pivotal adjoints?

19 Jul 2007

      Dear category theorists,

Suppose we have chosen left and right adjoints for F:A->B and G:A->B

   F-| F* -| F     and     G-| G*-|G

  i_F: 1_B => FF*       i_G: 1_B => GG*
  e_F: F*F => 1_A       e_G: G*G => 1_A
  j_F: 1_A => F*F       j_G: 1_A => G*G
  k_F: FF* => 1_B       k_G: GG* => 1_B

Then given any 2-morphism a:F=>G there are two obvious duals (mates  
under adjunction) for the 2-morphsism 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?

The above condition appears to be related to the notion of pivotal  
category when we look at Hom(A,A) for any object A.

Thanks,
Aaron Lauda