Choiceless, Pointless, but not Useless: Dualities for Preframes

We provide the appropriate common ‘(pre)framework’ for various central results of domain theory and topology, like the Lawson duality of continuous domains, the Hofmann–Lawson duality between continuous frames and locally compact sober spaces, the Hofmann–Mislove theorems about continuous semilattices of compact saturated sets, or the theory of stably continuous frames and their topological manifestations. Suitable objects for the pointfree approach are quasiframes, i.e., up-complete meet-semilattices with top, and preframes, i.e., meet-continuous quasiframes. We introduce the pointfree notion of locally compact well-filtered preframes, show that they are just the continuous preframes (using a slightly modified definition of continuity) and establish several natural dualities for the involved categories. Moreover, we obtain various characterizations of preframes having duality. Our results hold in ZF set theory without any choice principles.