Extended Stone Duality via Monoidal Adjunctions (2401.08219v3)

Abstract: Extensions of Stone-type dualities have a long history in algebraic logic and have also been instrumental for proving results in algebraic language theory. We show how to extend abstract categorical dualities via monoidal adjunctions, subsuming various incarnations of classical extended Stone and Priestley duality as special cases. Guided by these categorical foundations, we investigate residuation algebras, which are algebraic models of language derivatives, and show the subcategory of derivation algebras to be dually equivalent to the category of profinite ordered monoids, restricting to a duality between boolean residuation algebras and profinite monoids. We further refine this duality to capture relational morphisms of profinite ordered monoids, which dualize to natural morphisms of residuation algebras. As a special case of the discrete version of the duality between monoids and complete residuation algebras, we obtain a novel concrete description for the dual of the category of all small categories and functors.