From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/1750 Path: news.gmane.org!not-for-mail From: Peter Freyd Newsgroups: gmane.science.mathematics.categories Subject: Re: Query about Ab[C] Date: Thu, 14 Dec 2000 09:44:57 -0500 (EST) Message-ID: <200012141444.eBEEiv005794@saul.cis.upenn.edu> NNTP-Posting-Host: main.gmane.org X-Trace: ger.gmane.org 1241018067 32738 80.91.229.2 (29 Apr 2009 15:14:27 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:14:27 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Thu Dec 14 16:18:41 2000 -0400 Return-Path: Original-Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id eBEJNpG12236 for categories-list; Thu, 14 Dec 2000 15:23:51 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 23 Original-Lines: 20 Xref: news.gmane.org gmane.science.mathematics.categories:1750 Archived-At: Bill Rowan asks if there is a simple characterization of those categories C for which Ab[C] is abelian. I doubt if there can be a useful necessary and sufficient condition. A sufficient condition can be found on page 91 of Cats and Alligators, to wit, that the category be effective regular (where "effective" means that every equivalence relation is effective,i.e. it appears as a pullback of a map against itself). Note that the conclusion (abelian) is self-dual but the condition (effective regular) is not. I'm pessimistic about a useful necessary and sufficient condition because of the following: Let C be a category with cartesian squares (needed to define abelian-group-object) such that Ab[C] is abelian. Let C' be a full subcategory closed under cartesian squaring that contains the image of the forgetful functor from Ab[C] back to C. Then Ab[C'] = Ab[C]. An example of the sort of pathological categories to be found among such C' is the category of all groups in which the commutator subgroup is a product of a finite number of simple groups each of which was described prior to 30 June 1973.