From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6174 Path: news.gmane.org!not-for-mail From: selinger@mathstat.dal.ca (Peter Selinger) Newsgroups: gmane.science.mathematics.categories Subject: Re: Canonical quotients Date: Mon, 13 Sep 2010 11:09:19 -0300 (ADT) Message-ID: Reply-To: selinger@mathstat.dal.ca (Peter Selinger) NNTP-Posting-Host: lo.gmane.org X-Trace: dough.gmane.org 1284594853 16493 80.91.229.12 (15 Sep 2010 23:54:13 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Wed, 15 Sep 2010 23:54:13 +0000 (UTC) Cc: categories@mta.ca To: pare@mathstat.dal.ca Original-X-From: majordomo@mlist.mta.ca Thu Sep 16 01:54:11 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 1Ow1nk-0005yB-MG for gsmc-categories@m.gmane.org; Thu, 16 Sep 2010 01:54:08 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:43937) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1Ow1md-0007Gi-2Y; Wed, 15 Sep 2010 20:52:59 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1Ow1mY-0002YL-Su for categories-list@mlist.mta.ca; Wed, 15 Sep 2010 20:52:55 -0300 X-Mailer: ELM [version 2.5 PL8] Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6174 Archived-At: [note from moderator: post delayed from Monday by a transmission error] Hi Bob, how about this: using a sufficiently strong version of the axiom of choice, assume a well-ordering on the class of all sets (the "global well-ordering"). (Someone will be able to tell me whether this is strictly stronger than the ordinary axiom of choice, or equivalent to it. In any case, assuming set theory is consistent in the first place, and sufficiently large cardinals exist, there certainly exist models of set theory with such a property). Given any equivalence relation ~ on a set X, say that an element x of X is a "canonical representative" of ~ if x is the (unique) least element in its equivalence class under the global well-ordering. Let X/~ be the set of canonical representatives, and define the map X -> X/~ to pick out the canonical representative of each class. I think this gives canonical quotients on (this version of) Set. -- Peter Robert Pare wrote: > > > 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? > > Bob [For admin and other information see: http://www.mta.ca/~cat-dist/ ]