From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/1762 Path: news.gmane.org!not-for-mail From: Michael Barr Newsgroups: gmane.science.mathematics.categories Subject: Re: whoops Date: Sun, 17 Dec 2000 14:10:04 -0500 (EST) Message-ID: References: <200012161622.eBGGMY222073@saul.cis.upenn.edu> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Trace: ger.gmane.org 1241018082 328 80.91.229.2 (29 Apr 2009 15:14:42 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:14:42 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Mon Dec 18 09:27:00 2000 -0400 Return-Path: Original-Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id eBICXMY23191 for categories-list; Mon, 18 Dec 2000 08:33:22 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f X-Authentication-Warning: triples.math.mcgill.ca: barr owned process doing -bs X-Sender: barr@triples.math.mcgill.ca In-Reply-To: <200012161622.eBGGMY222073@saul.cis.upenn.edu> Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 35 Original-Lines: 32 Xref: news.gmane.org gmane.science.mathematics.categories:1762 Archived-At: Sigh! Peter is, of course, correct. It actually occurred to me a couple days after I wrote my reply that Ab[C^op] is not the opposite of Ab[C] but I never looked into it. Michael On Sat, 16 Dec 2000, Peter Freyd wrote: > There was a little trap waiting for Mike and me to fall into. We were > seeking conditions on a category C that force Ab[C] to be abelian > and we came up with the same condition. I wrote, "Note that the > conclusion (abelian) is self-dual but the condition (effective > regular) is not." The trouble is: the definition of Ab[C] is not > self-dual. I suppose my assertion is true as it stands but the > implicit message is wrong. Mike had no such luck; he made it explicit: > "[I]t is sufficient that either C or C^op be exact." (Yes, my > "effective regular" and Mike's "exact" are equivalent -- it follows > just from regularity that the pullback of a cover against itself is > a pushout.) > > It took me a while to find a counterexample and I'm not happy with the > one I found. Adjoin to the equational theory of abelian groups a new > constant and no further axioms. Let C be the category of finite > models. As is the case for the finite models of any equational theory, > C is effective regular (and Ab[C] is isomorphic to the category of > finite abelian groups). But Ab[C^op] is not abelian. It's empty. > (C^op doesn't have a terminator.) > >