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Let $Prof$ the bicategory with profunctors (on small categories), arrows are like $D: \mathscr{A} \dashrightarrow \mathscr{B}$ and this means that $D: \mathscr{A}^{op} \times \mathscr{B}\to Set$.

Its well known that there are (faithful and locally full and faithful) immersions $(-)_\bullet: Cat \to Prof ,\ (-)^\bullet: Cat^{op} \to Prof $ (where $Cat$ is the usual 2-category, and $Cat^{op}$ dualizes (reverses) arrows only).

We can extend the (cartesian) monoidal products:

defining for $ \mathscr{A} \dashrightarrow B$ and $E: \mathscr{C} \dashrightarrow \mathscr{D}$ the profunctor $D \times E: \mathscr{A} \times \mathscr{C} \dashrightarrow \mathscr{B} \times \mathscr{D}$ as the composition $(\mathscr{A} \times \mathscr{C})^{op} \times (\mathscr{B} \times \mathscr{D}) \cong (\mathscr{A} \times \mathscr{B}^{op} )\times (\mathscr{C} \times \mathscr{D}^{op}) \xrightarrow{D\times E } Set \times Set \xrightarrow{\times } Set$

If I'm not wrong the immersions $(-)_\bullet,\ (-)^\bullet$ come out strict monoidal.

I ask if $Prof$ has also a (monoidal) closed structure (I tried in vain to get it..)

Edit: changed $\otimes$ in $\times$ (At first I thought the problem into enriched context).

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  • $\begingroup$ To clearify: Objects in Prof are profunctors? What are the morphisms you consider? There are several possibilities. Furthermore: I suggest you use another notation for the product of profunctors as '$\otimes$' is usually used for the composition of profunctors. I'd suggest '$\boxtimes$'. It is usually used for 'outer prodcts' like the one describe. $\endgroup$ Aug 2, 2015 at 8:57
  • $\begingroup$ Also: What is the functor $\otimes:Set\times Set \to Set$? The cartesian product? What is the functor $\otimes:Cat\times Cat\to Cat$ you use? $\endgroup$ Aug 2, 2015 at 9:06
  • $\begingroup$ objects are small categories, profunctors are arrow, the symbol $\otimes$ is the cartesian prodoct of categories (monoidal product in the cartesian monoidal structure). $\endgroup$ Aug 2, 2015 at 10:03

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If by monoidal closed structure you mean the appropriate 2-dimensional analogue of the familiar 1-dimensional concept, then unless I misunderstand your question, the answer is that the bicategory $Prof$ is in fact compact closed: the dual of a small category $A$ in the sense of compact closure is the ordinary dual $A^{op}$, meaning that there is a 2-natural equivalence $Prof(A \times B, C) \simeq Prof(B, A^{op} \times C)$. This is essentially obvious since both sides are equivalent to the category of functors $A^{op} \times B^{op} \times C \to Set$ in a 2-natural way.

This generalizes the familiar fact that the bicategory $Rel$ (of sets, relations, and inclusions between relations) is compact closed, where the dual of a set = discrete small category $A$ is itself. More examples and discussion of this type can be found in this paper by Mike Stay.

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