From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/1753 Path: news.gmane.org!not-for-mail From: "Dr. P.T. Johnstone" Newsgroups: gmane.science.mathematics.categories Subject: Re: Query about Ab[C] Date: Thu, 14 Dec 2000 16:25:39 +0000 (GMT) Message-ID: References: <200012140419.eBE4JSF19580@transbay.net> 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 1241018075 32757 80.91.229.2 (29 Apr 2009 15:14:35 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:14:35 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Thu Dec 14 16:20:59 2000 -0400 Return-Path: Original-Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id eBEJQgx06464 for categories-list; Thu, 14 Dec 2000 15:26:42 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f In-Reply-To: <200012140419.eBE4JSF19580@transbay.net> from "Bill Rowan" at Dec 13, 2000 08:19:28 PM X-Mailer: ELM [version 2.5 PL2] Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 26 Original-Lines: 23 Xref: news.gmane.org gmane.science.mathematics.categories:1753 Archived-At: > > > Hi all, > > Ab[C] is just my notation for the category of abelian group objects in > the category C. I was wondering if there is a simple characterization of > those categories C for which Ab[C] is abelian. > > Bill Rowan You can't hope to characterize them: knowing properties of Ab[C] can't tell you everything about C. For example, if C has a strict terminal object, then Ab[C] is abelian (because it's degenerate), but that gives you no information about what else C might contain. If you're looking for a sufficient condition on C, a canonical one is "Barr-exact" (= effective regular, in Freyd's terminology): Ab[C] inherits Barr-exactness from C, and abelian is equivalent to Barr-exact plus additive. Conversely, every abelian category A is isomorphic to Ab[C] for a suitable Barr-exact C, namely C = A. Peter Johnstone