From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4130 Path: news.gmane.org!not-for-mail From: Robin Houston Newsgroups: gmane.science.mathematics.categories Subject: Re: On defining *-autonomous categories Date: Tue, 18 Dec 2007 14:27:37 +0000 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii X-Trace: ger.gmane.org 1241019737 11836 80.91.229.2 (29 Apr 2009 15:42:17 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:42:17 +0000 (UTC) To: Categories list Original-X-From: rrosebru@mta.ca Tue Dec 18 21:56:11 2007 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 18 Dec 2007 21:56:11 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1J4o1H-0002cz-P2 for categories-list@mta.ca; Tue, 18 Dec 2007 21:46:47 -0400 Content-Disposition: inline Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 24 Original-Lines: 18 Xref: news.gmane.org gmane.science.mathematics.categories:4130 Archived-At: On Tue, Dec 18, 2007 at 01:08:07PM +0000, Robin Houston wrote: > Lemma 2. Let C be a category, and F: C -> C an endofunctor. If FF > is naturally isomorphic to 1_C, then F is an equivalence. > > Proof: Certainly F is essentially surjective, since every object > X in C is naturally isomorphic to FFX. It then follows by Lemma 1 > that F is full and faithful. > > > For the implication 1 => 2, take F = (- -o D) in Lemma 2. There is a silly mistake here, caused by the fact that the functor (- -o D) is contravariant. The error is really in the statement of Lemma 2; of course the proof still works for contravariant F. Robin