Relating models of impredicative type theories

Abstract

The object of study of this paper is the categorical semantics of three impredicative type theories, viz. Higher Order λ-calculus Fω, the Calculus of Constructions and Higher Order ML. The latter is particularly interesting because it is a two-level type theory with type dependency at both levels. Having described appropriate categorical structures for these calculi, we establish translations back and forth between all of them. Most of the research in the paper concerns the theory of fibrations and comprehension categories.