### 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 200111201650.fAKGol115859@foobar.pps.jussieu.fr):

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

and

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”.

## 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.