left Quillen.

The fact that $\kappa$ preserves weak equivalences follows from Ken Brown's Lemma \cite[Lemma 1.1.12]{hovey2007model} and the fact that all simplicial sets are cofibrant.

\end{proof}

induced by the forgetful functor from abelian groups to sets. The fact that $U$ is right Quillen follows then from \cite[Lemma 2.9 and Corollary 2.10]{goerss2009simplicial}.

\end{remark}

\begin{paragr}

In particular, $\kappa$ induces a morphism of localizers \[\kappa : (\Psh{\Delta},\W_{\Delta})\to(\Ch,\W_{\Ch}),\]

where we wrote $\W_{\Ch}$ for the class of quasi-isomorphisms.

...

...

@@ -1338,7 +1347,7 @@ Straightforward consequence of the fact that $N_n = N_{\oo} \circ \iota_n$ and t

Using the description of $\Or_0$, $\Or_1$ and $\Or_2$ from \ref{paragr:orientals}, we deduce that a morphism $F : i^*(N_{\oo}(C))\to i^*(N_1(D))$ in $\Psh{\Delta_{\leq2}}$ consists of a function $F_0 : C_0\to D_0$ and a function $F_1 : C_1\to D_1$ such that

\begin{itemize}

\begin{itemize}[label=-]

\item for every $x \in C_0$, we have $F_1(1_x)=1_{F_0(x)}$,

\item for every $x \in C_1$, we have $\src(F_1(x))=F_0(\src(x)))$ and $\trgt(F_1(x))=F_0(\trgt(x)))$,

\item for every $2$\nbd{}triangle

...

...

@@ -1412,7 +1421,7 @@ Finally, we obtain the result we were aiming for.

&\lambda(C)&

\end{tikzcd}.

\]

From Proposition \ref{prop:singhmlgylowdimension}, we know that $\alpha^{\sing}$ induces isomorphisms $H_k^{\sing}(C)\simeq H_k(\lambda(C))$ for $k \in\{0,1\}$ and from Corollary \ref{cor:polhmlgycofibrant} and Paragraph \ref{paragr:polhmlgylowdimension} we know that $\alpha^{\pol}$ induces isomorphisms $H_k^{\pol}(C)\simeq H_k(\lambda(C))$ for $k \in\{0,1\}$. The result follows then from an immediate 2-out-of-3 property.

From Proposition \ref{prop:singhmlgylowdimension}, we know that $\alpha^{\sing}$ induces isomorphisms \[H_k^{\sing}(C)\simeq H_k(\lambda(C))\] for $k \in\{0,1\}$ and from Corollary \ref{cor:polhmlgycofibrant} and Paragraph \ref{paragr:polhmlgylowdimension} we know that $\alpha^{\pol}$ induces isomorphisms $H_k^{\pol}(C)\simeq H_k(\lambda(C))$ for $k \in\{0,1\}$. The result follows then from an immediate 2-out-of-3 property.

\end{proof}

\begin{paragr}

A natural question following the above proposition is:

Now, it follows from \cite[Lemma 5.1]{street1987algebra} that the image of the inclusion $\partial\Delta_n \to\Delta_n$ by $c_{\omega}$ can be identified with the canonical inclusion

\[

(\Or_n)_{\leq n-1}\to\Or_n.

\sk_{n-1}(\Or_n)\to\Or_n.

\]

Since $\Or_n$ is free, this last morphism is by definition a push-out of a coproduct of folk cofibrations, hence a folk cofibration.

Since $\Or_n$ is free, this last morphism is by definition a push-out of a coproduct of folk cofibrations (see Definition \ref{def:nbasis}), hence a folk cofibration.

\end{proof}

As an immediate consequence of the previous lemma, we have the following proposition.

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.

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 non-negatively graded chain complexes (i.e.\ the localization of the category of non-negatively graded chain complexes 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 $\oo$\nbd{}categories most certainly started with the introduction by Street \cite{street1987algebra} of a nerve functor

...

...

@@ -44,7 +44,7 @@ This is precisely the question to be addressed in this work. Yet, the reader wil

\end{named}

\begin{named}[Another formulation of the problem]A first step of present work is a more abstract formulation of the question of comparison of Street and polygraphic homology of $\oo$-categories. %As often, the reward for abstraction is a much clearer understanding of the problem.

In order to do that, recall first that by a variation of the Dold-Kan equivalence (see for example \cite{bourn1990another}), the category of abelian group objects in $\oo\Cat$ is equivalent to the category of chain complexes in non-negative degree

In order to do that, recall first that by a variation of the Dold-Kan equivalence (see for example \cite{bourn1990another}), the category of abelian group objects in $\oo\Cat$ is equivalent to the category of non-negatively graded chain complexes