Re: 'Directed Algebraic Topology'

[Gaucher Philippe <Philippe.Gaucher@pps.jussieu.fr>]

Le mardi 22 septembre 2009 10:37:13, Marco Grandis a écrit :
> Dear Urs,
>
> There are various directed topological structures, which have
> directed homotopies
> and fundamental (higher) categories, like:
>
> - preordered topological spaces (simple but poor);
> - locally preordered topological spaces (in a suitable sense);
> - d-spaces = topological spaces equipped with distinguished paths;
> - spaces equipped with distinguished cubes;
> - cubical sets (in the combinatorial world);
> - generalised metric spaces (in the sense of Lawvere);
> - 'inequilogical spaces';
> - etc.
>
> I prefer d-spaces, which have also been studied by other authors.

Dear categorists,

Some remarks about topological and categorical models of directed algebraic 
topology.

Indeed, d-spaces have a lot of interesting features as topological model since 
i also work in a recent preprint with multipointed d-spaces (1) and i prove 
that from a homotopical point of view, they are the topological version (in 
the sense of concrete topological functors) of the category of flows (Or on the 
contrary, the flows are a categorical version of multipointed d-spaces). I had 
introduced them for studying branching and merging homology theories, which 
require a specific feature of the topological model that d-spaces do not have. 
This study can also be done with multipointed d-spaces in theory, but the 
theory remains to be written (I am working on that...). I also have a model 
structure on multipointed d-spaces preserving the homotopy type of path 
spaces. I'd like to mention in this mail that I do not know how to prove that 
it is left-proper and any idea would be really welcome.

Concerning cubical sets now. The usual degeneracy maps have no interest in 
computer science. But they do not "disturb". On the contrary, there is a new 
kind of degeneracy map that I call transverse degeneracy (2) which are of 
interest in computer science. The symmetric transverse precubical sets  are 
the only kind of precubical set such that the 1-dimensional coskeleton functor 
is well-behaved from a computer science point of view (see (2)). The base 
category \widehat{\square} appears also in the study of topological models of 
concurrency. Roughly speaking, the space of morphisms from a topological m-
cube to a topological n-cube preserving the labelling will be always homotopy 
equivalent to \widehat{\square}([m],[n]), if the m+n labels are the same (if 
they are not the same, it will be homotopy equivalent to the subset 
corresponding to maps preserving the labelling of course).

Precubical sets are enough to model all process algebras (3) but they are too 
poor for a mathematical treatment, even in the combinatorial world. For 
example, in the category of precubical sets, the labelled square corresponding 
to the concurrent execution of a and b is not isomorphic to the labelled 
square corresponding to the concurrent execution of b and a if a<>b ! 
Symmetric precubical sets are better (or less bad) because this drawback 
disappears. Transverse symmetric precubical sets are even better but they are 
more complicated to understand. I have an explicit combinatorial description 
of the symmetric precubical set of labels. I do not have such a description 
for the transverse symmetric precubical set of labels for example.
Symmetric precubical sets are also related to higher dimensional transition 
systems in a non-trivial way: the latter can be identified to a full reflective 
subcategory of the (labelled) former (4).

pg. http://www.pps.jussieu.fr/~gaucher/

(1) Homotopical interpretation of globular complex by multipointed d-space 
(preprint)
(2) Combinatorics of labelling in higher dimensional automata (preprint)
(3) Towards a homotopy theory of process algebra (HHA)
(4) Directed algebraic topology and higher dimensional transition system 
(preprint)











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