Skolem, Gödel, and Hilbert fibrations

Abstract

Grothendieck fibrations are fundamental in capturing the concept of dependency, notably in categorical semantics of type theory and programming languages. A relevant instance are Dialectica fibrations which generalise G\"odel's Dialectica proof interpretation and have been widely studied in recent years. We characterise when a given fibration is a generalised, dependent Dialectica fibration, namely an iterated completion of a fibration by dependent products and sums (along a given class of display maps). From a technical perspective, we complement the work of Hofstra on Dialectica fibrations by an internal viewpoint, categorifying the classical notion of quantifier-freeness. We also generalise both Hofstra's and Trotta et al.'s work on G\"odel fibrations to the dependent case, replacing the class of cartesian projections in the base category by arbitrary display maps. We discuss how this recovers a range of relevant examples in categorical logic and proof theory. Moreover, as another instance, we introduce Hilbert fibrations, providing a categorical understanding of Hilbert's $\epsilon$- and $\tau$-operators well-known from proof theory.