Lattice-Free and Point-Free: Vickers Duality for Subbases of Stably Locally Compact Spaces

TL;DR

This paper develops a categorical duality framework for subbases of stably locally compact spaces, extending classical dualities like Priestley-Stone through entailment relations inspired by prior foundational work.

Contribution

It introduces a novel encoding of subbases via entailment relations and generalizes existing dualities to a broader class of spaces.

Findings

Established a duality framework for subbases of stably locally compact spaces.

Unified various classical and recent dualities under a common categorical approach.

Extended the scope of duality theories to include more general topological structures.

Abstract

Inspired by classic work of Wallman and more recent work of Jung-Kegelmann-Moshier and Vickers, we show how to encode general subbases of stably locally compact spaces via certain entailment relations. We further build this up to a categorical duality encompassing the classic Priestley-Stone duality and its various extensions to stably locally compact spaces by Shirota, De Vries, Hofmann-Lawson (in the stable case), Jung-S\"underhauf, Hansoul-Poussart, Bezhanishvili-Jansana, van Gool and Bice-Starling.