Recursively Defined Types in Constructive Type Theory

Publisher Summary This chapter presents an account of recursive definitions in a constructive type theory. It discusses these definitions from the viewpoint of category theory. An intimate connection between logic and category theory is well-known. The chapter discusses the relationship between Martin-Lof's intuitionistic type theory (ITT) and categorical logic. To understand the presentation of recursive type definitions, it is first necessary to understand the relationship between ITT and categorical logic. One of the advantages of category theory is that one can give very general definitions of standard constructions. Initially, these definitions might seem a bit contrived because they are expressed entirely in terms of morphisms; however, the resulting definition is more widely applicable than the corresponding definitions given in terms of sets and membership. One can recover many set theoretic ideas in category theory. The resulting theory of set-like categories is called topos theory. Topos theory comes with its own logic and turns out to define an intuitionistic set theory.