The G\"odel Fibration

TL;DR

This paper introduces G"odel fibrations, a categorical structure capturing logical principles like Skolemization and prenex normal form, and characterizes the Dialectica construction within this framework, providing structural insights and potential applications.

Contribution

It defines G"odel fibrations, characterizes the Dialectica construction as an instance of these fibrations, and analyzes their structural properties and applications.

Findings

G"odel fibrations embody Skolemization and prenex normal form principles.

Dialectica construction is characterized as a G"odel fibration.

Structural analysis reveals how categorical structures interact with the Dialectica construction.

Abstract

We introduce the notion of a G\"odel fibration, which is a fibration categorically embodying both the logical principle of traditional Skolemization (we can exchange the order of quantifiers paying the price of a functional) and the existence of a prenex normal form presentation for every logical formula. Building up from Hofstra's earlier fibrational characterization of the de Paiva's categorical Dialectica construction, we show that a fibration is an instance of the Dialectica construction if and only if it is a G\"odel fibration. This result establishes an internal presentation of the dialectica construction. Then we provide a deep structural analysis of the Dialectica construction producing a full description of which categorical structure behaves well with respect to this construction, focusing on (weak) finite products and coproducts. We conclude describing the applications we envisage for this generalized fibrational version of the Dialectica construction.