Predicative theories of continuous lattices

The paper introduces strong proximity join-semilattice, a predicative notion of continuous lattice which arises as the Karoubi envelop of the category of algebraic lattices. Strong proximity join-semilattices can be characterised by the coalgebras of the lower powerlocale on the wider category of proximity posets (known as abstract bases and R-structure). Moreover, locally compact locales can be characterised in terms of strong proximity join-semilattices by the coalgebras of the double powerlocale on the category of proximity posets. The paper also introduces more logical characterisation of a strong proximity join-semilattice, called a strong continuous finitary cover, which uses an entailment relation to present the underlying join-semilattice. It is shown that this structure naturally corresponds to the notion of continuous lattice in the predicative pointfree topology. The result makes the predicative and finitary aspect of the notion of continuous lattice in point-free topology more explicit.