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Adjoints between posets are (monotone or antitone) Galois connections; monads correspond to closure operators; what is a two-variable adjunction in this low-categorical setting?

I'm able to write the bare definition, of course. What I seek is intuition, and possibly an instance of this structure, on a triple of poset $P,Q,R$, possibly under a different name.

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One important special case of a 2-variable adjunction is a biclosed monoidal structure on a category, such as a cartesian closed category. A cartesian closed poset (with finite joins as well) is called a Heyting algebra, and corresponds to intuitionistic logic in the same way that Boolean algebras correspond to classical logic. In particular, every Boolean algebra is a Heyting algebra and hence is equipped with a biclosed monoidal structure.

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  • $\begingroup$ This is sort of the paradigmatic example, and I have clear in mind why, but then I think the next step would be to say what a $\bf Heyt$-enriched category, which is tensored and cotensored, is called outside category theory. $\endgroup$
    – fosco
    Commented Jan 18, 2017 at 20:04
  • $\begingroup$ I presume you mean a category enriched over a particular Heyting algebra, rather than over the category of Heyting algebras (which is what "Heyt-enriched category" would normally mean). You might be interested in Joyal-Tierney On an extension of the Galois theory of Grothendieck where they study modules over monoids in suplattices as an analogue of rings; I don't remember exactly what they correspond to in the localic world in that generality, but I think there is something. $\endgroup$
    – Mike Shulman
    Commented Jan 19, 2017 at 12:43
  • $\begingroup$ Yes, I meant a category enriched over a single $H$, sorry :-) thanks for the ref! $\endgroup$
    – fosco
    Commented Jan 20, 2017 at 0:37

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