Constructing Strictly Positive Types

TL;DR: The notion of a Martin-L of category was introduced in this article, which is a locally cartesian closed category with disjoint coproducts and initial algebras of container functors (the categorical analogue of W-types).

Abstract: We introduce the notion of a Martin-L¨ of 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-L ¨ of category. Central to our development are the notions of containers and container functors, introduced in Abbott, Altenkirch, and Ghani (2003a). 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-L ¨ of 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.