From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4501 Path: news.gmane.org!not-for-mail From: Tom Fiore Newsgroups: gmane.science.mathematics.categories Subject: Re: biadjoint biequivalences and spans in 2-categories Date: Wed, 20 Aug 2008 04:04:12 -0500 (CDT) Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed X-Trace: ger.gmane.org 1241019985 13594 80.91.229.2 (29 Apr 2009 15:46:25 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:46:25 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Wed Aug 20 08:23:13 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 20 Aug 2008 08:23:13 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KVlmL-0001VQ-N2 for categories-list@mta.ca; Wed, 20 Aug 2008 08:23:05 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 36 Original-Lines: 71 Xref: news.gmane.org gmane.science.mathematics.categories:4501 Archived-At: Hello, Details of a part of John and Mike's post can be found in one of my publications. Theorem. 9.17 Let X and A be strict 2-categories, and G:A -> X a pseudo functor. There exists a left biadjoint for G if and only if for every object x of X there exists an object r of A and a biuniversal arrow x -> Gr from x to G. The proof is as one could expect. http://arxiv.org/abs/math.CT/0408298 Fiore, Thomas M. Pseudo limits, biadjoints, and pseudo algebras: categorical foundations of conformal field theory. Mem. Amer. Math. Soc. 182 (2006), no. 860, x+171 pp. Best greetings, Tom On Tue, 19 Aug 2008, Michael Shulman wrote: > 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 >> >> >> >> > >