From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6971 Path: news.gmane.org!not-for-mail From: Urs Schreiber Newsgroups: gmane.science.mathematics.categories Subject: Re: Simplicial versus (cubical with connections) Date: Tue, 18 Oct 2011 15:27:56 +0200 Message-ID: References: <33D5C4F9-416F-47E2-9CB3-C0109F977475@gmail.com> Reply-To: Urs Schreiber NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=windows-1252 Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1318976883 15159 80.91.229.12 (18 Oct 2011 22:28:03 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Tue, 18 Oct 2011 22:28:03 +0000 (UTC) To: categories Original-X-From: majordomo@mlist.mta.ca Wed Oct 19 00:27:59 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 1RGI8b-0006Qt-Sl for gsmc-categories@m.gmane.org; Wed, 19 Oct 2011 00:27:58 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:45912) by smtpx.mta.ca with esmtp (Exim 4.76) (envelope-from ) id 1RGI6a-0004l5-Pb; Tue, 18 Oct 2011 19:25:52 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1RGI6Z-0003Zo-4j for categories-list@mlist.mta.ca; Tue, 18 Oct 2011 19:25:51 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6971 Archived-At: Dear category theorists, a while back we had a discussion here about model structures on cubical sets. My colleague Dmitry Roytenberg sent the following message to the list, which, for some reason, did not seem to have gone through. Since it might be of interest to some members of the list I would like to repost it hereby. ---------- Forwarded message ---------- From: Dmitry Roytenberg Date: Thu, Sep 22, 2011 at 12:18 PM Subject: Re: categories: Re: Simplicial versus (cubical with connections) To: categories Dear colleagues, Thanks to everyone who has replied, either privately or on the list. Having read up on the subject a bit I've discovered that quite a lot is known by now about the homotopy theory of cubical sets, thanks mainly to the work of Cisinski and Jardine (Jardine's "Categorical homotopy theory" contains a good exposition of Cisinski's methods, especially useful for non-French speakers). The spatial model structure mentioned by Urs has inclusions as cofibrations and cubical Kan fibrations as fibrations; it is proper, combinatorial and monoidal with respect to the tensor product described by Ronnie, with weak equivalences stable under filtered colimits. There is a monoidal left Quillen equivalence from cubical to simplcial sets mapping the n-dimensional cube to the n-th power of the 1-simplex (the simplicial realization). This is enough to conclude that cubical sets form an excellent model category, in view of Lemma A.3.2.20 and Remark A.3.2.21 in HTT. It then follows from (HTT, Theorem A.3.2.24) that a cubical category is fibrant iff all its function complexes are Kan. As for presheaves on other cubical sites (i.e. with more than just the classical faces and degeneracies), Isaacson in his thesis http://www.ma.utexas.edu/users/isaacson/PDFs/diss.pdf constructs a cubical site containing Brown-Higgins connections (there called conjunctions and disjunctions, as in logic) as well as symmetries of hypercubes, closely related but different from Grandis and Mauri's site !K. Isaacson constructs a _symmetric_ monoidal models tructure on the resulting category of symmetric cubical sets, and equips it with a monoidla left Quillen equivalence from the ordinary cubical sets. However, this model category is not excellent, as not all monomorphisms are cofibrations. As far as I can tell, it is not known if there is an excellent model structure on cubical sets with connections (but without symmetries). 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. Finally, to my astonishment 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. Clearly the cubes are to be viewed as finite power sets, but which structure on the power sets is preserved by the morphisms in each case? Best, Dmitry On Fri, Sep 16, 2011 at 3:24 PM, Fernando Muro wrote: > > For a published reference see: > > MR2591923 (2010k:18022) > Maltsiniotis, Georges(F-PARIS7-IMJ) > La cat=E9gorie cubique avec connexions est une cat=E9gorie test stricte. = (French. English summary) [The category of cubes with connections is a stri= ct test category] > Homology, Homotopy Appl. 11 (2009), no. 2, 309=96326. > > Best, > > Fernando > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]