From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6332 Path: news.gmane.org!not-for-mail From: Peter May Newsgroups: gmane.science.mathematics.categories Subject: Re: terminology for simplicial sets Date: Thu, 21 Oct 2010 20:43:08 -0500 Message-ID: References: <4CBDAE22.8090609@math.uchicago.edu> 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 1287762943 19649 80.91.229.12 (22 Oct 2010 15:55:43 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Fri, 22 Oct 2010 15:55:43 +0000 (UTC) To: Categories mailing list Original-X-From: majordomo@mlist.mta.ca Fri Oct 22 17:55:42 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 1P9Jxz-00088b-I0 for gsmc-categories@m.gmane.org; Fri, 22 Oct 2010 17:55:39 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:44611) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1P9Jx5-00028k-Qg; Fri, 22 Oct 2010 12:54:43 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1P9Jx2-0005dc-Aw for categories-list@mlist.mta.ca; Fri, 22 Oct 2010 12:54:41 -0300 In-Reply-To: <4CBDAE22.8090609@math.uchicago.edu> Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6332 Archived-At: I agree with Peter J. that a good name is a good thing. However, he has suggested that ``regular'' is surely better than ``Property A'', and I have to say that it most assuredly is not. Regular CW complex has a standard meaning, and there is a related standard meaning for regular simplicial set, which is recalled on page 4 of the link I sent originally (http://www.math.uchicago.edu/~may Categories, posets, Alexandrov spaces, simplicial complexes, with emphasis on finite spaces. Buenos Aires, November 10, 2008) A nondegenerate n-simplex x in a simplicial set K is regular if the subcomplex [x] that it generates is the pushout of the last face inclusion \Delta_{n-1} \to \Delta_{n} along the last face d_n x : \Delta[n-1] \to [d_nx]. K itself is regular if all of its nondegenerate simplices are regular. The subdivision of any simplicial set is regular. This definition is standard because the realization of a regular simplicial set is a regular CW complex, and regular CW complexes are triangulable, that is homeomorphic to the realization of a simplicial set coming from a classical simplicial complex. This is all classical, and I could cite a number of sources. A modern one (1990) with a good treatment is Fritsch and Piccinini, Cellular structures in topology. See p. 208. I hold no particular brief for Property A (and B and C), but they will do until something definitely better comes along. Richard has suggested semisimplicial, but that to my mind is certainly not better (but then I'm old enough to remember when semisimplicial meant what we now call simplicial, to differentiate from classical simplicial complexes). Peter [For admin and other information see: http://www.mta.ca/~cat-dist/ ]