Commit fc9174e4 authored by Leonard Guetta's avatar Leonard Guetta
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gotta go

parent 81b381b1
......@@ -24,7 +24,7 @@ where $\ho(\oo\Cat^{\Th})$ is the localization of $\oo\Cat$ with respect to Thom
\[
f : P \to X
\]
that satisfies properties formally resembling those of fibrations of topological spaces (or of simplicial sets). Besides, every free $\oo$\nbd{}category can be ``abelianized'' to a chain complex $\lambda(P)$ and Métayer proved that for two different polygraphic resolutions of the same $\oo$\nbd{}category, $P \to C$ and $P' \to C$, the chain complexes $\lambda(P)$ and $\lambda(P')$ are quasi-isomorphic. Hence, we can define the \emph{$k$-th polygraphic homology group} of $C$, denoted by $H_k^{\pol}(C)$, as the $k$-th homology group of $\lambda(P)$ for any polygraphic resolution $P \to C$.
that satisfies properties formally resembling those of trivial fibrations of topological spaces (or of simplicial sets). Besides, every free $\oo$\nbd{}category can be ``abelianized'' to a chain complex $\lambda(P)$ and Métayer proved that for two different polygraphic resolutions of the same $\oo$\nbd{}category, $P \to C$ and $P' \to C$, the chain complexes $\lambda(P)$ and $\lambda(P')$ are quasi-isomorphic. Hence, we can define the \emph{$k$-th polygraphic homology group} of $C$, denoted by $H_k^{\pol}(C)$, as the $k$-th homology group of $\lambda(P)$ for any polygraphic resolution $P \to C$.
One is then lead to the following question:
\begin{center}
......@@ -122,7 +122,7 @@ This way of understanding polygraphic homology as a left derived functor has bee
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.
\end{named}
\begin{named}[Finding homologically coherent $\oo$-categories]
One of the main result presented in this dissertation is:
One of the main results presented in this dissertation is:
\begin{center}
Every (small) category $C$ is homologically coherent.
\end{center}
......@@ -150,7 +150,7 @@ This way of understanding polygraphic homology as a left derived functor has bee
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
$\oo$\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.
......@@ -175,12 +175,12 @@ with respect to Thomason equivalences? As a consequence, a substantial part of t
presented here consists in developping tools to detect homotopy cocartesian squares
of $2$\nbd{}categories with respect to Thomason equivalences. While it appears that
these tools do not allow to completely answer the above question, they still make it
possible to detect such homotopy cocartesian squares in many concrete situtations. In fact,
possible to detect such homotopy cocartesian squares in many concrete situations. In fact,
a whole section of the thesis is dedicated to giving examples of (free) $2$\nbd{}categories
and computing the homotopy type of their nerve using these tools. Amongst all these examples,
a particular class of well-behaved $2$\nbd{}categories, which I coined as ``bubble-free
$2$\nbd{}categories'', seems to stand oud. This class is easily characterized as follows.
Given a $2$\nbd{} category, let us call \emph{bubble} a non-trivial $2$\nbd{}cell
$2$\nbd{}categories'', seems to stand out. This class is easily characterized as follows.
Given a $2$\nbd{}category, let us call \emph{bubble} a non-trivial $2$\nbd{}cell
whose source and target are units on a $0$\nbd{}cell (necessarily the same).
A \emph{bubble-free $2$\nbd{}category} is then nothing but a $2$\nbd{}category that has no bubbles.
The archetypal example of a $2$\nbd{}category that is \emph{not} bubble-free is the $2$\nbd{}category $B$ introduced earlier (which is the commutative monoid $(\mathbb{N},+)$ seen as a $2$\nbd{}category). As already said, this $2$\nbd{}category is not \good{} and this does not seem to be a coincidence. It is indeed remarkable that of all the many examples of $2$\nbd{}categories studied in this work, the only ones that are not \good{} are exactly the ones that are \emph{not} bubble-free. This led to the conjecture below, which stands as a conclusion of the thesis.
......@@ -207,16 +207,29 @@ The archetypal example of a $2$\nbd{}category that is \emph{not} bubble-free is
on the category of weak $\oo$\nbd{}categories and that there should be a good notion of free
weak $\oo$\nbd{}category. In fact, this last notion should be defined as weak $\oo$\nbd{}categories
that are recursively obtained by freely adjoining cells, which is the formal analogue of the
strict version but in the weak context. The important point here is that there is no reason in general
that a free strict $\oo$\nbd{}category be free when considered as a weak $\oo$\nbd{}category. For
example, the $2$\nbd{}category $B$ we have introduced earlier, which is free as a strict
$\oo$\nbd{}category, seems to be \emph{not} free as
a weak $\oo$\nbd{}category. Moreover, there are good candidates for the polygraphic homology of
weak $\oo$\nbd{}categories, and when trying to compute the weak polygraphic homology groups
of $B$ (which needs to take a ``weak polygraphic resolution'' of $B$), it seems that it gives the
strict version but in the weak context. The important point here is that a free strict $\oo$\nbd{}category is \emph{never} free as a weak $\oo$\nbd{}category
(except for the empty $\oo$\nbd{}category).
% For
% example, the $2$\nbd{}category $B$ we have introduced earlier, which is free as a strict
% $\oo$\nbd{}category, seems to be \emph{not} free as
% a weak $\oo$\nbd{}category.
Moreover, there are good candidates for the polygraphic homology of
weak $\oo$\nbd{}categories obtained by mimicking the definition in the strict
case. But there is no reason in general that the polygraphic homology of a
strict $\oo$\nbd{}category is the same that its ``weak polygraphic homology''.
Indeed, since free strict $\oo$\nbd{}categories are not free as weak
$\oo$\nbd{}categories, taking a ``weak polygraphic resolution'' of a strict
$\oo$\nbd{}category is not the same as taking a polygraphic resolution. In
fact, when trying to compute the weak polygraphic homology of $B$, it seems that it gives the
homology groups of a $K(\mathbb{Z},2)$-space. From this observation, it is natural to wonder
whether the fact that polygraphic and singular homologies of strict $\oo$\nbd{}categories do not
coincide is a defect due to working in too narrow a setting.
It is then tempting to make the following conjecture:
\begin{center}
The weak polygraphic homology of a (strict) $\oo$\nbd{}category coincide
with it singular homology.
\end{center}
\end{named}
\begin{named}[Organization of the thesis]
In the first chapter, we review some aspects of the theory of $\oo$\nbd{}categories. In particular,
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......@@ -10,6 +10,11 @@
%%%
\fi
%%% For line numbering (used for proodreading purposes)
\usepackage[pagewise]{lineno}
\linenumbers
\title{Homology of strict $\omega$-categories}
\author{Léonard Guetta}
\begin{document}
......
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