From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6897 Path: news.gmane.org!not-for-mail From: Ronnie Brown Newsgroups: gmane.science.mathematics.categories Subject: Re: Simplicial versus (cubical with connections) Date: Wed, 14 Sep 2011 11:04:39 +0100 Message-ID: References: <33D5C4F9-416F-47E2-9CB3-C0109F977475@gmail.com> Reply-To: Ronnie Brown NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1316029232 26402 80.91.229.12 (14 Sep 2011 19:40:32 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Wed, 14 Sep 2011 19:40:32 +0000 (UTC) Cc: Categories list , Marco Grandis To: =?UTF-8?B?Sm9uYXRoYW4gQ0hJQ0hFIOm9iuato+iIqg==?= Original-X-From: majordomo@mlist.mta.ca Wed Sep 14 21:40:28 2011 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpy.mta.ca ([138.73.1.128]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1R3vJr-00059V-Bd for gsmc-categories@m.gmane.org; Wed, 14 Sep 2011 21:40:27 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:42074) by smtpy.mta.ca with esmtp (Exim 4.76) (envelope-from ) id 1R3vIJ-0008Uw-9n; Wed, 14 Sep 2011 16:38:51 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1R3vIH-0003YG-M3 for categories-list@mlist.mta.ca; Wed, 14 Sep 2011 16:38:49 -0300 In-Reply-To: <33D5C4F9-416F-47E2-9CB3-C0109F977475@gmail.com> Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6897 Archived-At: The result of Maltsiniotis referred to by Jonathan is very welcome. But=20 I wonder if there is still a problem with cubical sets with connection: the geometric realisation of a simplicial group is, in a convenient=20 category, a topological group, because of the homeomorphism f: |K \times Y| \to |K| \times |Y| . However in the case of cubical sets with connections this map f is a=20 homotopy equivalence but it seems is not a homeomorphism (?). As=20 Grothendieck wrote: `homotopically speaking' that is not a problem! For homotopies and higher homotopies cubes are nice and easy because of=20 the basic formula I^m \times I^n =3D I^{m+n}. This leads to monoidal closed structures on strict cubical higher=20 categories and groupoids. For a basic discussion of other issues such as algebraic inverses to=20 subdivision and commutative cubes I refer to my 2009 Liverpool seminar=20 on`What is and what should be `Higher dimensional group theory'?' http://pages.bangor.ac.uk/~mas010/pdffiles/liverpool-beamer-handout.pdf Ronnie [For admin and other information see: http://www.mta.ca/~cat-dist/ ]