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:

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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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