Esakia duality

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 categories

• HA of Heyting algebras and Heyting algebra homomorphisms

and

• Esa of Esakia spaces and Esakia morphisms.

Theorem:[1][2][3] HA is dually equivalent to Esa.