Re: pivotal adjoints?

[Aaron Lauda <lauda@math.columbia.edu>]

I would like to thank all those that I replied so far.

Quoting John Baez <baez@math.ucr.edu>:

> 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?

To answer John, I would like to know what condition is required on  
left and right adjoints in a 2-category K to ensure that a string  
diagram representing a 2-morphism in K is invariant under topological  
deformation restricting to the identity on the boundary.  I prefer not  
to use monoidal 2-categories, just ordinary 2-categories/bicategories.

If I take a monoidal bicategory with duals and forget the monoidal  
structure will this be what I am after?  2-tangles clearly have the  
property I am looking for, but what if we adjoin some new 2-morphism A  
to 2-tangles.  What condition would I need in order to ensure that any  
string diagram with the new morphism A was invariant under topological  
deformation?

> I'd be curious to know what if any replies you received.

Aside from the replies that have been posted, I have also received a  
pointer to the paper "Introduction to linear bicategories" by Cockett,  
Koslowski, and Seely.  The condition that *a=a* is studied in the  
context of linear bicategories and what are called cyclic adjoints.   
In particular, the discussion of cyclic mates seems to especially  
relevant.  But I have not finished reading the paper and am still  
trying to understand what implications the `linear' in linear  
bicatgories will have on the ordinary bicategory case.

Regards,
Aaron