A Model of Type Theory in Cubical Sets

Abstract: The intensional identity type is one if the most intricate concepts of dependent type theory. The recently discovered connection between homotopy theory and type theory gives a novel perspective on the identity type. Voevodsky's so-called Univalence Axiom furthermore explains the identity type for type theoretic universes as homotopy equivalences. This licentiate thesis is concerned with understanding these new developments from a computational point of view. While the Univalence Axiom has a model using Kan simplicial sets, this model inherently uses classical logic and thus can not be used to explain the axiom computationally. To preserve the computational properties of type theory it is, however, crucial to give a computational interpretation of the added constants. This thesis presents a model of dependent type theory with dependent products, sums, a universe, and identity types, based on cubical sets. The novelty of this model is that it is formulated in a constructive meta theory.