From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3842 Path: news.gmane.org!not-for-mail From: Aaron Lauda Newsgroups: gmane.science.mathematics.categories Subject: Re: pivotal adjoints? Date: Wed, 25 Jul 2007 09:27:11 -0400 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain;charset=ISO-8859-1;DelSp="Yes";format="flowed" Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1241019556 10540 80.91.229.2 (29 Apr 2009 15:39:16 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:39:16 +0000 (UTC) To: categories Original-X-From: rrosebru@mta.ca Wed Jul 25 14:22:23 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 25 Jul 2007 14:22:23 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IDkSx-0003PI-Hr for categories-list@mta.ca; Wed, 25 Jul 2007 14:16:03 -0300 Content-Disposition: inline Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 36 Original-Lines: 36 Xref: news.gmane.org gmane.science.mathematics.categories:3842 Archived-At: I would like to thank all those that I replied so far. Quoting John Baez : > Perhaps it would be good to pose a specific question. What would > you like to know about pivotal 2-categories? Or are you mainly > just looking for references? To answer John, I would like to know what condition is required on =20 left and right adjoints in a 2-category K to ensure that a string =20 diagram representing a 2-morphism in K is invariant under topological =20 deformation restricting to the identity on the boundary. I prefer not =20 to use monoidal 2-categories, just ordinary 2-categories/bicategories. If I take a monoidal bicategory with duals and forget the monoidal =20 structure will this be what I am after? 2-tangles clearly have the =20 property I am looking for, but what if we adjoin some new 2-morphism A =20 to 2-tangles. What condition would I need in order to ensure that any =20 string diagram with the new morphism A was invariant under topological =20 deformation? > I'd be curious to know what if any replies you received. Aside from the replies that have been posted, I have also received a =20 pointer to the paper "Introduction to linear bicategories" by Cockett, =20 Koslowski, and Seely. The condition that *a=3Da* is studied in the =20 context of linear bicategories and what are called cyclic adjoints. =20 In particular, the discussion of cyclic mates seems to especially =20 relevant. But I have not finished reading the paper and am still =20 trying to understand what implications the `linear' in linear =20 bicatgories will have on the ordinary bicategory case. Regards, Aaron