The $\mathfrak L$ functor on $\textsf{Prof}$

Question

$\def\L{\mathfrak{L}}\def\Prof{\mathsf{Prof}}$ Recall that Isbell duality $\text{Spec}\dashv {\cal O} : {\cal V}^{A^°} \leftrightarrows \big({\cal V}^A\big)^°$ allows us to define the functor $$ \L : \Prof(A,B) \to \Prof(B,A) $$ ($\cal V$ is a cosmos in which $A,B$ are enriched categories) sending $K : A^° \times B \to \cal V$ into $\L(K) : (b,a)\mapsto {\cal V}^{A^°}(K(-,b), \hom(-,a)) = {\cal O}(K_b)_a$.

Is it true that $\L(H \circ K) \cong \L(K)\circ \L(H)$, maybe under some additional assumptions on $H,K$? Or maybe in general, but that's not a formal proof.

Is there any relation between $\text{Ran}_FK$, as a profunctor $C^° \times B\to \cal V$ and the composition $K\circ \L(F)$?

How does this construction interact with the "tautological dualiser" of $\Prof$ that sends $A \mapsto A^°$?

Is there any reference that encompasses all, or almost all, these questions?

Answer 1

This is the right Kan extension of $\hom_A: A \nrightarrow A$ along $K: A \nrightarrow B$ in the bicategory of profunctors. Which is to say that for every profunctor (aka bimodule) $L: B \nrightarrow A$ there is a natural bijection between morphisms $LK \to \hom_A$ and morphisms $L \to \mathcal{L}(K)$. Since $\hom_A$ is a unit $1_A$ in this bicategory, it may be more suggestive to write $\mathcal{L}(K) = Ran_K 1_A$.

Thus for $H: A \nrightarrow B$ and $K: B \nrightarrow C$, we have a canonical map $\mathcal{L}(K)K \to 1_B$, and whiskering on the right by $H$ and on the left by $\mathcal{L}(H)$, we get a composite

$$\mathcal{L}(H)\mathcal{L}(K)KH \to \mathcal{L}(H)1_B H \cong \mathcal{L}(H)H \to 1_A.$$

By the universal property of $\mathcal{L}(KH)$, we now obtain a canonical map $\mathcal{L}(H)\mathcal{L}(K) \to \mathcal{L}(KH)$.

But this map is typically not an isomorphism. The simplest type of example that comes to mind is just the classical case of bimodules over rings, where for a right-$B$ left-$C$ bimodule $K$, we have $\mathcal{L}(K) = \hom_A(K, B)$ regarded as a right-$C$ left-$B$ bimodule. We have in this situation a canonical map

$$\hom_A(H, A) \otimes_B \hom_B(K, B) \to \hom_A(K \otimes_B H, A)$$

but normally this won't be an isomorphism. Indeed, even in the humble case of vector spaces over a ground field $k$, the canonical map $V^\ast \otimes_k W^\ast \to (V \otimes W)^\ast$ isn't generally an isomorphism. Of course we do get an isomorphism here in some special cases, such as if $K$ is finitely generated projective over $B$. More abstractly, this is the situation where $K$ has a left adjoint bimodule, and here we may recall that in a 2-category or bicategory, if an arrow $K: B \nrightarrow C$ has a left adjoint $L$, then it is necessarily $L = Ran_K 1_B$, i.e., $L = \mathcal{L}(K)$ in our situation. In that case, the asserted inverse $\mathcal{L}(KH) \to \mathcal{L}(H)\mathcal{L}(K)$ of the canonical map is mated (by the adjunction $\mathcal{L}(K) \dashv K$) to an arrow $\mathcal{L}(KH)K \to \mathcal{L}(H)$, which in turn is mated to the canonical arrow $\mathcal{L}(KH)KH \to 1_A$ using the definition of right Kan extension.

[For the bicategory of profunctors or bimodules, such right adjoints $K$ are induced by functors $F: C \to \bar{B}$ where $\bar{B}$ is the Cauchy completion of $B$ (which we may think of as analogous to the category of finitely generated projective $B$-modules); specifically, $K(b, c) = \bar{B}(b, Fc)$ and its left adjoint is given by $\mathcal{L}(K)(c, b) = \bar{B}(Fc, b)$.]

By similar reasoning as above, one may calculate that if $K: B \nrightarrow C$ has a left adjoint $L$ (in a bicategory $\mathbf{B}$), then for every $F: B \nrightarrow A$ we have an isomorphism $Ran_F(K) \cong K \circ Ran_F(1_B)$, assuming these right Kan extensions exist. This is purely formal of course: for every $H: A \nrightarrow C$ we have isomorphisms

$$\mathbf{B}(A, C)(H, Ran_F(K)) \cong \mathbf{B}(B, C)(HF, K) \cong \mathbf{B}(B, B)(LHF, 1_B) \cong \mathbf{B}(A, B)(LH, Ran_F(1_B)) \cong \mathbf{B}(A, C)(H, K Ran_F(1_B)).$$

I'm not sure what could be said about interaction with the "tautological dualizer" (where now are viewing $\textbf{Prof}$ as a compact closed bicategory): essentially all of the above has to do with the bicategory structure, not the monoidal bicategory structure.