From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2569 Path: news.gmane.org!not-for-mail From: Steve Lack Newsgroups: gmane.science.mathematics.categories Subject: re: Equivalences and psuedo-equivalences (two items) Date: Wed, 25 Feb 2004 09:34:18 +1100 Message-ID: <16443.53610.557110.403907@milan.maths.usyd.edu.au> Reply-To: s.lack@uws.edu.au 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 1241018753 4779 80.91.229.2 (29 Apr 2009 15:25:53 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:25:53 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Tue Feb 24 19:20:03 2004 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 24 Feb 2004 19:20:03 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1Avlpi-0004Mh-00 for categories-list@mta.ca; Tue, 24 Feb 2004 19:19:22 -0400 X-Mailer: VM 6.90 under 21.1 (patch 7) "Biscayne" XEmacs Lucid Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 31 Original-Lines: 48 Xref: news.gmane.org gmane.science.mathematics.categories:2569 Archived-At: Carl Futia asks the following questions: (1) If F:A-->B is a biequivalence of bicategories, is it equivalent to a strict homomorphism? (2) As in (1), but suppose that A and B are 2-categories. The answer to both questions is no. Here's an example. Let A be the 2-element group {0,1}, seen as a 2-category with one object, two arrows, and no non-trivial 2-cells. Let B be the 2-category with one object, with the integers as arrows (and composition given by addition) and with a unique, invertible 2-cell between arrows m and n if m-n is even, and no other 2-cells. The only strict homormorphism (i.e. 2-functor) from A to B sends both arrows of A to the identity arrow 0 of B. There is, however, a biequivalence F:A-->B, sending 0 to 0 and 1 to 1. Note also that there is an obvious 2-functor G:B-->A which is a biequivalence. So the example also illustrates that for a 2-functor which is a biequivalence it may not be possible to choose an ``inverse biequivalence'' which is a 2-functor. This example appeared in: Stephen Lack, A Quillen model structure for 2-categories, K-Theory 26:171-205, 2002 as Example 3.1 on page 178. In that context G:B-->A is actually a trivial fibration, so the fact that there is no 2-functor F with GF=1 also shows that the 2-category A is not cofibrant. Those interested in the rest of the paper should also look at the sequel: Stephen Lack, A Quillen model structure for bicategories, available from http://www.maths.usyd.edu.au/u/stevel/papers/qmcbicat.html which corrects an error in the model structure definition given in the earlier paper, and also extends the model structure to bicategories, giving a Quillen equivalence between the two model categories. Steve Lack.