A four for the price of one duality principle for distributive spaces

TL;DR

This paper explores a duality principle for distributive spaces viewed as generalized ordered topological spaces, establishing a categorical equivalence with a specific class of frames through idempotent splitting.

Contribution

It introduces a new duality principle for distributive spaces and characterizes their categorical relationship with frames via idempotent split completion.

Findings

Distributive spaces satisfy a specific topological distributive law.

The category of these spaces is dually equivalent to a category of frames.

Both categories are the idempotents split completion of the same base category.

Abstract

In this paper we consider topological spaces as generalised orders and characterise those spaces which satisfy a (suitably defined) topological distributive law. Furthermore, we show that the category of these spaces is dually equivalent to a certain category of frames by simply observing that both sides represent the idempotents split completion of the same category.