Heyting frames and Esakia duality

TL;DR

This paper introduces Heyting frames and Brouwerian frames, establishing their categorical equivalences with Heyting and Brouwerian algebras, respectively, and providing a frame-theoretic perspective on Esakia duality and its generalizations.

Contribution

It defines new categories of frames for Heyting and Brouwerian algebras, extending duality results and offering a unified frame-theoretic approach.

Findings

Category of Heyting frames is equivalent to Heyting algebras.

Category of Brouwerian frames extends duality to Brouwerian structures.

Provides a generalized duality framework for these algebraic structures.

Abstract

We introduce the category of Heyting frames and show that it is equivalent to the category of Heyting algebras and dually equivalent to the category of Esakia spaces. This provides a frame-theoretic perspective on Esakia duality for Heyting algebras. We also generalize these results to the setting of Brouwerian algebras and Brouwerian semilattices by introducing the corresponding categories of Brouwerian frames and extending the above equivalences and dual equivalences. This provides a frame-theoretic perspective on generalized Esakia duality for Brouwerian algebras and Brouwerian semilattices.