Localic Esakia Duality via Conic Frames

Abstract

Esakia duality is the dual equivalence between Heyting algebras and Esakia spaces. However, the traditional proof uses the Prime Ideal Theorem to recover the algebra from its spectrum, a choice principle that is not constructively valid. We build on Townsend's localic Priestley duality to describe a fully constructive, localic Esakia duality. On the algebraic side, we use the recently introduced Heyting frames as the point-free version of Heyting algebras. On the spatial side, Esakia spaces are modeled by two point-free alternatives. First, Esakia locales are defined as a subclass of the ordered Stone locales introduced by Townsend, and it is shown directly that Townsend's equivalence restricts to a duality between Heyting frames and Esakia locales. Second, using the theory of conic frames, in which join-preserving closure operators on frames model localic preorders, we introduce Esakia frames as a frame-theoretic analogue of Esakia spaces. It is shown that Esakia frames are equivalent to Heyting frames and dually equivalent to Esakia locales, and that this factorises the restriction of Townsend's equivalence. This yields a fully constructive, localic Esakia duality.