On the symmetry behind duality

Abstract

Dualities such as Stone duality and the duality between sober spaces and spatial frames hinge on an interaction between open sets and compact saturated sets. In several important classes of spaces-Stone spaces, spectral spaces, and stably compact spaces-this interaction forms a perfect symmetry, reflected dually as order self-duality. But the class of sober spaces, despite being central to Stone-like dualities, exhibits only a partial symmetry between openness and compactness. This raises a central question: can we enlarge the setting enough to recover a perfect symmetry, while still retaining sober spaces and preserving the conditions that make the sober-spatial-frame duality work? We answer this question affirmatively. We introduce ko-spaces, whose families of open and compact saturated sets satisfy the compatibility needed for duality, and bi-dcpos, a pointfree companion generalizing both spatial frames and continuous domains. We prove that the categories of ko-spaces and distributive bi-dcpos are equivalent (and dually equivalent, too), and that each category carries a symmetry in the form of a self-duality. On spaces, this extends de Groot duality; on domains, it extends Lawson duality. Classical results fall out as special cases: the sober-spatial-frame duality reappears inside our symmetric framework, and continuous domains acquire a presentation akin to that of d-frames. Our work suggests that an appropriate home for Stone-like duality is a fully symmetric two-sorted world in which openness and compactness play on equal footing.