On the semantics of second order lambda calculus: From bruce-meyer-mitchell models to hyperdoctrine models and vice-versa

Abstract

In the literature there are two notions of model for the second order polymorphic λ-calculus: one by Bruce, Meyer and Mitchell (the BMM-model, for short) in set-theoretical formulation and one category-theoretical by Seely, based on hyperdoctrines. Using notions from Hayashi [1985], we adapt Seely's definition in such a way that a model of (non-extensional) 2^nd order λ-calculus is obtained that satisfies the ζ-rules for the interpretation with valuation functions. This hyperdoctrine model is called a (categorical) λ₂-model; it gives rise to a BMM-model. The other way around, we show that a BMM-model gives rise to a categorical λ₂-model. If that λ₂-model is transformed into a BMM-model again, the result is essentially the same as what we started with.

Research partially performed at the University of Pisa, Italy, during the first half of 1989, supported by the “Jumelage” project ST2J-0374-C of the European Communitee.