Commit 81b381b1 authored by Leonard Guetta's avatar Leonard Guetta
Browse files

dodo

parent 78721a44
......@@ -194,8 +194,9 @@ The archetypal example of a $2$\nbd{}category that is \emph{not} bubble-free is
all in the rest of the dissertation. It ought to be thought of as a guideline for future work.
In the same way that (strict) $2$\nbd{}categories are particular cases of bicategories,
strict $\oo$\nbd{}categories are in fact particular cases of weak $\oo$\nbd{}categories.
Such mathematical objects have been defined, for example, by Batanin using globular operads
strict $\oo$\nbd{}categories are in fact particular cases of what is usually called
\emph{weak $\oo$\nbd{}categories}. Such mathematical objects have been defined, for example,
by Batanin using globular operads
\cite{batanin1998monoidal} or by Maltsiniotis following ideas of Grothendieck
\cite{maltsiniotis2010grothendieck}. Similarly to the fact that the theory of quasi-categories
(which is a homotopical model for the theory weak $\oo$\nbd{}categories whose cells are invertible
......@@ -207,15 +208,27 @@ The archetypal example of a $2$\nbd{}category that is \emph{not} bubble-free is
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
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 polygraphic homology groups
of $B$ (which needs to take a ``weak polygraphic resolution'' 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 the fact that we work in too narrow a setting.
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
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.
\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,
we study with great care free $\oo$\nbd{}categories, which are at the heart of the present work. It
is the only chapter of the thesis that does not contain any reference to homotopy theory whatsoever.
It is also there that we introduce the notion of discrete Conduché $\oo$\nbd{}functor and study
their relation with free $\oo$\nbd{}categories. The culminating point of the chapter is Theorem
\ref{thm:conduche}, which states that, given a discrete Conduché $\oo$\nbd{}functor $F : C \to D$,
if $D$ is free, then so is $C$. The proof of this theorem is long and technical and is broke down
into several distinct parts.
The second chapter is devoted to
\end{named}
%% Let us come back to the canonical $2$-triangle
%% \[
......
No preview for this file type
Markdown is supported
0% or .
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!
Please register or to comment