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Let $C, D $ be Cartesian closed categories, and suppose that $F : C\to D$ is a functor. What is a common name for the property of such a functor which is the conjunction of the following?

  1. $F$ preserves finite joins.
  2. There is a natural transformation $\eta_{AB} : F(A \Rightarrow B) \to F(A) \Rightarrow F(B)$. In other words, $F$ does not quite preserve exponentials, but there is some connection between an exponential and its image.

The property above corresponds to the notion of frame homomorphisms as opposed to that of complete Heyting homomorphisms in point-free topology, and I think that there are other situations that one does not need an actual preservation and can do with the weaker version above. What are these functors commonly called?

Clarification. The following is what I meant when I mentioned frame homomorphism: Frame homomorphisms have Condition 2. (but Condition 2 is not the defining property of such maps.) It is literally Condition 2 that I am interested in, rather than the straightforward categorical correspondent of the defining properties of frame homomorphisms.

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Permit me to use square brackets $[-,-]$ to denote the internal hom in a cartesian closed category.

A closed functor between cartesian closed categories $\mathcal{C}$ and $\mathcal{D}$ consists of:

  • a functor $F \colon \mathcal{C} \longrightarrow \mathcal{D}$
  • a natural transformation $\psi_{A,B} \colon F[A,B] \longrightarrow [FA,FB]$
  • a morphism $\psi_0 \colon 1 \longrightarrow F1$

subject to certain axioms.

Closed functors were originally defined by Eilenberg and Kelly between the more general class of closed categories. For a more modern and easily accessible reference, see the paper of Ross Street titled Skew-closed categories wherein closed functors are defined between the even more general class of skew-closed categories. (Note that these generalities do not complicate the definition of closed functor at all.)

Any finite product preserving functor $F \colon \mathcal{C} \longrightarrow \mathcal{D}$ between cartesian closed categories can be canonically equipped with the structure of a closed functor. A natural transformation $\psi$ as in the definition above may be found by applying $F$ to the ''evaluation'' morphism $$[A,B]\times A \longrightarrow B$$ thereby yielding a morphism $$F[A,B]\times FA \cong F([A,B]\times A) \longrightarrow FB$$ whose transpose under the product/hom adjunction is a morphism $$F[A,B] \longrightarrow [FA,FB]$$ which is indeed part of a closed structure on the functor $F$.

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