pivotal adjoints?

[John Baez <baez@math.ucr.edu>]

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