From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9510 Path: news.gmane.org!.POSTED!not-for-mail From: Peter Johnstone Newsgroups: gmane.science.mathematics.categories Subject: Re: Products of epimorphisms Date: Sat, 20 Jan 2018 10:57:30 +0000 (GMT) Message-ID: References: Reply-To: Peter Johnstone NNTP-Posting-Host: blaine.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed X-Trace: blaine.gmane.org 1516468484 21053 195.159.176.226 (20 Jan 2018 17:14:44 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Sat, 20 Jan 2018 17:14:44 +0000 (UTC) Cc: "categories@mta.ca" To: David Yetter Original-X-From: majordomo@mlist.mta.ca Sat Jan 20 18:14:40 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 1ecwip-0004ZY-CR for gsmc-categories@m.gmane.org; Sat, 20 Jan 2018 18:14:27 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:59983) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1ecwlE-00012g-GO; Sat, 20 Jan 2018 13:16:56 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1ecwjx-0005wQ-9D for categories-list@mlist.mta.ca; Sat, 20 Jan 2018 13:15:37 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9510 Archived-At: Dear David, Yes, this is true for regular epis in any cartesian closed category. I don't know a published proof, but here is the argument: if h: A x C --> E factors through 1_A x g, then its transpose C --> E^A factors through g, and so coequalizes any pair R ==> C of which g is a coequalizer. But if h also factors through f x 1_C, then its transpose factors through E^B --> E^A, which is monic (since E^(-), being self-adjoint on the right, sends epis to monos), so the induced C --> E^B also coequalizes R ==> C and hence factors through g. Hence h factors through f x g. In fact, as is clear from the above argument, we need only one of f and g to be regular epic. Best regards, Peter Johnstone On Fri, 19 Jan 2018, David Yetter wrote: > 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/ ]