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
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
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
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Aaron Lauda -
John Baez