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An Isbell duality theorem for type refinement systems

Published online by Cambridge University Press:  20 March 2017

PAUL-ANDRÉ MELLIÈS  and
NOAM ZEILBERGER
PAUL-ANDRÉ MELLIÈS
Affiliation:
CNRS, Université Paris Diderot, Sorbonne Paris Cité, Paris, France Email: mellies@pps.univ-paris-diderot.fr
NOAM ZEILBERGER
Affiliation:
School of Computer Science, University of Birmingham, Birmingham, U.K. Email: noam.zeilberger@gmail.com
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Abstract

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.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 28 , Issue 6 , June 2018 , pp. 736 - 774
Copyright
Copyright © Cambridge University Press 2017 

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