Some notes on Esakia spaces

TL;DR

This paper explores the duality between Esakia spaces and Heyting algebras, extending the concept to stably locally compact spaces and connecting it with split algebra structures.

Contribution

It provides a simplified proof of a general duality result using idempotent split completion and extends Esakia spaces to broader topological contexts.

Findings

Extended Esakia spaces to stably locally compact spaces.

Connected Esakia spaces with split algebras for related monads.

Provided a simpler proof of duality involving relations as morphisms.

Abstract

Under Stone/Priestley duality for distributive lattices, Esakia spaces correspond to Heyting algebras which leads to the well-known dual equivalence between the category of Esakia spaces and morphisms on one side and the category of Heyting algebras and Heyting morphisms on the other. Based on the technique of idempotent split completion, we give a simple proof of a more general result involving certain relations rather then functions as morphisms. We also extend the notion of Esakia space to all stably locally compact spaces and show that these spaces define the idempotent split completion of compact Hausdorff spaces. Finally, we exhibit connections with split algebras for related monads.