Polyadic approximations, fibrations and intersection types

Starting from an exact correspondence between linear approximations and non-idempotent intersection types, we develop a general framework for building systems of intersection types characterizing normalization properties. We show how this construction, which uses in a fundamental way Mellies and Zeilbergerfs "type systems as functors" viewpoint, allows us to recover equivalent versions of every well known intersection type system (including Coppo and Dezanifs original system, as well as its non-idempotent variants independently introduced by Gardner and de Carvalho). We also show how new systems of intersection types may be built almost automatically in this way.