From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4493 Path: news.gmane.org!not-for-mail From: Michael Shulman Newsgroups: gmane.science.mathematics.categories Subject: Re: biadjoint biequivalences and spans in 2-categories Date: Tue, 19 Aug 2008 13:11:10 -0500 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii X-Trace: ger.gmane.org 1241019980 13559 80.91.229.2 (29 Apr 2009 15:46:20 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:46:20 +0000 (UTC) To: categories Original-X-From: rrosebru@mta.ca Tue Aug 19 22:02:23 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 19 Aug 2008 22:02:23 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KVc4Z-0004fG-Vk for categories-list@mta.ca; Tue, 19 Aug 2008 22:01:16 -0300 Content-Disposition: inline Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 28 Original-Lines: 47 Xref: news.gmane.org gmane.science.mathematics.categories:4493 Archived-At: Hi John, Answers to both this and your previous question about biadjoint biequivalences are at least asserted in Street's "Fibrations in Bicategories". At the end of section 1, he defines a functor (=homomorphism) to have a left biadjoint if each object has a left bilifting, to be a biequivalence if it is biessentially surjective and locally fully faithful, and states that "clearly a biequivalence T has a left biadjoint S which is also a biequivalence". At the beginning of section 3 he defines a bicategory of spans from A to B in any bicategory, and given finite bilimits, essentially describes how to construct what one might call an "unbiased tricategory" of spans (of course, the definition of tricategory didn't exist at the time). He doesn't give any details of the proofs, but one could probably construct a detailed proof from these ideas without much more than tedium. I don't know whether anyone has written them out. Best, Mike On Tue, Aug 19, 2008 at 07:45:12AM -0700, John Baez wrote: > Dear Categorists - > > Given a category C with pullbacks we can define a bicategory Span(C) > where objects are objects of C, morphisms are spans - composed > using pullback - and 2-morphisms are maps between spans. > > Have people tried to categorify this yet? > > Suppose we have a 2-category C with pseudo-pullbacks. Then we should > be able to define a tricategory Span(C). Has someone done this? > > Or maybe people have gotten some partial results, e.g. in the case > where C = Cat. I'd like to know about these! > > Best, > jb > > > >