Categorical Models of Polymorphism

Andrea Asperti, Simone Martini

Abstract: We present and discuss the relations between two classes of categorical models of the second order (or polymorphic) lambda-calculus, namely those based on internal categories (internal models) and those based on indexed categories (external models). We start, in Part I, with a detailed introduction to internal categories and their relations to indexed categories; the presentation is by means of equations between arrows in an ambient category with finite limits. In Part II we recall the definition of the two classes of models and we present the “externalization process” that given an internal model yields an external model. We show how one can go back in a straightforward way, and that, by making a full round trip (from an internal model to an internal model via an external one, or vice-versa), one does obtain equivalent models. Part III discusses three major examples of models (provable retractions inside a PER model, PER inside ω-Set, PL-categories inside their Grothendieck completions). The appendix contains an account of internal adjunctions and internal CCCs.