Re: pivotal adjoints?

Re: pivotal adjoints?

[Aaron Lauda]

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

pivotal adjoints?

[John Baez]

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

pivotal adjoints?

[Aaron Lauda]

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