An Isbell Duality Theorem for Type Refinement Systems

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  • Université Paris Denis Diderot


Any refinement system (= functor) has a fully faithful representation in the refinement system of presheaves, by interpreting types as relative slice categories, and refinement types as presheaves over those categories. Motivated by an analogy between side effects in programming and context effects in linear logic, we study logical aspects of this “positive” (covariant) representation, as well as of an associated “negative” (contravariant) representation. We establish several preservation properties for these representations, including a generalization of Day’s embedding theorem for monoidal closed categories. Then we establish that the positive and negative representations satisfy an Isbell-style duality. As corollaries, we derive two different formulas for the positive representation of a pushforward (inspired by the classical negative translations of proof theory), which express it either as the dual of a pullback of a dual, or as the double dual of a pushforward.
Besides explaining how these constructions on refinement systems generalize familiar category-theoretic ones (by viewing categories as special refinement systems), our main running examples involve representations of Hoare logic and linear sequent calculus.


Original languageEnglish
Pages (from-to)736-774
JournalMathematical Structures in Computer Science
Issue number6
Early online date20 Mar 2017
Publication statusPublished - Jun 2018