Two-dimensional models of type theory

We describe a non-extensional variant of Martin-L"of type theory which we call two-dimensional type theory, and equip it with a sound and complete semantics valued in 2-categories.

💡 Research Summary

The paper introduces a novel variant of Martin‑Löf type theory called two‑dimensional type theory (2DTT), which extends the traditional syntax by allowing not only terms and identity proofs but also higher‑dimensional identifications between those proofs. In 2DTT, a type is interpreted as an object of a 2‑category, a term as a 1‑cell, and an identity proof as a 2‑cell. This extra layer of structure makes the theory intrinsically non‑extensional: two terms are considered equal only when there exists a proof of equality, and two proofs of equality are considered equal only when there is a higher‑dimensional proof witnessing their equivalence. Consequently, the theory can express “identifications between identifications,” a feature absent from ordinary Martin‑Löf type theory.

The authors develop a sound semantics by mapping the syntactic constructs of 2DTT into any bicategory (or strict 2‑category). They verify that each inference rule—formation, introduction, elimination, and computation—preserves the categorical structure: type formation corresponds to the existence of objects, term formation to the existence of 1‑cells, and identity formation to the existence of 2‑cells. The soundness proof shows that any derivable judgment in 2DTT is interpreted as a valid diagram in the chosen 2‑category, respecting composition and coherence laws.

The central technical contribution is a completeness theorem. To prove it, the authors construct a free 2‑category generated by the syntax of 2DTT. They then demonstrate a bijective correspondence between derivations in the type theory and composites of generating 2‑cells in this free 2‑category. Every 2‑cell in the semantic model can be represented by a proof term, and conversely, every proof term yields a well‑defined 2‑cell. This establishes that the semantics is not only sound but also fully expressive for the intended class of models.

The paper situates its work within the broader landscape of higher‑dimensional type theories, especially Homotopy Type Theory (HoTT). While HoTT interprets identity types as paths in spaces, 2DTT captures a similar intuition at the categorical level, allowing a direct algebraic treatment of 2‑dimensional homotopies. The authors argue that their 2‑categorical semantics provides a bridge between syntactic type‑theoretic formalisms and the algebraic topology of bicategories, opening avenues for future extensions to n‑dimensional theories.

Finally, the authors discuss potential applications. The richer equality structure could improve proof assistants by enabling automated reasoning about higher‑dimensional equalities, and the categorical semantics may guide the design of programming languages with built‑in support for homotopical reasoning. The paper concludes that two‑dimensional type theory, together with its 2‑categorical models, offers a robust foundation for exploring non‑extensional, higher‑dimensional aspects of type theory and for integrating logical and homotopical perspectives.