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.

We prove a conservativity result for extensional type theories over propositional ones, i.e. dependent type theories with propositional computation rules, or computation axioms, using insights from homotopy type theory. The argument exploits a notion of canonical homotopy equivalence between contexts, and uses the notion of a category with attributes to phrase the semantics of theories of dependent types. Informally, our main result asserts that, for judgements essentially concerning h-sets, reasoning with extensional or propositional type theories is equivalent.

Abstract Gödel’s Dialectica interpretation was designed to obtain the consistency of Peano arithmetic via a proof of consistency of Heyting arithmetic and double negation. In recent years, proof theoretic transformations (the so-called proof interpretations) based on Gödel’s Dialectica interpretation have been used systematically to extract new content from proofs and so the interpretation has found relevant applications in several areas of mathematics and computer science. Following our previous work on ‘Gödel fibrations’, we present a (hyper)doctrine characterization of the Dialectica, which corresponds exactly to the logical description of the interpretation. To show that, we derive the soundness of the interpretation of the implication connective, as expounded on by Spector and Troelstra, in the categorical model. This requires extra logical principles, going beyond intuitionistic logic, namely Markov Principle and the Independence of Premise principle, as well as some choice. We show how these principles are satisfied in the categorical setting, establishing a tight (internal language) correspondence between the logical system and the categorical framework. We make sure that this tight correspondence extends to the use of the principles above, instead of the weaker rules we had proved earlier on. This tight correspondence should come handy not only when discussing the traditional applications of the Dialectica but also when dealing with newer uses in modelling games or concurrency theory.

We introduce the notion of a Gödel fibration, which is a fibration categorically embodying both the logical principle of traditional Skolemization (we can exchange the order of quantifiers paying the price of a functional) and the existence of a prenex normal form presentation for every logical formula. Building up from Hofstra's earlier fibrational characterization of the de Paiva's categorical Dialectica construction, we show that a fibration is an instance of the Dialectica construction if and only if it is a Gödel fibration. This result establishes an internal presentation of the dialectica construction. Then we provide a deep structural analysis of the Dialectica construction producing a full description of which categorical structure behaves well with respect to this construction, focusing on (weak) finite products and coproducts. We conclude describing the applications we envisage for this generalized fibrational version of the Dialectica construction.