Dear David, There is a way, but more structure, and possibly a change of perspective is required. I don't know how to describe opposites in terms of the category Cat. But if you work instead with the monoidal bicategory Prof/Mod/Dist (all three names are used) you can. I'll write A* for the opposite of a category A. I'll write Prof for the bicategory whose objects are (small) categories, and whose morphisms from A to B are functors A-->[B*,Set]. These are called profunctors (or modules or distributors) from A to B. They are composed using colimits; one easy way to see the composition is that up to equivalence, we can regard such profunctors as being cocontinuous functors [A*,Set]-->[B*,Set] and from this latter point of view we simply use ordinary composition of (cocontinuous) functors. Prof is monoidal, via the cartesian product of categories AxB. Note that this is not a cartesian monoidal structure on Prof, although it does have some features of cartesianness; it is an example of what is called a cartesian bicategory (studied by Carboni, Walters, Wood, and others). Anyway, in Prof, the opposite of a category B is dual to B, in the sense of monoidal (bi)categories, since functors AxB-->[C*,Set] correspond to functors A-->[BxC*,Set], and so to functors A-->[(CxB*)*,Set]. Thus in Prof, morphisms AxB-->C correspond to morphisms A-->CxB*. Steve Lack. On 7/03/10 11:28 PM, "David Leduc" <david.leduc6@googlemail.com> wrote:
Dear all,
The same way monads on categories can be generalized to monads on objects of a bicategory, is there a way to generalize opposites of categories to opposites of objects in a bicategory?
David
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