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