Covariant Types

Abstract

The covariant type system is an impredicative system that is rich enough to represent some polymorphism on inductive types, such as lists and trees, and yet is simple enough to have a set-theoretic semantics. Its chief novelty is to replace function types by transformation types, which denote parametric functions. Their free type variables are all in positive positions, and so can be modelled by covariant functors. Similarly, terms denote natural transformations. There is a translation from the covariant type system to system F which preserves non-trivial reductions. It follows that covariant reduction is strongly normalising and confluent. This work suggests a new approach to the semantics of system F, and new ways of basing type systems on the categorical notions of functor and natural transformation.