It is known that one can construct non-parametric functions by assuming classical axioms. Our work is a converse to that: we prove classical axioms in dependent type theory assuming speci€c instances of non-parametricity. We also address the interaction between classical axioms and the existence of automorphisms of a type universe. We work over intensional Martin-Löf dependent type theory, and ...

The recent discovery of an interpretation of constructive type theory into abstract homotopy theory suggests a new approach to the foundations of mathematics with intrinsic geometric content and a computational implementation. Voevodsky has proposed such a program, including a new axiom with both geometric and logical significance: the Univalence Axiom. It captures the familiar aspect of inform...

We present a game semantics for intuitionistic type theory. Specifically, we propose a new variant of games that provides a uniform treatment of games and strategies, and enables us to interpret the hierarchy of universes without a Russell-like paradox. We then formulate categories with families of the games for both extensional and intensional variants of the type theory, which support ∏ -, ∑ ...

We discuss the homotopy type theory library in the Lean proof assistant. The library is especially geared toward synthetic homotopy theory. Of particular interest is the use of just a few primitive notions of higher inductive types, namely quotients and truncations, and the use of cubical methods.

Cubical type theory provides a constructive justification to certain aspects of homotopy type theory such as Voevodsky’s univalence axiom. This makes many extensionality principles, like function and propositional extensionality, directly provable in the theory. This paper describes a constructive semantics, expressed in a presheaf topos with suitable structure inspired by cubical sets, of some...

Inductive types can be cleanly represented internally as W-types [14] [20], that is, as initial algebras of containers [1]. In this paper, we give a similar presentation that extends the notion of W-type to more general forms of induction, including mutually defined data types and higher inductive types.

Capretta’s delay monad can be used to model partial computations, but it has the “wrong” notion of built-in equality, strong bisimilarity. An alternative is to quotient the delay monad by the “right” notion of equality, weak bisimilarity. However, recent work by Chapman et al. suggests that it is impossible to define a monad structure on the resulting construction in common forms of type theory...

In this paper, we analyze and compare three of the many algebraic structures that have been used for modeling dependent type theories: categories with families, split type-categories, and representable maps of presheaves. We study these in the setting of univalent foundations, where the relationships between them can be stated more transparently. We also introduce relative universes, generalizi...