# 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.

## References

- Esakia, Leo (1974). "Topological kripke models".
*Soviet Math*.**15**(1): 147–151. - Esakia, L (1985). "Heyting Algebras I. Duality Theory".
*Metsniereba, Tbilisi*. - Bezhanishvili, N. (2006).
*Lattices of intermediate and cylindric modal logics*(PDF). Amsterdam Institute for Logic, Language and Computation (ILLC). ISBN 978-90-5776-147-8.