In mathematics, Esakia duality is the dual equivalence between the category of Heyting algebras and the category of Esakia spaces. Esakia duality provides an order-topological representation of Heyting algebras via Esakia spaces.
Let Esa denote the category of Esakia spaces and Esakia morphisms.
Let H be a Heyting algebra, X denote the set of prime filters of H, and ≤ denote set-theoretic inclusion on the prime filters of H. Also, for each a∈ H, let φ(a) = {x∈ X : a∈ x} , and let τ denote the topology on X generated by {φ(a), X − φ(a) : a∈ H}.
Theorem[1]: (X,τ,≤) is an Esakia space, called the Esakia dual of H. Moreover, φ is a Heyting algebra isomorphism from H onto the Heyting algebra of all clopen up-sets of (X,τ,≤). Furthermore, each Esakia space is isomorphic in Esa to the Esakia dual of some Heyting algebra.
This representation of Heyting algebras by means of Esakia spaces is functorial and yields a dual equivalence between the category HA of Heyting algebras and Heyting algebra homomorphisms and the category Esa of Esakia spaces and Esakia morphisms.
Theorem[2]: HA is dually equivalent to Esa.