Skip to the Main Content

Note:These pages make extensive use of the latest XHTML and CSS Standards. They ought to look great in any standards-compliant modern browser. Unfortunately, they will probably look horrible in older browsers, like Netscape 4.x and IE 4.x. Moreover, many posts use MathML, which is, currently only supported in Mozilla. My best suggestion (and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source Mozilla browser.

February 8, 2006

Special Ambidextrous Adjunctions

Posted by Urs Schreiber

Here is a question about adjunctions.

While making a web search I came across an old message by Paul Levy to the category theory mailing list. He had originally asked (in message

If P and Q are objects in a 2-category C, and there is an equivalence between them, must there be an adjoint equivalence (an adjunction whose unit and counit are both isomorphisms) between them?

After the answer was given he revealed the motivation for his question:

I’m trying to make an argument that the natural 2-categorical analogue of isomorphism is adjoint equivalence rather than equivalence, but your result suggests that it doesn ‘t matter.

I am currently wondering about a closely related observation. While playing around with the notion of 2-transport, I noticed that, contrary to my original assumption, in order for a certain 2-functor to be expressible in terms of another 2-functor (to be “locally trivializable” in my application) it suffices for both 2-functors to be related by a “special ambidextrous adjunction”.

By a special ambidextrous adjunction I mean an ambidextrous adjunction


such that




are identity 2-morphisms.

This is strictly weaker than an adjoint equivalence.

(I have chosen the adjective “special” in order to allude to the fact that the Frobenius algebras obtained from these adjunctions are called “special Frobenius algebras”.)

I would like to know if there are any well known insights concerning such “special ambidextrous adjunctions”.

Posted at February 8, 2006 3:20 PM UTC

TrackBack URL for this Entry:

1 Comment & 3 Trackbacks

Re: Special Ambidextrous Adjunctions

I don’t know anything highly memorable about these “special” ambidextrous adjunctions, but I have done a lot of string diagram calculations with special Frobenius algebras here:

Quantum Gravity Seminar - Fall 2004,
week 7

and neighboring weeks.

Right now James Dolan and I are doing wonderful things with ambidextrous adjunctions between 2-vector spaces - they are related to Jones’ work on subfactors, quivers and the McKay correspondence. We are specially interested in ambidextrous adjunctions with fixed “bubble value” - meaning that the two closed loops you can draw (in string diagram notation) equal the same fixed number b. These include your “special” ambidextrous adjunctions when b = 1. But, I’m not sure the things we’re discovering are what you’re interested in.

Posted by: John Baez on June 29, 2006 7:17 PM | Permalink | Reply to this
Read the post Local Transition of Transport, Anafunctors and Descent of n-Functors
Weblog: The n-Category Café
Excerpt: Conceps and examples of what would be called transition data or descent data for n-functors.
Tracked: December 8, 2006 7:29 AM
Read the post Towards the FFRS Description of 2dCFT (B)
Weblog: The n-Category Café
Excerpt: "2-processes": 2-morphisms, adjunctions, the exchange law and the Frobenius property
Tracked: February 1, 2007 6:27 PM
Read the post Some Notes on Local QFT
Weblog: The n-Category Café
Excerpt: Some aspects of the AQFT description of 2d CFT.
Tracked: April 1, 2007 5:34 AM

Post a New Comment