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) [For admin and other information see: http://www.mta.ca/~cat-dist/ ]