From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6972 Path: news.gmane.org!not-for-mail From: Marco Grandis Newsgroups: gmane.science.mathematics.categories Subject: Re: Simplicial versus (cubical with connections) Date: Wed, 19 Oct 2011 10:35:01 +0200 Message-ID: References: <33D5C4F9-416F-47E2-9CB3-C0109F977475@gmail.com> Reply-To: Marco Grandis NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 (Apple Message framework v753.1) Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed Content-Transfer-Encoding: 7bit X-Trace: dough.gmane.org 1319025748 22362 80.91.229.12 (19 Oct 2011 12:02:28 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Wed, 19 Oct 2011 12:02:28 +0000 (UTC) To: categories@mta.ca, Urs Schreiber Original-X-From: majordomo@mlist.mta.ca Wed Oct 19 14:02:23 2011 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.4]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1RGUqj-0007bf-Bw for gsmc-categories@m.gmane.org; Wed, 19 Oct 2011 14:02:21 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:45585) by smtpx.mta.ca with esmtp (Exim 4.76) (envelope-from ) id 1RGUp4-0001Lg-2d; Wed, 19 Oct 2011 09:00:38 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1RGUp2-0001Qq-4n for categories-list@mlist.mta.ca; Wed, 19 Oct 2011 09:00:36 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6972 Archived-At: This is about two points of a recent message of Dmitry Roytenberg, forwarded by Urs Schreiber. > I have not been able to find an abstract > description of any of the cubical sites, in the spirit of the simplex > category being the category of non-empty finite ordinals I do not know of any such description. But there is a nice abstract description of the site of cubical sets with connections, parallel to a well-known characterisation of the simplicial site: - the free strict monoidal category with an assigned dioid. See [GM], Thm. 5.2. (There are analogous results for the other cubical sites.) A `dioid' is a set with two monoid operations, where the unit of each operation is absorbant for the other. Typically, an abstract interval has such a structure, and a cylinder functor has the structure of a 'diad'. See [Gr]. (I was also using the terms 'cubical monoid' and 'cubical monad', for an obvious analogy; later I abandoned them because they could be misleading - obviously again). Every lattice is an idempotent dioid, but idempotency is - apparently - of no interest in homotopy. This leads us to the second point: smooth homotopy. > In any case, using these connections in a differential-geometric > context is problematic, not (just) because of a clash with established > terminology, but because the max and min maps are only piecewise > smooth. For smooth homotopy one should use a different (non-idempotent) dioid, still commutative and involutive: NOT the standard interval with min, max, linked by the involution t' = 1 - t, BUT the standard interval with multiplication and *, linked by the same involution: x*y = (x'.y')' = x + y - xy. See [Gr]. [Gr] M. Grandis, Cubical monads and their symmetries, in: Proc. of the Eleventh Intern. Conf. on Topology, Trieste 1993, Rend. Ist. Mat. Univ. Trieste 25 (1993), 223-262. http://www.dmi.units.it/~rimut/volumi/25/index.html [GM] M. Grandis - L. Mauri, Cubical sets and their site, Theory Appl. Categ. 11 (2003), No. 8, 185-211. Best regards Marco Grandis [For admin and other information see: http://www.mta.ca/~cat-dist/ ]