Containers: constructing strictly positive types

  • Authors:

  • Michael Abbott;Thorsten Altenkirch;Neil Ghani

  • Affiliations:

  • Diamond Light Source, Rutherford Appleton Laboratory, UK;School of Computer Science and Information Technology, Nottingham University, UK;Department of Mathematics and Computer Science, University of Leicester, UK

  • Venue:
  • Theoretical Computer Science - Applied semantics: Selected topics
  • Year:
  • 2005

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Abstract

We 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.