@@ -86,11 +86,11 @@ This way of understanding polygraphic homology as a left derived functor has bee

&\ho(\Ch)

\end{tikzcd}.

\]

This triangle is \emph{not} commutative (even up to an isomorphism), since this would imply that the Street and polygraphic homology groups coincide for every $\oo$-category. However, since both functors $\LL\lambda^{\folk}$ and $\LL\lambda^{\Th}$ are left derived functors of the same functor $\lambda$, it can be shown by universal property the existence of a natural transformation $\LL\lambda^{\Th}\Rightarrow\LL\lambda^{\folk}\circ\J$. Since $\J$ is the identity on objects, for every $\oo$-category $C$, this natural transformation yields a map

This triangle is \emph{not} commutative (even up to an isomorphism), since this would imply that the Street and polygraphic homology groups coincide for every $\oo$-category. However, since both functors $\LL\lambda^{\folk}$ and $\LL\lambda^{\Th}$ are left derived functors of the same functor $\lambda$, it can be shown by universal property the existence of a natural transformation $\pi : \LL\lambda^{\Th}\Rightarrow\LL\lambda^{\folk}\circ\J$. Since $\J$ is the identity on objects, for every $\oo$-category $C$, this natural transformation yields a map

\[

\sH^{\St}(C)\to\sH^{\pol}(C),

\pi_C : \sH^{\St}(C)\to\sH^{\pol}(C),

\]

which we refer to as the \emph{canonical comparison map}. Let us say that $C$ is \emph{homologically coherent} if the canonical comparison map is an isomorphism (which means exactly that for every $k\geq0$, the induced map $H^{\Th}_k(C)\to H_k^{\pol}(C)$ is an isomorphism). The question of study then becomes:

which we refer to as the \emph{canonical comparison map}. Let us say that $C$ is \emph{homologically coherent} if $\pi_C$ is an isomorphism (which means exactly that for every $k\geq0$, the induced map $H^{\Th}_k(C)\to H_k^{\pol}(C)$ is an isomorphism). The question of study then becomes:

\begin{center}

\textbf{(Q')} Which $\oo$-categories are homologically coherent ?

\end{center}

...

...

@@ -183,6 +183,39 @@ This way of understanding polygraphic homology as a left derived functor has bee

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.

What \emph{would} it mean that the natural transformation $\pi$ be an isomorphism (i.e.\ that all $\oo$-categories be homologically coherent) ?

\end{center}

For simplification, let us assume that the conjectured Thomason-like model structure on $\oo\Cat$ was established and that $\lambda$ was left Quillen with respect to this model structure (which is also conjectured).

Now, the conjectured cofibrations of the Thomason-like model structure (see \cite{ara2014vers}) are particular cases of folk cofibrations and thus, all Thomason cofibrant objects are folk cofibrant objects. The converse, on the other hand, is not true. Consequently, Quillen's theory of derived functors tells us that for a \emph{Thomason} cofibrant object $P$, we have

(and the resulting isomorphism is obviously the canonical comparison map). Now, \emph{if} the natural transformation $\pi$ were an isomorphism, then a quick 2-out-of-3 reasoning would show that \eqref{equationintro} would also be true when $P$ is only \emph{folk} cofibrant. Hence, intuitively speaking, if $\pi$ were an isomorphism, then folk cofibrant objects would be \emph{sufficiently cofibrant} for the homology, even though there are not Thomason cofibrant. (And in fact, using cofibrant replacements, it can be shown that this condition is sufficient to ensure that $\pi$ be an isomorphism).

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