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:

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On 05/09/2010, at 5:41 AM, Aleks Kissinger wrote:

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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.

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