Dialectica Categories over Heyting Algebras

2026-07-25

AI Summary

This study investigates the algebraic structures and logical properties induced by the categorification of the Dialectica interpretation over Heyting algebras. By specializing de Paiva’s categorical framework to the posetal setting, the authors reconstruct the original construction in purely algebraic terms and demonstrate that the resulting embedding of Heyting algebras into residuated lattices admits a definable adjoint. Key contributions include clarifying the distinct behavior of the single Dialectica tensor with respect to contraction in intuitionistic versus classical logic, and proving that, within ZF set theory, the collapse of the propositional Dialectica construction PD(Set) to PD(2) is equivalent to the Axiom of Choice. The work synthesizes methods from category theory, algebraic logic, and axiomatic set theory, thereby extending the algebraic foundations of Dialectica models and deepening their connections to foundational axiomatic systems.

Abstract

Categorification---the process of constructing a categorical model of a piece of mathematics---often identifies a common abstraction that connects formerly unrelated but known structures. In the case of de Paiva's categorification of Gödel's Dialectica interpretation, we find that its specialization to partial orders produces (functorial) embeddings of Heyting algebras into residuated lattices that appear to have been overlooked. For the non-categorical audience, we present this specialization and take care to reproduce the original proofs in the algebraic setting. Along the way we obtain results particular to this algebraic setting: an embedding lacking an evident adjoint in de Paiva's general construction acquires a definable one here; a single Dialectica tensor validates contraction in the intuitionistic construction D yet refutes it in the classical variant G; and, over ZF, the poset reflection PD(Set) collapses onto the four-element algebra PD(2) exactly when the Axiom of Choice holds.

Problem

• Dialectica interpretation
• Heyting algebras
• residuated lattices
• categorification
• partial orders

Innovation

• Dialectica interpretation
• Heyting algebras
• residuated lattices
• categorification
• adjoint functors