Intersection Type Distributors

We study a family of distributors-induced bicategorical models of λ-calculus, proving that they can be syntactically presented via intersection type systems. We first introduce a class of 2-monads whose algebras are monoidal categories modelling resource management. We lift these monads to distributors and define a parametric Kleisli bicategory, giving a sufficient condition for its cartesian closure. In this framework we define a proof-relevant semantics: the interpretation of a term associates to it the set of its typing derivations in appropriate systems. We prove that our model characterize solvability, adapting reducibility techniques to our setting. We conclude by describing two examples of our construction.

1 For a general survey on relational semantics we refer to [51]. See also [7] for results on the lambda-theories induced by this kind of models.

2 Another popular name for this kind of structures is profunctor.