ABSTRACT. The use of intersection type systems in the the study of denotational semantics of the untyped λ-calculus has been a fruitful line of research over the past thirty-five years. In this extended abstract we present a research project aiming at broaden the boundaries of such type-theoretical approach to semantics, possibly to other idealized programming languages. The key idea is to ex- ploit a more abstract view on what a type system is. This view finds its root in higher-dimensional category theory.