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

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.