Kurt Gödel's 1958 Dialectica interpretation translated intuitionistic logic into computable functional types; extending Dialectica categories over arbitrary Heyting algebras creates rich mathematical structures for categorical proof theory.
In mathematical logic, constructive and intuitionistic proofs carry computational content: proving a theorem constructively is equivalent to writing an algorithm that executes the claim.
Gödel's Dialectica interpretation extracted computational witness programs from classical arithmetic proofs, but extending this translation to general category theory required generalizing beyond standard Boolean and Heyting truth values.
This paper establishes a rigorous category-theoretic construction of Dialectica categories over arbitrary Heyting algebras, proving sound categorical duality theorems and clarifying the relationship between constructive logic and functional programming types.
These Dialectica categories provide foundational tools for automated program synthesis, compiler optimization, and the verified extraction of bug-free software from mathematical proofs.
Dialectica Categories over Heyting Algebras
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.
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