The biequivalence of path categories and axiomatic Martin-Löf type theories

The semantics of extensional type theory has an elegant categorical description: models of extensional =-types, 1-types, and Sigma-types are biequivalent to finitely complete categories, while adding Pi-types yields locally Cartesian closed categories. We establish parallel results for axiomatic type theory, which includes systems like cubical type theory, where the computation rule of the =-types only holds as a propositional axiom instead of a definitional reduction. In particular, we prove that models of axiomatic =-types, and standard 1- and Sigma-types are biequivalent to certain path categories, while adding axiomatic Pi-types yields dependent homotopy exponents. This biequivalence simplifies axiomatic =-types, which are more intricate than extensional ones since they permit higher dimensional structure. Specifically, path categories use a primitive notion of equivalence instead of a direct reproduction of the syntactic elimination rules and computation axioms. We apply our correspondence to prove a coherence theorem: we show that these weak homotopical models can be turned into equivalent strict models of axiomatic type theory. In addition, we introduce a more modular notion, that of a display map path category, which only models axiomatic =-types by default, while leaving room to add other axiomatic type formers such as 1-, Sigma-, and Pi-types.

💡 Research Summary

The paper establishes a precise 2‑categorical biequivalence between models of axiomatic Martin‑Löf type theory (ATT) and a class of homotopical structures known as path categories. ATT is characterised by the presence of propositional equality (=‑types) without any definitional reduction rules; the usual β‑rule for =‑types is weakened to a β‑axiom, and the η‑rule is omitted. This setting permits higher‑dimensional phenomena while retaining decidable type checking, and it encompasses systems such as cubical type theory.

The authors first relate ATT to display map categories (DMCs), a semantic framework that identifies a type in a context with a display map (a projection Γ.A → Γ) and a term with a context morphism. While DMCs capture the syntax closely, they only guarantee weak stability of type formers. To obtain a more robust semantics, the paper introduces path categories, originally defined by van den Berg and Moerdijk, which take equivalences as primitive. In a path category, the elimination and computation rules for =‑types need not be explicitly interpreted; instead, the existence of a suitable equivalence suffices.

The central technical contribution is a two‑step biequivalence. First, the authors prove that the 2‑category of (cloven) path categories is isomorphic to the 2‑category of suitably structured DMCs. “Cloven” means that each display map comes equipped with a chosen substitution, providing enough data to reconstruct the path‑category structure. Second, assuming the logical framework (LF) condition, they apply the left‑adjoint splitting technique of Lumsdaine‑Warren and Bocquet to strictify any cloven path category into a genuine model of ATT, i.e., a strict DMC satisfying the usual substitution equations on the nose. This yields a coherence theorem: every weak homotopical model can be replaced by an equivalent strict model.

When dependent function types (Π‑types) are added, the corresponding path categories must possess dependent homotopy exponentials, which exactly capture the semantics of Π‑types in a locally Cartesian closed setting. Thus the biequivalence extends to the full suite of type formers: =‑types, 1‑types, Σ‑types, and Π‑types.

A further innovation is the notion of a display‑map path category, which separates display maps from fibrations. Such categories model only axiomatic =‑types by default, leaving 1‑, Σ‑, and Π‑type formers as optional extensions. This modular approach yields a minimal semantics for ATT and facilitates the incremental addition of further type constructors without increasing categorical complexity.

The paper situates its results among prior work: the classic biequivalence between extensional type theory and finitely complete categories (Seely, Clairambault‑Dybjer), the universal‑property characterisations of Maietti, and the homotopy‑theoretic extensions of van der Weide. It shows that finitely complete categories are degenerate path categories where the only equivalences are isomorphisms and every morphism is a fibration.

The structure of the paper proceeds as follows. Section 2 presents the syntax of axiomatic =‑types, distinguishing between based and unbased path induction and explaining why the based version is preferable in the absence of Π‑types. Section 3 reviews display map categories, defines weak and strict stability, and introduces the LF condition. Section 4 recalls the definition and basic properties of path categories. Section 5 proves that path categories are rooted display map categories with weakly stable =‑, 1‑, and Σ‑types, and conversely that such display map categories reconstruct a path category, establishing a 2‑categorical isomorphism. Section 6 introduces display‑map path categories and shows an analogous biequivalence. Section 7 exploits the biequivalence to strictify cloven path categories, yielding genuine models of ATT and proving the coherence theorem.

In summary, the work provides a clean categorical semantics for axiomatic Martin‑Löf type theory, demonstrates how weak homotopical models can be systematically strictified, and offers a modular framework (display‑map path categories) for extending the core theory with additional type formers. This bridges the gap between homotopy‑theoretic models and traditional syntactic presentations of type theory, and it lays groundwork for further investigations into higher‑dimensional type constructors and their categorical counterparts.