On the symmetry behind duality

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.