An Isbell Duality Theorem for Type Refinement Systems

TL;DR

This paper develops a duality theorem for type refinement systems, representing types as presheaves and exploring logical and categorical properties, with applications to programming semantics and logic systems.

Contribution

It introduces an Isbell duality for refinement systems, generalizing classical category theory results and connecting to logic and programming semantics.

Findings

Establishes preservation properties for positive and negative representations.

Generalizes Day's embedding theorem for monoidal closed categories.

Derives dual formulas for pushforward in refinement systems.

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.