On fixpoint objects and gluing constructions

Abstract

This article has two parts: in the first part, we present some general results about fixpoint objects. The minimal categorical structure required to model soundly the equational type theory which combines higher order recursion and computation types (introduced by Crole and Pitts (1992)) is shown to be precisely a let-category possessing a fixpoint object. Functional completeness for such categories is developed. We also prove that categories with fixpoint operators do not necessarily have a fixpoint object.In the second part, we extend Freyd's gluing construction for cartesian closed categories to cartesian closed let-categories, and observe that this extension does not obviously apply to categories possessing fixpoint objects. We solve this problem by giving a new gluing construction for a limited class of categories with fixpoint objects; this is the main result of the paper. We use this category-theoretic construction to prove a type-theoretic conservative extension result.Mathematics Subject Classification (1991). 18C10.