Re: Dual category and dual object

[Aleks Kissinger <aleks0@gmail.com>]

I originally thought it was Set, but this doesn't seem to make sense
using the cartesian product of categories as the monoidal product in
Prof. If this were the case, candidates for the unit and counit would
be the upper and lower stars of the hom functor.

However, let the 1-object category be the monoidal product. Then,
since 1 X C^op X C is isomorphic to C^op X C, we can regard the hom
functor of C as a profunctor 1 --|--> C^op X C,

Hom_C : 1 X C^op X C --> Set

and similarly, the hom functor of C^op as a profunctor C X C^op --|--> 1.

I saw this briefly mentioned here:
http://ncatlab.org/nlab/show/trace+of+a+category

Does anyone know if there is a more fleshed out treatment of this
somewhere? Also, do the "lifted" versions of the hom functor (upper
and lower star) serve some structural purpose in Prof?


Aleks


On Sat, Sep 4, 2010 at 8:50 PM, David Leduc <david.leduc6@googlemail.com> wrote:
> Could you please spell out what is the unit in Prof (It is not Set, is
> it?) and what are the units and counits for dual objects?
>
> Thanks for your help.
>
>
> On Sun, Sep 5, 2010, Ross Street <ross.street@mq.edu.au> wrote:
>> On 05/09/2010, at 5:41 AM, Aleks Kissinger wrote:
>>> In the (bi)category Prof of categories and profunctors, the dual of an
>>> object is the dual category. Profunctors most certainly came later
>>> than the notions of categorical dual and dual objects (or at least
>>> their concrete counterparts, dual spaces), so this might just be a
>>> happy coincidence.
>>
>> Very well put!
>> I might add that an extra point needed is that Prof is a monoidal bicategory
>> where the tensor product is the cartesian product of categories (it is not
>> the cartesian product in Prof). And yes, Prof is compact, autonomous,
>> rigid, whichever word you prefer, and the dual in Prof of a category A
>> is A^{op}. In reading the literature, note that other names for Prof are
>> Dist, Bimod and Mod.
>


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]