Bifunctors, bimodules, and profunctors

Preliminary draft.

Let $\calV$ be a symmetric monoidal category, and $\calA$, $\calB$, and $\calC$ be categories enriched in $\calV\,$.
We consider bifunctors (functors of two variables) which are (contravariant on the left) and (covariant on the right):
$$\begin{array}{cccl} \calA\op \tensor \calB & \xrightarrow{\textstyle \boxed\functorF} & \calC & & \text{the bifunctor} \\ \objectA,\objectB & \mapsto & \boxed{\hom \objectA \functorF \objectB} & & \text{the bifunctor's action on objects} \\ \hom {\objectA,\objectB} {(\calA\op \tensor \calB)} {\objectAp,\objectBp} \xlongequal[\textstyle \text{defn.} \tensor]{} \hom \objectA {\calA\op} \objectAp \tensor \hom \objectB \calB \objectBp \xlongequal[\textstyle \text{defn.} \op]{} \hom \objectAp \calA \objectA \tensor \hom \objectB \calB \objectBp & \xrightarrow[\textstyle \boxed{\hom {\objectA,\objectB} \functorF {\objectAp,\objectBp}}]{} & \hom {\hom \objectA\functorF\objectB} \calC {\hom \objectAp\functorF\objectBp} & & \text{the bifunctor's action on hom-objects} \\ \end{array}$$ For (the general bifunctor), just (drop the $\op$ from $\calA\op$).

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The next display is a schematic representation of the last line of the above display,
showing the components of (the bifunctor's action $\boxed{\hom {\objectA,\objectB} \functorF {\objectAp,\objectBp}}$ on hom-objects).
$$\begin{array} {} \objectA & \xrightarrow{\textstyle \hom \objectA \functorF \objectB \in \calC} & \objectB \\ \llap{ \calV \ni \hom \objectA {\calA\op} \objectAp = \hom \objectAp \calA \objectA } \Bigg\uparrow & \llap{\hom {\hom \objectA\functorF\objectB} \calC {\hom \objectAp\functorF\objectBp}} \Bigg\downarrow \rlap{{} \in \calV} & \Bigg\downarrow \rlap{\hom \objectB \calB \objectBp \in \calV} \\ \objectAp & \xrightarrow[\textstyle \hom \objectAp \functorF \objectBp \in \calC]{\mkern10em} & \objectBp \\ \end{array}$$ Note how this display "spreads out", literally diagrams, those components of the one-line notation,
while also showing explicitly some (otherwise implicit) typing information.
Not explicitly shown are (the tensor product, $\tensor$), and (the arrow $\hom {\objectA,\objectB} \functorF {\objectAp,\objectBp}$ itself).
The tensor product is of (what the two outer vertical arrows represent),
while the arrow $\hom {\objectA,\objectB} \functorF {\objectAp,\objectBp}$ itself is from (that tensor product) to (what is represented by the inner vertical arrow).

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We now consider several important special cases of such contra-co bifunctors.

First, consider the case where (the target category $\calC$) is (the base for enrichment, $\calV$), so the bifunctor is $\boxed{\functorF : \calA\op \tensor \calB \longrightarrow \calV}$.
This particular form of bifunctor, contravariant on the left, covariant on the right, and taking values in $\calV$,
may be (and usually is) called a $\calV$-valued bimodule.

Second, we specialize even more to the case where $\calV$ is not merely symmetric monoidal, but symmetric monoidal closed.
This is actually the case which the nLab article considers.

Third are the cases where $\calV$ is cartesian closed.
We display below the explicit data for $\functorF$ in this situation,
replacing (the tensor product) with (the cartesian product)
and (the $\hom \setX \calV \setY$ notation for $\calV$'s hom) with (the $[\objectX,\objectY]$ notation (often written $\objectY^\objectX$) often used for internal homs). $$\begin{array}{cccl} \calA\op \times \calB & \xrightarrow{\textstyle \boxed\functorF} & \calV & & \text{the bifunctor} \\ \objectA,\objectB & \mapsto & \boxed{\hom \objectA \functorF \objectB} & & \text{the bifunctor's action on objects} \\ \hom {\objectA,\objectB} {(\calA\op \times \calB)} {\objectAp,\objectBp} \xlongequal[\textstyle \text{defn.} \times]{} \hom \objectA {\calA\op} \objectAp \times \hom \objectB \calB \objectBp \xlongequal[\textstyle \text{defn.} \op]{} \hom \objectAp \calA \objectA \times \hom \objectB \calB \objectBp & \xrightarrow[\textstyle \boxed{\hom {\objectA,\objectB} \functorF {\objectAp,\objectBp}}]{} & [\hom \objectA\functorF\objectB , \hom \objectAp\functorF\objectBp] & & \text{the bifunctor's action on hom-objects} \\ \end{array}$$ Then the schematic representation becomes: $$\begin{array} {} \objectA & \xrightarrow{\textstyle \hom \objectA \functorF \objectB \in \calV} & \objectB \\ \llap{ \calV \ni \hom \objectA {\calA\op} \objectAp = \hom \objectAp \calA \objectA } \Bigg\uparrow & \llap{ [\hom \objectA\functorF\objectB , \hom \objectAp\functorF\objectBp] } \Bigg\downarrow \rlap{{} \in \calV} & \Bigg\downarrow \rlap{\hom \objectB \calB \objectBp \in \calV} \\ \objectAp & \xrightarrow[\textstyle \hom \objectAp \functorF \objectBp \in \calV]{\mkern12em} & \objectBp \\ \end{array}$$

In this ($\calV$ cartesian-closed case) we can take (the exponential transpose of $\hom {\objectA,\objectB} \functorF {\objectAp,\objectBp}$), to give (an equivalent form of it): $$\boxed{ \hom \objectAp \calA \objectA \times \hom \objectA\functorF\objectB \times \hom \objectB \calB \objectBp \xrightarrow[\textstyle \boxed{\hom {\objectA,\objectB} \functorF {\objectAp,\objectBp}}]{} \hom \objectAp\functorF\objectBp } \; ,$$ (the customary form) for (the two-sided action map of a bimodule).
Note how we use the same notation, $\hom {\objectA,\objectB} \functorF {\objectAp,\objectBp}\;$, for two different arrows in $\calV$,
which correspond under $\calV\mkern2mu$'s closedness adjunction.

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Work in progress:

Monoidal, cartesian, and closed categories

	closed?
cartesian?	monoidal $\; \tensor$ $\hom R \Mod R$ for $R$ a noncommutative ring	monoidal closed $\; \tensor, \; [{-},{-}]$ $\AbGrp$
	cartesian $\; \times$ $\Top$	cartesian closed $\; \times, \; [{-},{-}]$ $\Set$

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References

[CKVW] Carboni, A.; Kelly, G. M.; Verity, D.; Wood, R. J. (1998). "A 2-Categorical Approach To Change Of Base And Geometric Morphisms II". Theory and Applications of Categories. 4 (5): 82–136.
This reference defines and extensively discusses a variation on the above definition,
namely, the case where the target category is $\CAT$, considered as a bicategory, and the actions are not necessarily strict, but are allowed to be pseudo.
Numerous examples of such structures are given, and the general theory is extensively studied.

[CIC] Street, Ross (1980). "Cosmoi of internal categories". Trans. Amer. Math. Soc. 258 (2): 278–318. doi:10.1090/S0002-9947-1980-0558176-3. MR 0558176.
Section 1 of this gives the appropriate form of the Grothendieck construction when $\calC=\calV=\Cat$.
This is a 2-functor $\boxed{ \calG : [\calA\op \times \calB, \Cat] \to \Cat\downarrow (\calB\times\calA) }$ which has a left adjoint $\boxed\calM$ --
see (1.5)-(1.7) of Street's paper for this, and the rest of its Section 1 for the relation of this adjunction to split fibrations.