From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6147 Path: news.gmane.org!not-for-mail From: Thorsten Palm Newsgroups: gmane.science.mathematics.categories Subject: Re: Canonical quotients Date: Sun, 12 Sep 2010 23:18:10 +0200 (MEST) Message-ID: References: <20100912120438.14DF55C186@chase.mathstat.dal.ca> Reply-To: Thorsten Palm NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Trace: dough.gmane.org 1284402068 18226 80.91.229.12 (13 Sep 2010 18:21:08 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Mon, 13 Sep 2010 18:21:08 +0000 (UTC) Cc: categories@mta.ca To: Robert Pare Original-X-From: majordomo@mlist.mta.ca Mon Sep 13 20:21:07 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 1OvDeM-0002zd-H3 for gsmc-categories@m.gmane.org; Mon, 13 Sep 2010 20:21:06 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:37354) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1OvDct-0007qI-Vl; Mon, 13 Sep 2010 15:19:35 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1OvDcq-0008Sr-AG for categories-list@mlist.mta.ca; Mon, 13 Sep 2010 15:19:32 -0300 In-Reply-To: <20100912120438.14DF55C186@chase.mathstat.dal.ca> Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6147 Archived-At: 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/ ]