Topology via logic
The Hoare and Smyth power domain constructors commute under composition
Journal of Computer and System Sciences
Information systems for continuous posets
Theoretical Computer Science
Mathematical Structures in Computer Science
Presenting locale pullback via directed complete posets
Theoretical Computer Science - Logic, semantics and theory of programming
Entailment systems for stably locally compact locales
Theoretical Computer Science - Logic, semantics and theory of programming
A Cartesian closed extension of the category of locales
Mathematical Structures in Computer Science
Presenting Dcpos and Dcpo Algebras
Electronic Notes in Theoretical Computer Science (ENTCS)
Electronic Notes in Theoretical Computer Science (ENTCS)
De groot duality and models of choice: Angels, demons and nature†
Mathematical Structures in Computer Science
Observationally-induced Effects in Cartesian Closed Categories
Electronic Notes in Theoretical Computer Science (ENTCS)
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The double powerlocale P(X) (found by composing, in either order, the upper and lower powerlocale constructions PU and PL) is shown to be isomolphic in [Locop, Set] to the double exponential SSX where S is the Sierpinski locale. Further PU(X) and PL(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 SX (the presheaf Loc(_ × X, S)) to SY (i.e. Loc(_ × Y, S)) are equivalent to dcpo morphisms (Scott continuous maps) from the flame ΩX to ΩY. It is also shown that SX has a localic reflection in [Locop, Set] whose frame is the Scott topology on ΩX.The reasoning is constructive in the sense of topos validity.