A duality involution is an abstract operation that generalizes the notion of opposite category. What this means precisely depends on the underlying abstract structure.
A duality involution on a 2-category is a 2-functor 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.
In the bicategory Prof of categories and profunctors, is the dual object of relative to a monoidal structure, making 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.
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 that are pseudonatural in . But perhaps in general some compatibility with the composition of profunctors should also be assumed?
Created on June 17, 2016 at 19:23:48. See the history of this page for a list of all contributions to it.