  We introduce a notion of 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 (also known as abstract bases or R-structures). 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. We also provide 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. We show
that this structure naturally corresponds to the notion of continuous lattice
in the predicative point-free topology. Our result makes the predicative and
finitary aspect of the notion of continuous lattice in point-free topology more
explicit.
