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.

which is canonically isomorphic to $\emptyset\to X/-$ (see Lemma \ref{lemma:filtration}), is also a cofibration for the projective model structure.

\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 $\ho(\oo\Cat^{\folk})$

@@ -82,8 +82,13 @@ The functor $\kappa : \Psh{\Delta} \to \Ch$ is left Quillen and sends weak equiv

%% Recall that we consider localization functors as identity on objects (see \ref{paragr:loc}). Hence, the homology of $X$ is simply $\kappa(N_{\oo}(X))$, only considered as a chain complex up to quasi-isomorphism, i.e. an object of $\ho(\Ch)$. The homology groups of $X$ are the homology groups of the chain complex $\kappa(N_{\oo}(X))$. However, with our definition, the \emph{homology of $X$} means something more precise than the mere sequence of homology groups. An alternative terminology would be to call $\sH(X)$ the \emph{homology type of $X$}, in reference to the homotopy type of a topological space.

%% \end{paragr}

\begin{remark}

The adjective ``singular'' is there to avoid future confusion with another homological invariant for $\oo$\nbd{}categories that will be introduced later. As a matter of fact, the underlying point of view adopted in this thesis is that \emph{singular homology of $\oo$-categories} ought to be simply called \emph{homology of $\oo$\nbd{}categories} as it is the only ``correct'' definition of homology. This assertion will be justified later. \todo{Le faire !}

\begin{remark}\label{remark:singularhmlgyishmlgy}

The adjective ``singular'' is there to avoid future confusion with another

homological invariant for $\oo$\nbd{}categories that will be introduced later.

As a matter of fact, the underlying point of view adopted in this thesis is

that \emph{singular homology of $\oo$-categories} ought to be simply called

\emph{homology of $\oo$\nbd{}categories} as it is the only ``correct''

definition of homology. This assertion will be justified in Remark \ref{remark:polhmlgyisnotinvariant}.

\end{remark}

\begin{remark}

We could also have defined the homology of $\oo$\nobreakdash-category with $K : \Psh{\Delta}\to\Ch$ instead of $\kappa : \Psh{\Delta}\to\Ch$ since these two functors are quasi-isomorphic (see \cite[Theorem 2.4]{goerss2009simplicial} for example). An advantage of the later one is that it is left Quillen.

...

...

@@ -710,7 +715,14 @@ Another consequence of the above counter-example is the following result, which

It follows from the previous result that if we think of $\oo$\nbd{}categories as a

model for homotopy types (see Theorem \ref{thm:gagna}), then the polygraphic

homology of an $\oo$-category is \emph{not} a well defined invariant. This

justifies what we said in remark \ref{remark:singularhmlgyishmlgy}, which is

that \emph{singular homology} is the only ``correct'' homology of $\oo$\nbd{}categories.

\end{remark}

\begin{paragr}\label{paragr:defcancompmap}

Even though triangle \eqref{cmprisontrngle} is not commutative (even up to an isomorphism), it can be filled up with a $2$-morphism. Indeed, consider the following $2$\nbd{}square

The present chapter sticks out from the others as it contains no original results. Its goal is simply to introduce the language and tools of homotopical algebra that we shall need in the rest of the dissertation. In consequence, most of the results are simply asserted and the reader will find references to literature for the proofs. The main notion of homotopical algebra we are aiming for is the one of \emph{homotopy colimits} and the language we chose to express this notion is the one given by the theory of Grothendieck's \emph{derivators}\cite{grothendieckderivators}. We do not assume that the reader is familiar with this theory and will quickly recall the basics. If needed, gentle introductions can be found in \cite{maltsiniotis2001introduction} and in a letter from Grothendieck to Thomason \cite{grothendieck1991letter}; more detailed introductions can be found in \cite{groth2013derivators} and in the first section of \cite{cisinski2003images}; finally, a rather complete (yet unfinished and unpublished) textbook on the subject is \cite{groth2013book}.

\todo{Expliquer le choix des dérivateurs ?}%This theory has the advantage of being much more elementary than the theory of \emph{weak $(\infty,1)$-category}

\iffalse Let us quickly motive this choice for the reader unfamiliar with this theory.

From an elementary point of view, a homotopy theory is given (or rather \emph{presented by}) by a category $\C$ and a class $\W$ of arrows of $\C$, which we traditionally refer to as \emph{weak equivalences}. The point of homotopy theory is to consider that the objects of $\C$ connected by a zigzag of weak equivalences should be indistinguishable. From a category theorist perspective, a most natural One of the most basic invariant associated to such a data is the localisation of $\C$ with respect to $\W$. That is to say, the category $\ho^{\W}(\C)$ obtained from $\C$ by forcing the arrows of $\W$ to become isomorphisms. While the ``problem'' is that the category $\ho^{\W}(\C)$ is poorly behaved. For example, \fi

@@ -6,7 +6,12 @@ The general framework in which the work to be presented in this dissertation tak

\[

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

\]

that associates to any $\oo$-category $C$ a simplicial set $N_{\oo}(C)$ called the \emph{nerve of $C$}, generalizing the usual nerve of (small) categories. Using this functor, we can pull back the homotopy theory of simplicial sets to $\oo$-categories, as it is done for example in the articles \cite{ara2014vers}, \cite{ara2018theoreme}, \cite{gagna2018strict}, \cite{ara2019quillen} and \cite{ara2020theoreme}. Following the terminology of these articles, a morphism of $\oo\Cat$, $f : C \to D$, is a \emph{Thomason weak equivalence} if $N_{\omega}(f)$ is a (Quillen) weak equivalence of simplicial sets. By definition, the nerve functor induces a functor at the level of homotopy categories

that associates to any $\oo$-category $C$ a simplicial set $N_{\oo}(C)$ called

the \emph{nerve of $C$}, generalizing the usual nerve of (small) categories.

Using this functor, we can pull back the homotopy theory of simplicial sets to

$\oo$-categories, as it is done for example in the articles \cite{ara2014vers},

\cite{ara2020theoreme} and \cite{ara2020comparaison}. Following the terminology of these articles, a morphism of $\oo\Cat$, $f : C \to D$, is a \emph{Thomason weak equivalence} if $N_{\omega}(f)$ is a (Quillen) weak equivalence of simplicial sets. By definition, the nerve functor induces a functor at the level of homotopy categories

@@ -532,7 +532,8 @@ Furthermore, this function satisfies the condition

\begin{paragr}\label{paragr:weight}

Let $C$ be an $n$-category with an $n$-basis $E$. For an $n$-cell $x$ of $C$,

we refer to the integer $w_{\alpha}(x)$ as the \emph{weight of $\alpha$ in

$x$}. The reason for such a name will become clearer later once we give an

$x$}. The reason for such a name will become clearer after Remark

\ref{remark:weightexplicitly} where we give an

explicit construction of $w_{\alpha}$ as a function that ``counts the number

of occurrences of $\alpha$ in an $n$-cell''.

...

...

@@ -1075,7 +1076,6 @@ We can now prove the expected result.

\[

ab = ba.

\]

\todo{À finir ? Il faudrait peut-être dire qu'en rajoutant des hypothèses on doit pouvoir quand même avoir une forme normale. Dans le cas n=2, quand on fait l'hypothèse que le 2-polygraphe est positif, je sais que c'est vrai.}

@@ -1847,7 +1847,18 @@ We end this section with yet another characterisation of $n$-basis of $\oo$-cate

where $F$ is the morphism of $(n+1)$-magmas from the proof of Lemma \ref{lemma:rhoequiv}. Since $\sim$ is the smallest categorical congruence on $\E_{\Sigma}^+$, it follows from Proposition \ref{prop:Estarasquotient} that $\E_{\Sigma}^+/{\sim}$ is (canonically isomorphic to) $\E_{\Sigma}^*$ and it is easily seen that the functor $\overline{F}$ is nothing but the $(n+1)$-functor $\E_{\Sigma}^*\to\tau_{\leq n+1}^s$ as obtained in \ref{paragr:cextfromsubset}. Hence, from Proposition \ref{prop:criterionnbasis}, and the fact that $\overline{F}$ is the identity on cells of dimension non greater than $n$, we deduce that $\Sigma$ is an $(n+1)$-basis of $C$ if and only if $\rho_{\Sigma}$ is an isomorphism.

The result follows immediately from the fact that equivalence classes of $\sim$ are in bijection with maximal connected components of $\G[\Sigma]$.

\end{proof}

\todo{Rajouter la construction explicite de la fonction de comptage ?}

\begin{remark}\label{remark:weightexplicitly}

Let $n>0$, $C$ be an $n$\nbd{}category with an $n$\nbd{}basis $\Sigma$ and

$\alpha\in\Sigma$. It is immediate to check from the definition of

elementary moves that for two equivalent well formed words $u \sim u'$ of

$\T[\Sigma]$, the number of occurences of $\cc_{\alpha}$ in $u$ and $u'$ are

the same. In particular, for any $a \in C_n$, we can define the integer

$w_{\alpha}(a)$ to be the number of occurences of $\cc_{\alpha}$ in any well

formed word $u$ such that $\rho_{\sigma}(u)=a$. An immediate induction using

the properties of $\rho_{\Sigma}$ shows that this function $w_{\alpha} : C_n \to\mathbb{N}$ is the same

as the one whose existence was established in Proposition

\ref{prop:countingfunction}.

\end{remark}

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

In this section, we finally go back to Conduché $\oo$-functors. The first two parts might have been considered as preliminaries and the key points of the proof of Theorem \ref{thm:conduche} lie within this third and last part.