From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9511 Path: news.gmane.org!.POSTED!not-for-mail From: Clemens.BERGER@unice.fr Newsgroups: gmane.science.mathematics.categories Subject: Re: Products of epimorphisms Date: Sat, 20 Jan 2018 14:49:56 +0100 Message-ID: References: Reply-To: Clemens.BERGER@unice.fr NNTP-Posting-Host: blaine.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 8bit X-Trace: blaine.gmane.org 1516468552 3666 195.159.176.226 (20 Jan 2018 17:15:52 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Sat, 20 Jan 2018 17:15:52 +0000 (UTC) Cc: categories@mta.ca To: David Yetter Original-X-From: majordomo@mlist.mta.ca Sat Jan 20 18:15:48 2018 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp2.mta.ca ([198.164.44.40]) by blaine.gmane.org with esmtp (Exim 4.84_2) (envelope-from ) id 1ecwk4-0000UT-6H for gsmc-categories@m.gmane.org; Sat, 20 Jan 2018 18:15:44 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:59989) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1ecwmX-0001GY-6q; Sat, 20 Jan 2018 13:18:17 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1ecwlG-0005xd-QO for categories-list@mlist.mta.ca; Sat, 20 Jan 2018 13:16:58 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9511 Archived-At: Dear David, your property holds in any regular category provided f and g are regular epis, and so in any elementary topos provided f and g are mere epis. Indeed, your square is a pullback square of regular epis, and in a regular category, any such is also a pushout square. A slightly more general context where your property holds is a finitely complete category with a strong epi-mono factorisation system for which strong epis are closed under pullback along monos and closed under cartesian product. All the best, Clemens. Le 2018-01-19 20:20, David Yetter a ??crit??: > Dear fellow category theorists, > > I'm interested in finding out at what level of generality a result > that plainly holds in Sets (and Sets^op) is true: > > Given two epimorphisms f:A-->>B and g:C-->D, the square formed by f x > 1_C, 1_A x g, f x 1_D and > 1_B x D is a pushout. > > I'd like it to be true (at least) in toposes and I think I have an > element-wise proof (but don't remember the details of the semantics > given by, for instance, Osius, well enough to be sure I've really > proven the result in all toposes -- it's been years since I thought > seriously about that sort of thing). > > And, is there anywhere in the literature that this occurs? It feels > like the sort of thing that would have been known long ago. > > Best Thoughts, > David Yetter > Professor of Mathematics > Kansas State University > > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]