From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6163 Path: news.gmane.org!not-for-mail From: Thorsten Palm Newsgroups: gmane.science.mathematics.categories Subject: Re: Canonical quotients Date: Tue, 14 Sep 2010 17:06:29 +0200 (MEST) Message-ID: 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 1284505643 5689 80.91.229.12 (14 Sep 2010 23:07:23 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Tue, 14 Sep 2010 23:07:23 +0000 (UTC) Cc: Robert Pare , categories@mta.ca To: Michael Barr Original-X-From: majordomo@mlist.mta.ca Wed Sep 15 01:07:21 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpy.mta.ca ([138.73.1.139]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1Oveau-0006vC-9e for gsmc-categories@m.gmane.org; Wed, 15 Sep 2010 01:07:20 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:47660) by smtpy.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1OveZf-0004dS-Cc; Tue, 14 Sep 2010 20:06:03 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1OveZU-0002Uh-03 for categories-list@mlist.mta.ca; Tue, 14 Sep 2010 20:05:52 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6163 Archived-At: Michael Barr hat am 13.09.10 geschrieben: > But that's only an equivalent category; everyone knows that can be done. Yes, as far as that category of partitions is concerned. But note my last sentence: > On Sun, 12 Sep 2010, Thorsten Palm wrote: > >> For the remaining sets as sources, additionally choose the >> identity in case of the trivial quotient, the canonical map otherwise. Perhaps I should have put the emphasis differently: for a proper quotient of a non-partition, choose the canonical map (its range is a partition); for a quotient of a partition, use the union construction. Of course the case distinction is somewhat unsatisfactory, and so I was trying to hint that there would be a nicer solution if all sets were partitions in the first place. It might be worth mentioning that a similar idea serves to equip the very category of sets with a (binary-)product functor to make it strictly monoidal. Thorsten [For admin and other information see: http://www.mta.ca/~cat-dist/ ]