In recent years Esakia duality for Heyting algebras has been extended in two directions. First to weak Heyting algebras, namely distributive lattices with an implication with weaker properties than that of the implication of a Heyting algebra, and secondly to implicative semilattices. The first algebras correspond to subintuitionistic logics, the second ones to the conjunction and implication fragment of intuitionistic logic. Esakia duality has also been complemented with dualities for categories whose objects are Heyting algebras and whose morphisms are maps that preserve less structure than homomorphisms of Heyting algebras. In this chapter we survey these developments.
Celani, S. A., & Jansana, R. (2014). Easkia duality and its extensions. In Outstanding Contributions to Logic (Vol. 4, pp. 63–98). Springer. https://doi.org/10.1007/978-94-017-8860-1_4