Location of Repository

An Isbell Duality Theorem for Type Refinement Systems

By Paul-André Melliès and Noam Zeilberger

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

Topics: [ INFO.INFO-LO ] Computer Science [cs]/Logic in Computer Science [cs.LO], [ MATH.MATH-CT ] Mathematics [math]/Category Theory [math.CT]
Publisher: HAL CCSD
Year: 2015
OAI identifier: oai:HAL:hal-01214119v1
Provided by: Hal-Diderot
Download PDF:
Sorry, we are unable to provide the full text but you may find it at the following location(s):
  • https://hal.archives-ouvertes.... (external link)
  • https://hal.archives-ouvertes.... (external link)
  • https://hal.archives-ouvertes.... (external link)
  • Suggested articles


    To submit an update or takedown request for this paper, please submit an Update/Correction/Removal Request.