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Feb 23, 2017 at 9:08 vote accept fosco
Jan 20, 2017 at 0:37 comment added fosco Yes, I meant a category enriched over a single $H$, sorry :-) thanks for the ref!
Jan 19, 2017 at 12:43 comment added Mike Shulman 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.
Jan 18, 2017 at 20:04 comment added fosco 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.
Jan 18, 2017 at 5:38 history answered Mike Shulman CC BY-SA 3.0