From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5158 Path: news.gmane.org!not-for-mail From: Marco Grandis Newsgroups: gmane.science.mathematics.categories Subject: Re: 'Directed Algebraic Topology' Date: Tue, 22 Sep 2009 10:37:13 +0200 Message-ID: Reply-To: Marco Grandis NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 (Apple Message framework v752.2) Content-Type: text/plain; charset=ISO-8859-1; delsp=yes; format=flowed Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1253622195 16010 80.91.229.12 (22 Sep 2009 12:23:15 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Tue, 22 Sep 2009 12:23:15 +0000 (UTC) To: Urs Schreiber , categories@mta.ca Original-X-From: categories@mta.ca Tue Sep 22 14:23:08 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 1Mq4Oi-0003NS-7W for gsmc-categories@m.gmane.org; Tue, 22 Sep 2009 14:23:08 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Mq3z3-0002pF-2Y for categories-list@mta.ca; Tue, 22 Sep 2009 08:56:37 -0300 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5158 Archived-At: Dear Urs, There are various directed topological structures, which have =20 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. =20 (Notice that the one-dimensional information which is added to a topological space has effects in all =20 dimension.) However, directed homology works much better for cubical sets, or spaces with =20 distinguished cubes. In my web page you can find references to many papers of mine on this =20= domain, and such papers have many references to other authors. You could begin by: - M. Grandis, Directed homotopy theory, I. The fundamental category, =20 Cah. Topol. G=E9om. Diff=E9r. Cat=E9g. 44 (2003), 281-316. -, The shape of a category up to directed homotopy, Theory Appl. =20 Categ. 15 (2005/06), No. 4, 95-146. A more complete study can be found in my book. The latter does not cover higher fundamental categories, which - in =20 dimension 2 - can be found in: -, Modelling fundamental 2-categories for directed homotopy, Homology =20= Homotopy Appl. 8 (2006), 31-70. -, Lax 2-categories and directed homotopy, Cah. Topol. G=E9om. Diff=E9r. = =20 Cat=E9g. 47 (2006), 107-128. -, Absolute lax 2-categories, Appl. Categ. Struct. 14 (2006), 191-214. Marco Grandis http://www.dima.unige.it/~grandis/ On 21 Sep 2009, at 11:44, Urs Schreiber wrote: > Marco Grandis wrote: > >> My book >> >> 'Directed Algebraic Topology' >> Models of non-reversible worlds >> >> has appeared, at Cambridge University Press. > > In that context I am wondering about the following: > > it would be nice to have a notion of directed topological space that > would extend the relation between (nice) topological spaces and > oo-groupoids to one between (nice) directed topological spaces and > (oo,1)-categories. > > ... > Has anything like this been considered? [For admin and other information see: http://www.mta.ca/~cat-dist/ ]