Entailment systems for stably locally compact locales

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The category SCFr U of stably continuous frames and preframe homomorphisms (preserving finite meets and directed joins) is dual to the Karoubi envelope of a category Ent whose objects are sets and whose morphisms X→ Y are upper closed relations between the finite powersets FX and FY . Composition of these morphisms is the “cut composition” of Jung et al. that interfaces disjunction in the codomains with conjunctions in the domains, and thereby relates to their multi-lingual sequent calculus. Thus stably locally compact locales are represented by “entailment systems” ( X,⊢) in which ⊢, a generalization of entailment relations, is idempotent for cut composition. Some constructions on stably locally compact locales are represented in terms of entailment systems: products, duality and powerlocales. Relational converse provides Ent with an involution, and this gives a simple treatment of the duality of stably locally compact locales. If A and B are stably continuous frames, then the internal preframe hom A⋔B is isomorphic to A ̃ ⊗B where A ̃ is the Hofmann–Lawson dual. For a stably locally compact locale X, the lower powerlocale of X is shown to be the dual of the upper powerlocale of the dual of X.