From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6982 Path: news.gmane.org!not-for-mail From: Todd Trimble Newsgroups: gmane.science.mathematics.categories Subject: Re: Simplicial versus (cubical with connections) Date: Sat, 22 Oct 2011 09:07:59 -0400 Message-ID: Reply-To: Todd Trimble NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; format=flowed; charset=iso-8859-1; reply-type=response Content-Transfer-Encoding: 7BIT X-Trace: dough.gmane.org 1319371688 1751 80.91.229.12 (23 Oct 2011 12:08:08 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sun, 23 Oct 2011 12:08:08 +0000 (UTC) Cc: Categories list To: Ross Street , Vaughan Pratt Original-X-From: majordomo@mlist.mta.ca Sun Oct 23 14:08:04 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 1RHwqR-0005Bj-I4 for gsmc-categories@m.gmane.org; Sun, 23 Oct 2011 14:08:03 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:47061) by smtpx.mta.ca with esmtp (Exim 4.76) (envelope-from ) id 1RHwpL-0004mk-LW; Sun, 23 Oct 2011 09:06:55 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1RHwpK-0003nf-3l for categories-list@mlist.mta.ca; Sun, 23 Oct 2011 09:06:54 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6982 Archived-At: My impression is that there are at least two distinct notions of cubical set which have entered this discussion. One version describes cubical sets as presheaves on the Lawvere theory generated by two constants or 0-ary operations; this is close to what Vaughan described. More precisely, instead of taking the category whose objects are finite sets equipped with two distinct points (which is opposite to the Lawvere theory), he adds in a terminal object (where the two constants are forced to coincide), giving a category C. Anyway, whether one takes the Lawvere theory or C^{op}, the result is a category with finite cartesian products and an interval object, and one notion of cubical set is that of presheaf on this category. Whereas cubical sets in the sense described by Ross are different: they are presheaves on the free *monoidal* category with an interval object. This category does not include diagonal maps. I expect this is the notion of cubical set that Dmitry and Urs were actually concerned with, but in any event, both the cartesian version and the monoidal version of the cubical site appear in the literature, and it is important to clarify which notion is meant. Todd Trimble [For admin and other information see: http://www.mta.ca/~cat-dist/ ]