Containers: Constructing strictly positive types

Abstract

AbstractWe introduce the notion of a Martin-Löf category—a locally cartesian closed category with disjoint coproducts and initial algebras of container functors (the categorical analogue of W-types)—and then establish that nested strictly positive inductive and coinductive types, which we call strictly positive types, exist in any Martin-Löf category.Central to our development are the notions of containers and container functors. These provide a new conceptual analysis of data structures and polymorphic functions by exploiting dependent type theory as a convenient way to define constructions in Martin-Löf categories. We also show that morphisms between containers can be full and faithfully interpreted as polymorphic functions (i.e.natural transformations) and that, in the presence of W-types, all strictly positive types (including nested inductive and coinductive types) give rise to containers