nLab duality involution

Idea

A duality involution is an abstract operation that generalizes the notion of opposite category. What this means precisely depends on the underlying abstract structure.

On 2-categories

A duality involution on a 2-category $K$ is a 2-functor $(-)^\circ : K^{co} \to K$ that is coherently (or strictly) self-inverse. A precise definition, and coherence theorem, can be found in Shulman 2016. This sort of duality involution is structure rather than a property, but it can be given a universal property relative to a 2-category with contravariance.

On bicategories

In the bicategory Prof of categories and profunctors, $A^{op}$ is the dual object of $A$ relative to a monoidal structure, making $Prof$ a compact closed bicategory. Thus, in any compact closed bicategory, the operation taking each object to its dual may be called a duality involution. Note that being compact closed is a property and not a structure on a bicategory.

On equipments

It should be possible to combine a duality involution on a 2-category and on a bicategory to obtain a notion of duality involution on a proarrow equipment, but it is not clear exactly what the coherence conditions should be. Weber 2007 extends a duality involution on a 2-category to its virtual equipment of discrete two-sided fibrations by asking for equivalences $DFib(A\times B,C) \simeq DFib(A,B^{\circ}\times C)$ that are pseudonatural in $A,B,C$. But perhaps in general some compatibility with the composition of profunctors should also be assumed?

References

• Mike Shulman, Contravariance through enrichment, arXiv, 2016

• Mark Weber, Yoneda structures from 2-toposes, Applied Categorical Structures, Vol. 15, p259-323, 2007, web