On the symmetry behind duality

TL;DR

This paper introduces a symmetric duality framework for open sets and compact saturated sets in certain spaces, extending existing dualities to a broader class including sober spaces, using structures similar to bitopological spaces and d-frames.

Contribution

It develops a new duality framework that symmetrically relates open sets and compact saturated sets, extending dualities like sober spaces and frames to a more general setting.

Findings

Extends duality between sober spaces and frames to a symmetric duality framework.

Introduces a self-duality extending de Groot and Lawson dualities.

Provides a new presentation of continuous domains akin to d-frames.

Abstract

Open sets and compact saturated sets enjoy a perfect formal symmetry, at least for classes of spaces such as Stone spaces or spectral spaces. For larger classes of spaces, a perfect symmetry may not be available, although strong signs of it may remain. These signs appear especially in the classes of spaces involved in Stone-like dualities (such as sober spaces). In this article, we introduce a framework with a perfect symmetry between open sets and compact saturated sets, and which includes sober spaces. Our main result is an extension of the duality between sober spaces and spatial frames to a duality between two categories, each equipped with a self-duality. On the spatial side, the self-duality extends de Groot self-duality for stably compact spaces, which swaps open sets with complements of compact saturated sets; this self-duality is made possible using structures reminiscent of bitopological spaces. On the pointfree side, the self-duality is an extension of Lawson self-duality for continuous domains and is achieved via a framework analogous to that of d-frames. We show how to derive from our main result the well-known duality between locally compact sober spaces and locally compact frames. In doing so, we provide a presentation of continuous domains in a manner akin to d-frames.