On duality and model theory for polyadic spaces

TL;DR

This paper develops a duality framework for first-order coherent logic using open polyadic Priestley spaces, leading to new completeness, interpolation, and omitting types theorems with a modular, syntax-free approach.

Contribution

It introduces a duality theorem connecting coherent hyperdoctrines with open polyadic Priestley spaces, enabling new model-theoretic results for coherent and intuitionistic logics.

Findings

Proved duality between coherent hyperdoctrines and open polyadic Priestley spaces

Established completeness, interpolation, and omitting types theorems for coherent and intuitionistic logic

Extended the approach to constant domain and G"odel-Dummett intuitionistic predicate logics

Abstract

This paper is a study of first-order coherent logic from the point of view of duality and categorical logic. We prove a duality theorem between coherent hyperdoctrines and open polyadic Priestley spaces, which we subsequently apply to prove completeness, omitting types, and Craig interpolation theorems for coherent or intuitionistic logic. Our approach emphasizes the role of interpolation and openness properties, and allows for a modular, syntax-free treatment of these model-theoretic results. As further applications of the same method, we prove completeness theorems for constant domain and G\"odel-Dummett intuitionistic predicate logics.