whose image by $u/c_0$ is equivalent for the relation $\sim_{\oo}$ to the above $(n+1)$\nbd{}cell of $B/c_0$ . In particular, the source and target of $\alpha$ are respectively $f$ and $f'$. Finally, we obtain that $\alpha\sim_{\oo}\beta$ by applying the canonical $\oo$\nbd{}functor $A/b_0\to A$, .
\end{enumerate}
\end{proof}
\todo{Il faudrait vérifier que je n'ai pas écrit de bêtises dans la preuve précédente.}
%\todo{Il faudrait vérifier que je n'ai pas écrit de bêtises dans la preuve précédente.}
\begin{paragr} The name ``folk Theorem A'' is an explicit reference of Quillen's Theorem A \cite[Theorem A]{quillen1973higher} and its $\oo$\nbd{}categorical generalization by Ara and Maltsiniotis \cite{ara2018theoreme,ara2020theoreme}. For the sake of comparison we recall below the latter one.
@@ -118,7 +118,7 @@ To end this section, we recall a derivability criterion due to Gonzalez, which w
\[\begin{tikzcd} F' : \ho(\C)\ar[r,shift left]&\ho(\C')\ar[l,shift left] : \RR G, \end{tikzcd}\]
then $F$ is absolutely totally left derivable and the pair $(F', \alpha)$, with $\alpha$ defined as in the previous paragraph, is its left derived functor.
\end{proposition}
\todo{Gonzalez ne formule pas son théorème exactement de cette manière. Il faudrait vérifier que je n'ai pas dit de bêtises en le reformulant.}
%\todo{Gonzalez ne formule pas son théorème exactement de cette manière. Il faudrait vérifier que je n'ai pas dit de bêtises en le reformulant.}
\section{(op)Derivators and homotopy colimits}
\begin{notation}We denote by $\CCat$ the $2$-category of small categories and $\CCAT$ the $2$-category of big categories. For a $2$-category $\underline{A}$, the $2$-category obtained from $\underline{A}$ by switching the source and targets of $1$-cells is denoted by $\underline{A}^{op}$.
@@ -18,7 +18,7 @@ $\oo$\nbd{}categories, as it is done for example in the articles \cite{ara2014ve
where $\ho(\oo\Cat^{\Th})$ is the localization of $\oo\Cat$ with respect to Thomason 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 $\oo$\nbd{}categories induced by Thomason equivalences is the same as the homotopy theory of spaces. Gagna's result is in fact a generalization of the analogous 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 analogous 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 $\oo$\nbd{}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{singular homology groups} of $C$ for these homology groups and use the notation $H^{\sing}_k(C)$.
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 $\oo$\nbd{}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{singular homology groups} of $C$ for these homology groups and the notation $H^{\sing}_k(C)$.
On the other hand, Métayer gives a definition in \cite{metayer2003resolutions} of other homology groups for $\oo$\nbd{}categories. This definition is based on the notion of \emph{free $\oo$\nbd{}category on a polygraph} (also known as \emph{free $\oo$\nbd{}category on a computad}), which are $\oo$\nbd{}categories that are obtained from the empty category by recursively freely adjoining cells. From now on, we simply say \emph{free $\oo$\nbd{}category}. Métayer observed that every $\oo$\nbd{}category admits what we call a \emph{polygraphic resolution}, which means that there exists a free $\oo$\nbd{}category $P$ and a morphism of $\oo\Cat$
\[
...
...
@@ -28,7 +28,7 @@ where $\ho(\oo\Cat^{\Th})$ is the localization of $\oo\Cat$ with respect to Thom
One is then lead to the following question:
\begin{center}
Do we have $H_{\bullet}^{\pol}(C)\simeq H_{\bullet}^{\sing}(C)$ for any $\oo$\nbd{}category $C$?
Do we have $H_{\bullet}^{\pol}(C)\simeq H_{\bullet}^{\sing}(C)$ for any $\oo$\nbd{}category $C$?
@@ -47,7 +47,7 @@ where $\ho(\oo\Cat^{\Th})$ is the localization of $\oo\Cat$ with respect to Thom
\end{center}
This is precisely the question around which revolves this dissertation. Yet, the reader will also find several new notions and results within this document that, although primarily motivated by the above question, are of interest in the theory of $\oo$\nbd{}categories and whose \emph{raisons d'être} go beyond the above consideration.
\end{named}
\begin{named}[Another formulation of the problem] A fundamental point of the present work is a more abstract reformulation of the question of comparison of singular and polygraphic homology of $\oo$\nbd{}categories. %As often, the reward for abstraction is a much clearer understanding of the problem.
\begin{named}[Another formulation of the problem] One of the achievement of the present work is a more abstract reformulation of the question of comparison of singular and polygraphic homology of $\oo$\nbd{}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 non-negatively graded chain complexes
\[
...
...
@@ -73,7 +73,10 @@ This way of understanding polygraphic homology as a left derived functor has bee
\[
\LL\lambda^{\Th} : \ho(\oo\Cat^{\Th})\to\ho(\Ch).
\]
This left derived functor being such that $ H_k^{\sing}(C)= H_k(\LL\lambda^{\Th}(C))$ for every $\oo$\nbd{}category $C$ and every $k \geq0$. Contrary to the ``folk'' case, this result is new (at least to my knowledge) and first appears within this document. Note that since, as said earlier, the existence of a Thomason-like model structure on $\oo\Cat$ is still a conjecture, usual tools from Quillen's theory of model categories were unavailable here to prove the left derivability of $\lambda$ and the difficulty was to find a workaround solution.
This left derived functor being such that $ H_k^{\sing}(C)= H_k(\LL
\lambda^{\Th}(C))$ for every $\oo$\nbd{}category $C$ and every $k \geq0$.
Contrary to the ``folk'' case, this result is new and first appears within
this document (at least to my knowledge). Note that since, as said earlier, the existence of a Thomason-like model structure on $\oo\Cat$ is still a conjecture, usual tools from Quillen's theory of model categories were unavailable here to prove the left derivability of $\lambda$ and the difficulty was to find a workaround solution.
From now on, we now set
\[
...
...
@@ -101,15 +104,22 @@ This way of understanding polygraphic homology as a left derived functor has bee
\end{center}
Note that, in theory, question \textbf{(Q')} is less general than question \textbf{(Q)} since we impose which morphism has to be an isomorphism in the comparison of homology groups. However, in practice, when we show that the polygraphic and singular homology groups of an $\oo$\nbd{}category are isomorphic, it is always via the above canonical comparison map and conversely when we show that they are not isomorphic it always rules out any isomorphism possible (not only the canonical comparison map).
Another consequence of the reformulation of the problem in terms of derived functors is that it shows that polygraphic homology is \emph{not} invariant by Thomason equivalences. This means that there exists at least one Thomason equivalence $f : C \to D$ such that the induced map
A formal consequence of the reformulation of the problem in terms of derived functors is that it shows that polygraphic homology is \emph{not} invariant by Thomason equivalences. This means that there exists at least one Thomason equivalence $f : C \to D$ such that the induced map
\[
\sH^{\pol}(C)\to\sH^{\pol}(D)
\]
is \emph{not} an isomorphism. Indeed, if this was not the case, then $\LL\lambda^{\folk}$ would factor through $\J$, yielding a functor ${\ho(\oo\Cat^{\Th})\to\ho(\Ch)}$, which can easily be proved by universal property to be (canonically isomorphic to) $\LL\lambda^{\Th}$. In particular, this would imply that every $\oo$\nbd{}category is homologically coherent, which, as we have already seen, is not true. In other words, if we think of $\oo\Cat$ as a model of homotopy types (via the localization by Thomason equivalences), then polygraphic homology is \emph{not} a well-defined invariant. Another point of view would be to consider that polygraphic homology is an intrinsic invariant of $\oo$\nbd{}categories (and not up to Thomason equivalences) and in that way is finer that singular homology. This is not the point of view adopted here, and the reason will be motivated at the end of this introduction. The slogan to retain is:
is \emph{not} an isomorphism.
Indeed, if this was not the case, then $\LL
\lambda^{\folk}$ would factor through $\J$, yielding a functor
${\ho(\oo\Cat^{\Th})\to\ho(\Ch)}$, which can easily be proved by universal
property to be (canonically isomorphic to) $\LL\lambda^{\Th}$. In
particular, this would imply that every $\oo$\nbd{}category is homologically
coherent, which, as we have already seen, is not true.
In other words, if we think of $\oo\Cat$ as a model of homotopy types (via the localization by Thomason equivalences), then polygraphic homology is \emph{not} a well-defined invariant. Another point of view would be to consider that polygraphic homology is an intrinsic invariant of $\oo$\nbd{}categories (and not up to Thomason equivalences) and in that way is finer that singular homology. This is not the point of view adopted here, and the reason will be motivated at the end of this introduction. The slogan to retain is:
\begin{center}
Polygraphic homology is a way of computing singular homology groups of a homologically coherent $\oo$\nbd{}category.
\end{center}
The point is that given a \emph{free}$\oo$\nbd{}category $P$ (which is thus its own polygraphic resolution), the chain complex $\lambda(P)$ is much ``smaller'' than the chain complex associated to the nerve of $P$ and hence the polygraphic homology groups of $P$ are much easier to compute than its singular homology groups. The situation is much comparable to using cellular homology for computing singular homology of a CW-complex. The difference is that in this last case, such thing is always possible while in the case of $\oo$\nbd{}categories, one must ensure that the (free) $\oo$\nbd{}category is homologically coherent. %Intuitively speaking, this means that some free $\oo$\nbd{}categories are not ``cofibrant enough'' for homology.
The point is that given a \emph{free}$\oo$\nbd{}category $P$ (which is thus its own polygraphic resolution), the chain complex $\lambda(P)$ is much ``smaller'' than the chain complex associated to the nerve of $P$ and hence the polygraphic homology groups of $P$ are much easier to compute than its singular homology groups. The situation is comparable to using cellular homology for computing singular homology of a CW-complex. The difference is that in this last case, such thing is always possible while in the case of $\oo$\nbd{}categories, one must ensure that the (free) $\oo$\nbd{}category is homologically coherent. %Intuitively speaking, this means that some free $\oo$\nbd{}categories are not ``cofibrant enough'' for homology.
One of the main result presented in this dissertation is:
...
...
@@ -118,7 +128,51 @@ This way of understanding polygraphic homology as a left derived functor has bee
\end{center}
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 $\oo$\nbd{}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 isomorphism). 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.
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 isomorphism). But the novelty lies in the proof which is more
conceptual that the one of Lafont and Métayer. It required the development of
several new concepts and results which in the end combine together smoothly,
yielding the above result on the homology of small categories. In the way this
dissertation has been written, all the elements needed to prove this result are
spread over several chapters; a more condensed version of it is the object of
the article \cite{guetta2020homology}. Amongst the
new notions developped in the course of proving the result, the one of discrete Conduché $\oo$\nbd{}functor is probably the most
significant. An $\oo$\nbd{}functor $f : C \to D$ is a \emph{discrete Conduché
$\oo$\nbd{}functor} when for every cell $x$ of $C$, if $f(x)$ can be written as
\[
f(x)=y\comp_k y',
\]
then there exists a unique pair $(x,x')$ of cells of $C$ that are
$k$\nbd{}composable and such that
\[
f(x)=y,\, f(x')=y \text{ and } x=x\comp_k x'.
\]
The main result concerning these $\oo$\nbd{}functors that we prove is that for a discrete
$\oo$\nbd{} functor $f : C \to D$, if the $\oo$\nbd{}category $D$ is free, then
$C$ is also free. The proof of this result is long and tedious, though not
extremely hard conceptually, and first appears in the paper
\cite{guetta2020polygraphs}, which is dedicated to it.
After having settled the case of ($1$\nbd{})categories, it was natural to move
on to $2$\nbd{}categories. The question of understanding which are the \good{}
$2$\nbd{}categories is not trivial, because on the one hand, not all $2$\nbd{}categories are
\good{}, and on the other hand, it is possible to construct many examples of \good{}
$2$\nbd{}categories. As a first step, we will
focus in this dissertation only on $2$\nbd{}categories which are free (as
$\oo$\nbd{}categories). With this simplication, the problem of characterization
of \good{} free $2$\nbd{}categories may be reduced to the following question:
given a cocartesian square of the form
\[
\begin{tikzcd}
\sS_2\ar[r]\ar[d]& P \ar[d]\\
\sD_2\ar[r]& P',
\ar[from=1-1,to=2-2,"\ulcorner",phantom,very near end]
\end{tikzcd}
\]
where $P$ is a free $2$\nbd{}category, when is it \emph{homotopy cocartesian}
\abstract{In this dissertation, we study the homology of strict $\oo$-categories. More precisely, we intend to compare the ``classical'' homology of an $\oo$-category (defined as the homology of its Street nerve) with its polygraphic homology. Along the way, we prove several important result concerning free strict $\oo$\nbd-categories over polygraphs-or-computad and concerning the homotopy theory of strict $\oo$\nbd-categories. }
\abstract{In this dissertation, we study the homology of strict
$\oo$\nbd{}categories. More precisely, we intend to compare the ``classical''
homology of an $\oo$\nbd{}category (defined as the homology of its Street
nerve) with its polygraphic homology. Along the way, we prove several
important results concerning free strict $\oo$\nbd{}categories on polygraphs
(also known as computads) and concerning the homotopy theory of strict $\oo$\nbd{}categories. }
