A universal characterization of the double powerlocale

Abstract

The double powerlocale P(X) (found by composing, in either order, the upper and lower powerlocale constructions P U and P L) is shown to be isomorphic in [ Loc op , Set ] to the double exponential S S X where S is the Sierpiński locale. Further P U( X) and P L( X) are shown to be the subobjects of P(X) comprising, respectively, the meet semilattice and join semilattice homomorphisms. A key lemma shows that, for any locales X and Y, natural transformations from S X (the presheaf Loc (_ × X, S) ) to S Y (i.e. Loc (_ × Y, S) ) are equivalent to dcpo morphisms (Scott continuous maps) from the frame ΩX to ΩY. It is also shown that S X has a localic reflection in [ Loc op , Set ] whose frame is the Scott topology on ΩX. The reasoning is constructive in the sense of topos validity.