This is a version from 29 Sept 2003 of the paper published under the same name in Theoretical Computer Science 316 (2004) 297–321. The double powerlocale P(X) (found by composing, in either order, the upper and lower powerlocale constructions PU and PL) is shown to be isomorphic in [Loc op, Set] to the double exponential S SX where S is the Sierpiński locale. Further PU (X) and PL(X) are shown to be the subobjects 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
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