# About a General Definition of Profunctor

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 ? - I think the answer is yes, but you need to consider \mathcal{E}^A and \mathcal{E}^B as \mathcal{E}-indexed categories, with the appropriate notion of cocontinuity. I don't recall a reference off the top of my head, though. – Mike Shulman Jan 3 '12 at 18:14 ## 1 Answer 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.

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Thank you. Searching about this question I just find this article, but I have to work more about it.. –  Buschi Sergio Apr 29 '13 at 11:12