Canonical extensions, Esakia spaces, and universal models

In this paper we survey some recent developments in duality for lattices with additional operations paying special attention to Heyting algebras and the connections to Esakia’s work in this area. In the process we analyse the Heyting implication in the setting of canonical extensions both as a property of the lattice and as an additional operation. We describe Stone duality as derived from canonical extension and derive Priestley and Esakia duality from Stone duality for maps. In preparation for this we show that the categories of Heyting and modal algebras are both equivalent to certain categories of maps between distributive lattices and Boolean algebras. Finally we relate the N-universal model of intuitionistic logic to the Esakia space of the corresponding Heyting algebra via bicompletion of quasi-uniform spaces.