Extended Stone Duality via Monoidal Adjunctions
Fabian Lenke, Henning Urbat, Stefan Milius
TL;DR
The paper develops a general, compositional framework for extending Stone-type dualities via strong monoidal adjunctions, producing a duality between $U$-operators on a category $\mathbf{C}$ and $J_T$-operators on its dual $\widehat{\mathbf{C}}$, where $T=\widehat{F}\widehat{U}$. It applies this framework to derive two key concrete dualities: (i) a duality between derivation algebras (residuation algebras) and profinite ordered monoids, refined to RelDer/RelProfOrdMon via relational morphisms, and (ii) a discrete duality yielding a concrete description of the dual of the category of all small categories as residuation CABAs. It further extends Priestley duality to relational morphisms and relates these dualities to modal correspondence theory in a categorical setting, providing a versatile toolkit for automata theory and language derivatives. The work links algebraic language theory, topological dualities, and category theory, offering new perspectives on profinite structures, derivation algebras, and the categorical structure of small categories with potential applications to data languages and beyond.
Abstract
Extensions of Stone-type dualities have a long history in algebraic logic and have also been instrumental in 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, and providing the foundation for two new concrete dualities: First, we investigate residuation algebras, which are lattices with additional residual operators modeling language derivatives algebraically. We show that the subcategory of derivation algebras is 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. Second, we apply the categorical extended duality to the discrete setting of sets and complete atomic Boolean algebras to obtain a concrete description for the dual of the category of all small categories.