From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2999 Path: news.gmane.org!not-for-mail From: "Reinhard Boerger" Newsgroups: gmane.science.mathematics.categories Subject: re: Normal quotients of categories Date: Wed, 18 Jan 2006 09:36:09 +0200 Organization: FernUniversitaet Message-ID: <43CE0C08.32295.302D02@localhost> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=US-ASCII Content-Transfer-Encoding: 7BIT X-Trace: ger.gmane.org 1241019031 6828 80.91.229.2 (29 Apr 2009 15:30:31 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:30:31 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Thu Jan 19 13:25:31 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 19 Jan 2006 13:25:31 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EzdP2-0004VD-Gc for categories-list@mta.ca; Thu, 19 Jan 2006 13:16:52 -0400 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 25 Original-Lines: 50 Xref: news.gmane.org gmane.science.mathematics.categories:2999 Archived-At: Hello, let me add some remarks to Marco Grandis' posting. > 1. Generalised quotients of categories. > > A very general notion of generalised congruence in a category - > also involving objects - can be found in a paper by Bednarczyk, > Borzyszkowski and Pawlowski [BBP]. I have not yet looked at that paper, but I think the "natural" thing is to consider equivalence relations R on a category C, which are subcategories of CxC (i.e. closed under composition, not necessary full; identities are in R by relexivity of R). In my diplomarbeit "Kongruenzrelationen auf Kategorien" from 1977, I considered that, but I was not the first one. Some years earlier there was a paper by Jacques Mersch from Liege (Belgium), which unfortunately appeared only in an internal publication of the university of Liege. Moreover, I think I remember that John Isbell did something on this subject. > The quotient p: X -> X/A is determined by the obvious > universal property: The universal property is also obvious in the general situation. For small categories, a functor with this property always exists, let's call it a quotient functor. For large categories it my happen that the hom-sets of the quotient become large, even if the hom-sets of the original category are small. In general, a congruence as above is not a kernel of a functor; the quotient functor may identify more morphisms. In my diplomarbeit, I rediscovered an example, which had already been found by Mersch. The quotient functors are exactly the regular epis in CAT. But unfortunately, they are not closed under compositon, so a quotient of a quotient of C need not be a quotient of C. > It is interesting to note that p automatically satisfies a 2- > dimensional universal property, as one can easily deduce from the fact > that natural transformations can be viewed as functors X -> Y^2, > with values in the category of morphisms of Y. Of course, this argument also works in the general situation. Greetings Reinhard Boerger