From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3840 Path: news.gmane.org!not-for-mail From: John Baez Newsgroups: gmane.science.mathematics.categories Subject: pivotal adjoints? Date: Wed, 25 Jul 2007 04:45:16 -0700 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii X-Trace: ger.gmane.org 1241019555 10535 80.91.229.2 (29 Apr 2009 15:39:15 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:39:15 +0000 (UTC) To: categories Original-X-From: rrosebru@mta.ca Wed Jul 25 09:38:28 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 25 Jul 2007 09:38:28 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IDg42-0004jW-Ra for categories-list@mta.ca; Wed, 25 Jul 2007 09:34:02 -0300 Content-Disposition: inline Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 34 Original-Lines: 65 Xref: news.gmane.org gmane.science.mathematics.categories:3840 Archived-At: Aaron Lauda writes: >Suppose we have chosen left and right adjoints for F:A->B and G:A->B > >Then given any 2-morphism a:F=>G there are two obvious duals (mates >under adjunction) for the 2-morphism a: > > a+ :G*=>F* := (e_G F*).(G*aF*).(G* i_F) > +a :G*=>F* := (F* k_G).(F*aG*).(j_F G*) > >or for those who like pictures: > > +a a+ > __ __ > / \ | | / \ > | | | | | | > | a | | a | > | | | | | | > | \__/ \__/ | > | | > > In general a+ is not equal to +a because if is was we could always twist one > of the units and counits so that it does not hold. Has the condition > that a+ = +a been investigated in the literature anywhere? In > particular, if a 2-category is such that all 1-morphisms F have a > simultaneous left and right adjoint then has anyone studied the > context where the adjoints are such that a+ = +a is always > satisfied? Perhaps, this notion has been studied in the language of > duals for 1-morphisms? I'd be curious to know what if any replies you received. As you already hinted, the special case of a monoidal category with this property has been studied: it's called "pivotal". Strict pivotal categories were studied here: P.J. Freyd and D.N. Yetter, Braided compact closed categories with applications to low dimensional topology, Adv. Math. 77 (1989), 156--182 and there's more discussion here: John W. Barrett and Bruce W. Westbury, Spherical Categories, Adv. Math. 143 (1999) 357-375. http://arxiv.org/abs/hep-th/9310164 I don't know who has studied more general (strict or weak) 2-categories with this pivotal property, though it's a natural generalization. Street should have bumped into it in his work on 2-categorical string diagrams. I've written about "2-categories with duals" in my work on the Tangle Hypothesis. These are pivotal, but they also have more structure, which you may not want. (You may want it if you're studying things like tangles!) 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? Best, jb