From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5167 Path: news.gmane.org!not-for-mail From: Gaucher Philippe Newsgroups: gmane.science.mathematics.categories Subject: Re: 'Directed Algebraic Topology' Date: Tue, 22 Sep 2009 15:12:09 +0200 Message-ID: Reply-To: Gaucher Philippe NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: Text/Plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1253668135 9868 80.91.229.12 (23 Sep 2009 01:08:55 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 23 Sep 2009 01:08:55 +0000 (UTC) To: categories@mta.ca Original-X-From: categories@mta.ca Wed Sep 23 03:08:48 2009 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.50) id 1MqGLg-0001ok-0B for gsmc-categories@m.gmane.org; Wed, 23 Sep 2009 03:08:48 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1MqFoP-0007cU-6m for categories-list@mta.ca; Tue, 22 Sep 2009 21:34:25 -0300 Content-Disposition: inline Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5167 Archived-At: Le mardi 22 septembre 2009 10:37:13, Marco Grandis a =E9crit : > 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 =3D 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= =20 topology. Indeed, d-spaces have a lot of interesting features as topological model si= nce=20 i also work in a recent preprint with multipointed d-spaces (1) and i prove= =20 that from a homotopical point of view, they are the topological version (in= =20 the sense of concrete topological functors) of the category of flows (Or on= the=20 contrary, the flows are a categorical version of multipointed d-spaces). I = had=20 introduced them for studying branching and merging homology theories, which= =20 require a specific feature of the topological model that d-spaces do not ha= ve.=20 This study can also be done with multipointed d-spaces in theory, but the=20 theory remains to be written (I am working on that...). I also have a model= =20 structure on multipointed d-spaces preserving the homotopy type of path=20 spaces. I'd like to mention in this mail that I do not know how to prove th= at=20 it is left-proper and any idea would be really welcome. Concerning cubical sets now. The usual degeneracy maps have no interest in= =20 computer science. But they do not "disturb". On the contrary, there is a ne= w=20 kind of degeneracy map that I call transverse degeneracy (2) which are of=20 interest in computer science. The symmetric transverse precubical sets are= =20 the only kind of precubical set such that the 1-dimensional coskeleton func= tor=20 is well-behaved from a computer science point of view (see (2)). The base=20 category \widehat{\square} appears also in the study of topological models = of=20 concurrency. Roughly speaking, the space of morphisms from a topological m- cube to a topological n-cube preserving the labelling will be always homoto= py=20 equivalent to \widehat{\square}([m],[n]), if the m+n labels are the same (i= f=20 they are not the same, it will be homotopy equivalent to the subset=20 corresponding to maps preserving the labelling of course). Precubical sets are enough to model all process algebras (3) but they are t= oo=20 poor for a mathematical treatment, even in the combinatorial world. For=20 example, in the category of precubical sets, the labelled square correspond= ing=20 to the concurrent execution of a and b is not isomorphic to the labelled=20 square corresponding to the concurrent execution of b and a if a<>b !=20 Symmetric precubical sets are better (or less bad) because this drawback=20 disappears. Transverse symmetric precubical sets are even better but they a= re=20 more complicated to understand. I have an explicit combinatorial descriptio= n=20 of the symmetric precubical set of labels. I do not have such a description= =20 for the transverse symmetric precubical set of labels for example. Symmetric precubical sets are also related to higher dimensional transition= =20 systems in a non-trivial way: the latter can be identified to a full reflec= tive=20 subcategory of the (labelled) former (4). pg. http://www.pps.jussieu.fr/~gaucher/ (1) Homotopical interpretation of globular complex by multipointed d-space= =20 (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=20 (preprint) [For admin and other information see: http://www.mta.ca/~cat-dist/ ]