Deriving dualities in pointfree topology from Priestley duality

TL;DR

This paper demonstrates how to derive key duality results in pointfree topology from Priestley duality for distributive lattices, providing new insights into classical dualities.

Contribution

It introduces a method to obtain several prominent dualities in pointfree topology from Priestley duality, unifying and clarifying their relationships.

Findings

Derives Hofmann-Lawson duality from Priestley duality.

Shows how to obtain Isbell duality via this approach.

Provides a new perspective on classical dualities in topology.

Abstract

There are several prominent duality results in pointfree topology. The Hofmann-Lawson duality establishes that the category of continuous frames is dually equivalent to the category of locally compact sober spaces. This restricts to a dual equivalence between the categories of stably continuous frames and stably locally compact spaces, which further restricts to Isbell duality between the categories of compact regular frames and compact Hausdorff spaces. We show how to derive these dualities from Priestley duality for distributive lattices, thus shedding new light on these classic results.