From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6327 Path: news.gmane.org!not-for-mail From: Peter May Newsgroups: gmane.science.mathematics.categories Subject: Re: terminology for simplicial sets Date: Tue, 19 Oct 2010 09:41:38 -0500 Message-ID: References: Reply-To: Peter May NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Trace: dough.gmane.org 1287540295 18337 80.91.229.12 (20 Oct 2010 02:04:55 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Wed, 20 Oct 2010 02:04:55 +0000 (UTC) Cc: Categories mailing list , rina@uchicago.edu To: "Prof. Peter Johnstone" Original-X-From: majordomo@mlist.mta.ca Wed Oct 20 04:04:52 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.114]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1P8O2u-0003Yb-2L for gsmc-categories@m.gmane.org; Wed, 20 Oct 2010 04:04:52 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:59694) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1P8O1F-0002zg-6D; Tue, 19 Oct 2010 23:03:09 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1P8O1C-0001N2-1X for categories-list@mlist.mta.ca; Tue, 19 Oct 2010 23:03:06 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6327 Archived-At: Dear Peter, Ah, that is a condition I know well, thanks to work of Rina Foygel, a one-time student of mine now in our Statistics department. I had asked her to study the combinatorics of subdivision of categories. You can find a link to a talk that discusses the condition (starting on page 5) on my web page: http://www.math.uchicago.edu/~may Categories, posets, Alexandrov spaces, simplicial complexes, with emphasis on finite spaces. Buenos Aires, November 10, 2008 (dvi)(pdf) The property you ask about is called property A there, and using certain related properties B and C one can prove Theorem. A simplicial set K has property A if and only if its second barycentric subdivision Sd^2(K) is the simplicial set associated to a classical (ordered) simplicial complex. Another result is that if K does not have A, then Sd(K) cannot be a quasi-category. Still another is that if K has A, then Sd(K) is the nerve of a category. One transfers properties A, B, and C to categories via the nerve functor N. Using them, one proves Theorem. The second subdivision sd^2(C) of any category C is a poset. Theorem. For any category C, sd(C) is isomorphic to the `fundamental category' \tau_1(Sd(NC)). Theorem. A category C has property A if and only if Sd(NC) is isomorphic to N(sdC). Moreover, posets are equivalent to Alexandrov $T_0$-spaces, which have weakly homotopy equivalent classical simplicial complexes. These results, and others related to them, shed light on the Thomason model structure on Cat. Peter May On 10/19/10 5:54 AM, Prof. Peter Johnstone wrote: > In something I've been thinking about recently, the condition > on a simplicial set that all faces of non-degenerate simplices > are non-degenerate seems to play a significant role. Does anyone > know whether this condition has been considered previously, and > if so whether it has a standard name? > > The condition is of course satisfied by those simplicial sets > which are derived from simplicial complexes in the standard way, > but it's more general: it allows the possibility that two > (formally) different faces of a non-degenerate simplex might > coincide, as long as they're not degenerate. > > Peter Johnstone > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]