From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6156 Path: news.gmane.org!not-for-mail From: Michael Barr Newsgroups: gmane.science.mathematics.categories Subject: Re: Canonical quotients Date: Mon, 13 Sep 2010 14:56:17 -0400 (EDT) Message-ID: References: Reply-To: Michael Barr NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed X-Trace: dough.gmane.org 1284466663 2855 80.91.229.12 (14 Sep 2010 12:17:43 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Tue, 14 Sep 2010 12:17:43 +0000 (UTC) Cc: Robert Pare , categories@mta.ca To: Thorsten Palm Original-X-From: majordomo@mlist.mta.ca Tue Sep 14 14:17:41 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.138]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1OvUS9-0000EM-JB for gsmc-categories@m.gmane.org; Tue, 14 Sep 2010 14:17:37 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:58525) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1OvUPH-0003ZB-PQ; Tue, 14 Sep 2010 09:14:39 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1OvUPE-00039B-O5 for categories-list@mlist.mta.ca; Tue, 14 Sep 2010 09:14:36 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6156 Archived-At: But that's only an equivalent category; everyone knows that can be done. Even easier is the dual category of complete atomic boolean algebras, which has canonical subobjects. On Sun, 12 Sep 2010, Thorsten Palm wrote: > > Robert Pare hat am 12.09.10 geschrieben: > >> >> Peter Freyd's and John Kennison's examples definitively settled >> Mike Barr's question about canonical subobjects that compose. But >> I had started thinking about it and had what I thought would be a >> nice example. The category of sets has canonical quotients (equivalence >> classes) but they don't compose. I think there is no choice that do, >> but so far I haven't been able to prove or disprove this. Anybody? > > There is. First consider the full subcategory of partitions; that is, > sets whose elements happen to be non-empty, pairwise disjoint sets. It > has an obvious choice of quotient maps that does the trick, namely > those maps for which each element of the target is the union of its > fibre. For the remaining sets as sources, additionally choose the > identity in case of the trivial quotient, the canonical map otherwise. > > Thorsten > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]