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.

To ensure decidability and consistency of its type theory, a proof assistant should only accept terminating recursive functions and productive corecursive functions. Most proof assistants enforce this through syntactic conditions, which can be restrictive and non-modular. Sized types are a type-based alternative where (co)inductive types are annotated with additional size information. Well-founded induction on sizes can then be used to prove termination and productivity. An implementation of sized types exists in Agda, but it is currently inconsistent due to the addition of a largest size. We investigate an alternative approach, where intensional type theory is extended with a large type of sizes and parametric quantifiers over sizes. We show that inductive and coinductive types can be constructed in this theory, which improves on earlier work where this was only possible for the finitely-branching inductive types. The consistency of the theory is justified by an impredicative realisability model, which interprets the type of sizes as an uncountable ordinal.