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Suraaj K S (Jun 18 2024 at 16:29):We know that a Heyting algebra satisfies the distributive law. I think this is simply an instance of distributivity in a CCC with coproducts.
Moreover, a complete lattice is a Heyting algebra if it satisfies the infinite distributive law.
I was wondering if this was an instance of some other abstract cateogory theoretic notion?
Graham Manuell (Jun 18 2024 at 16:46):The first is the fact that left adjoints preserve colimits. You can think of the second as a poset version of the adjoint functor theorem: a monotone map that preserves all joins has a right adjoint.
Todd Trimble (Jun 18 2024 at 17:03):Moreover, a complete lattice is a Heyting algebra if it satisfies the infinite distributive law.
A complete lattice in which finite meets distribute over arbitrary sups is also known as a [[frame]].
This isn't quite what you were asking, but there is a strong pull to saying that the notion [[Grothendieck topos]] categorifies the notion of frame or of its dual notion, [[locale]]. The sharpest expression of the analogy is probably Street's characterization. A frame is a poset such that its -enriched Yoneda embedding has a left exact (here, finite meet-preserving) left adjoint. Likewise, a Grothendieck topos is almost the same thing as a category such that its -enriched Yoneda embedding has a left exact (here, finite limit-preserving) left adjoint. To remove the "almost" you just need a mild size condition on the class of morphisms, working here under the assumption of one universe (i.e. a set vs. small set distinction), and assuming is locally small so that we can speak of its Yoneda embedding.
See also Giraud's characterization, which is closely related.