Re: Opposite of objects in a bicategory?

[Steve Lack <s.lack@uws.edu.au>]

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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