About a General Definition of Profunctor

Question

Given categories $\mathcal{A},\ \mathcal{B}$ let $\mathcal{A}^{<}:=Fun(\mathcal{A}, Set)$ the category of copresheaves on $\mathcal{A}$, let $\mathcal{A}^{>}:=(\mathcal{A}^{op})^{<}$ the category of presheaves on $\mathcal{A}$.

Let $coCont.Fun(\mathcal{A}, \mathcal{B})$ the category of colimit preserving functors (and natural transformations).

Given the categories $\mathcal{A},\ \mathcal{B}$, (small for simplicity) a profunctor $P: \mathcal{A}\dashrightarrow \mathcal{B}$ is (defined as) as an object of $(\mathcal{A}^{op}\times$$\mathcal{B})^{<}$.

We have the isomorphisms:

$(\mathcal{A}^{op}\times \mathcal{B})^<\cong (Fun({A}^{op}, \mathcal{B}^<)\cong (coCont.Fun({A}^<, \mathcal{B}^<)$

where the first isomorphism is the elementary trasposte, the second one is the left Kan extension by the yoneda contravariant $h^-: \mathcal{A}^{op}\to $$\mathcal{A}^{<}$. Then we can view the profunctor $P$ as a (cocontinuous) functor $\widetilde{P}: {A}^{<}\to \mathcal{B}^{<}$, and given another profunctor $Q: \mathcal{B}\dashrightarrow \mathcal{C}$ the composition $Q\otimes P$ (I use the left convenction) correspond to the functor composition $\widetilde{Q}\circ $$ \widetilde{P}$ . Quite similarly we can argue about enriched categories (on a fixed monoidal symmetric (closed) one).

Now what happen about internal categories as a topos $\mathcal{E}$?

Let $Cat(\mathcal{E})$ the (2-)category of internal categories of a topos $\mathcal{E}$.

For $A\in Cat(\mathcal{E})$ let $\mathcal{E}^A$ the category of internal copresheaves on $A$ (in the literature generally indicates the category of presheaves, but make more light our notations here).

An (internal) profunctor $P: A\dashrightarrow B$ is (defined as) an object $P$ of $\mathcal{E}^{A^{op}\times B}$, this is equivalent to $(\mathcal{E}^B)^{B^\ast(A^{op})}$ where $B^{\ast}: \mathcal{E}^B\to \mathcal{B}$ canonical, and this generalizes the first isomorphism above (related to small categories on $Set$ ) to the internal categories, but what about the second isomorphism? .

I wish to know: could the category $\mathcal{E}^{A^{op}\times B}$ be equivalent (in a natural way) to a category of functors (in some sense cocontinuous) of type: $P^*: \mathcal{E}^A\to \mathcal{E}^B$ ?

Answer 1

As Mike wrote in his comment, this statement is true with an appropriate notion of a category $\mathcal{E}^A$ and cocontinuity.

The category $\mathcal{E}^A$ from your definition is the underlying category of a locally internal category $\mathcal{E}^{\rightarrow^A}$. In more explicit terms, $\mathcal{E}^{\rightarrow}$ may be thought as of the codomain fibration over $\mathcal{E}$, $A$ as of the small fibration corresponding to the externalization of $\mathcal{E}$-internal category $A$, and $\mathcal{E}^{\rightarrow^A}$ as of the exponent $A \Rightarrow \mathcal{E}^\rightarrow$. Then your $\mathcal{E}^A$ is a mere fibre over terminal object in $\mathcal{E}^{\rightarrow^A}$.

Because for any internal category $A^{op}$, fibration $\mathcal{E}^{\rightarrow^{A}}$ is its internal free cocompletion, we get: $$\mathit{CoCont}(\mathcal{E}^{\rightarrow^{A}}, \mathcal{E}^{\rightarrow^{B}}) \approx \hom(A^{op}, \mathcal{E}^{\rightarrow^B})$$ The above statement may be found as a special case of Theorem 3.20 in Mark Weber's "Yoneda structures from 2-toposes".

Finally: $$\hom(A^{op}, \mathcal{E}^{\rightarrow^B}) \approx \hom(1, \mathcal{E}^{\rightarrow^{A^{op} \times B}}) \approx \mathcal{E}^{A^{op}\times B}$$ where the last equivalence is an instance of fibred Yoneda lemma.