Stone-type duality for varieties

Summary

A Stone-type duality for a variety or quasivariety of algebras is a dual equivalence of categories. It matches the algebras with Stone spaces that carry additional structure. Under it, homomorphisms of algebras correspond to continuous structure-preserving maps in the opposite direction. The prototype is Stone duality between Boolean algebras and Stone spaces; Priestley duality and Esakia duality for Heyting algebras are further examples. Such dualities turn algebraic questions into topological ones, and the theory of natural dualities gives a uniform method to build many of them.