  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.
