@@ -296,50 +296,103 @@ In particular, this means that we have a morphism of localizers $\kappa : (\Psh{

From \todo{Critère de Gonzalez version dérivateur à écrire !}, we conclude that $\lambda : \oo\Cat\to\Ch$ is strongly left derivable and that $\LL^{\Th}\lambda\simeq\overline{\kappa}\overline{N_{\oo}}$, which is, by definition, the ``true'' homology.

\end{proof}

\begin{remark}

Beware that neither $c_{\oo} : \Psh{\Delta}\to\oo\Cat$ nor $\lambda : \oo\Cat\to\Ch$ preserve weak equivalences \todo{Uniformiser appellations}, but this doesn't contradict the fact that $\lambda c_{\oo} : \Psh{\Delta}\to\Ch$ does.

Beware that neither $c_{\oo} : \Psh{\Delta}\to\oo\Cat$ nor $\lambda : \oo\Cat\to\Ch$ preserve weak equivalences \todo{Uniformiser appellations}, but this does not contradict the fact that $\lambda c_{\oo} : \Psh{\Delta}\to\Ch$ does.

\end{remark}

\section{Polygraphic homology vs. ``true'' homology}

\begin{paragr}\label{paragr:univmor}

Since $\sH : \Ho(\oo\Cat)\to\Ho(\Ch)$ is the left derived of the abelianization functor, it comes with a $2$-square

\[

\begin{tikzcd}

\oo\Cat\ar[d]\ar[r,"\lambda"]&\Ch\ar[d]\\

\Ho(\oo\Cat^{\Th})\ar[r,"\sH"]&\Ho(\Ch).

\ar[from=2-1,to=1-2,"\alpha^{\Th}",Rightarrow]

\end{tikzcd}

\]

The $2$-morphism $\alpha^{\Th}$ admits the following description. By post-composing the co-unit of the adjunction $c_{\oo}\dashv N_{\oo}$

Recall from Paragraph \ref{paragr:compweakeq} that the identity functor on $\oo\Cat$ induces a morphism of localizers $(\oo\Cat,\W^{\folk})\to(\oo\Cat,\W^{\Th})$, and hence a functor $\J : \ho(\oo\Cat^{\folk}\to\ho(\oo\Cat^{\Th})$. Together with the polygraphic and ``true'' homology functors, this yields a triangle of functors

Recall from Paragraph \ref{paragr:compweakeq} that the identity functor on $\oo\Cat$ induces a morphism of localizers

As we shall see later, this triangle is \emph{not} commutative, even up to an iso. However, it can be filled up with a natural transformation. Indeed, consider the following $2$-diagram

As we shall see later, this triangle is \emph{not} commutative, even up to an iso. However, it can be filled up with a $2$-morphism. Indeed, consider the following $2$-diagram

where the left square is commutative and the $2$-morphism on the right square is the one from Paragraph \ref{paragr:univmor}. Since the polygraphic homology is, by definition, the left derived of the abelianization functor when $\oo\Cat$ is equipped with canonical weak equivalences, the universal property of strongly derivable functor yields a canonical $2$-morphism

where the left square is commutative and the natural transformation in the right square is the canonical derivation morphism. \todo{Uniformiser appellations.}

From Proposition \ref{prop:hmlgyderived} we know that the ``true'' homology functor

\[\sH : \ho(\oo\Cat^{\Th})\to\ho(\Ch)\]

is the left derived functor of the abelianization functor

\[\lambda : \oo\Cat\to\Ch.\]

Hence, the universal property of left derived functors yields a unique natural transformation

is commutative \todo{Compléter diagramme précédent.}

For any $\oo$\nbd-category $X$, we shall refer to the map

Since $\J$ is nothing but the identity on objects, for any $\oo$\nbd-category $X$, the previous natural transformation yields a map

\[

\pi_X : \sH^{\pol}(X)\to\sH(X)

\pi_X : \sH(X)\to\sH^{\pol}(X),

\]

as the \emph{canonical comparison map.}

which we shall refer to as the \emph{canonical comparison map.}

\end{paragr}

This motivates the following definition.

\begin{definition}

An $\oo$\nbd-category $X$ is said to be \emph{\good} when the canonical comparison map

An $\oo$\nbd-category $X$ is said to be \emph{\good{}} when the canonical comparison map

\[

\pi_X : \sH^{\pol}(X)\to\sH(X)

\pi_X : \sH(X)\to\sH^{\pol}(X)

\]

is an isomorphism of $\ho(\Ch)$.

\end{definition}

...

...

@@ -354,7 +407,7 @@ In particular, this means that we have a morphism of localizers $\kappa : (\Psh{

\end{center}

The rest of this document is devoted to (partially) answering this question. We start by giving in the next paragraph an example due to Ara and Maltsiniotis of an $\oo$-category for which the comparison map is \emph{not} an isomorphism.

\end{paragr}\fi

We begin by spelling out an example due to Ara and Maltsiniotis of an $\oo$-category which is \emph{not}\good. Examples of \good$\oo$-categories will be presented later. The rest of this dissertation is devoted to the study of \good$\oo$\nbd-categories.

We begin by spelling out an example due to Ara and Maltsiniotis of an $\oo$-category which is \emph{not}\good{}. Examples of \good{}$\oo$-categories will be presented later. The rest of this dissertation is devoted to the study of \good{}$\oo$\nbd-categories.

\begin{paragr}

Let $B$ the commutative monoid $(\mathbb{N},+)$ considered as $2$-category.\footnote{The letter $B$ stands for ``bubble''.} That is,

\[

...

...

@@ -372,12 +425,12 @@ In particular, this means that we have a morphism of localizers $\kappa : (\Psh{

\[

\sH^{\pol}_k(B)=\begin{cases}\mathbb{Z}\text{ if } k=0,2\\0\text{ in other cases.}\end{cases}

\]

On the other hand, it was proven in \cite[Theorem 4.9 and Example 4.10]{ara2019quillen} that (the nerve of) $B$ is a $K(\mathbb{Z},2)$-space. In particular, it has non-trivial homology groups in every even dimension. This proves that $B$ is \emph{not}\good.

On the other hand, it was proven in \cite[Theorem 4.9 and Example 4.10]{ara2019quillen} that (the nerve of) $B$ is a $K(\mathbb{Z},2)$-space. In particular, it has non-trivial homology groups in every even dimension. This proves that $B$ is \emph{not}\good{}.

\end{paragr}

\iffalse\begin{remark}

The previous example of non \good$\oo$-category also proves that triangle \ref{cmprisontrngle} cannot be commutative up to an iso.

The previous example of non \good{}$\oo$-category also proves that triangle \ref{cmprisontrngle} cannot be commutative up to an iso.

\end{remark}\fi

We shall now proceed to give an abstract criterion to find \good$\oo$-categories. In the rest of this dissertation, we will exploit this criterion to exhibit the largest classes possible of \good$\oo$-categories.

We shall now proceed to give an abstract criterion to find \good{}$\oo$-categories. %In the rest of this dissertation, we will exploit this criterion to exhibit the largest classes possible of \good{} $\oo$-categories.

\begin{paragr}

Both the polygraphic homology

...

...

@@ -395,9 +448,31 @@ We shall now proceed to give an abstract criterion to find \good $\oo$-categorie

&\Ho(\Ch).\ar[from=A,to=1-2,"\pi",Rightarrow]

\end{tikzcd}

\]

We will see later that the top arrow of the previous diagram, which is induced by the identity functor on $\oo\Cat$, cannot be homotopy cocontinuous as if it were every $\oo$\nbd-category would be \good. \todo{Mettre ref interne de où il sera montré que le morphisme n'est pas cocontinu.}

However, as for any morphism of op-prederivators \todo{ref}, for any diagram

\[

d : I \to\oo\Cat

\]

We will see later that $\J$ cannot be homotopy cocontinuous as if it were, every $\oo$\nbd-category would be \good{}. \todo{Mettre ref interne de où il sera montré que le morphisme n'est pas cocontinu.}

\end{paragr}

\begin{proposition}

Let $X$ be an $\oo$\nbd-category. Suppose that there exists a diagram

\[

d : I \to\oo\Cat

\]

and a co-cone

\[

(\varphi_i : d(i)\to\oo\Cat)_{i \in\Ob(I)}

\]

such that:

\begin{enumerate}[label=\roman*)]

\item For every $i \in\Ob(I)$, the $oo$\nbd-category $d(i)$ is \good{}.

\item The canonical morphism

\[

\hocolim^{\folk}d \to X

\]

is an isomorphism of $\ho(\oo\Cat^{\folk})$.

\item The canonical morphism

\[

\hocolim^{\Th}d \to X

\]

is an isomorphism of $\ho(\oo\Cat^{\Th})$.

\end{enumerate}

Then the $\oo$\nbd-category $X$ is \good{}.

\end{proposition}

\todo{Preuve à écrire. Section à reprendre un peu}