Model structure on the universe in a two level type theory

There is an on-going connection between type theory and homotopy theory, based on the similarity between types and the notion of homotopy types for topological spaces. This idea has been made precise by exhibiting the category cSet of cubical sets as a model of homotopy type theory. It is natural to wonder, conversely, to what extend this model can be reflected in a type theory. The homotopy structure of cSet is given by a model structure; that is, a definition of three classes of maps—fibrations, cofibrations and weak equivalences—satisfying various properties. In this article, we internalize the notion of model structure in Martin-Löf type theory with a strict equality and formalize a model structure on the category of fibrant types in a type theory with two equalities (à la Voevodsky’s Homotopy Type System). This formalization is conducted in Coq, taking advantage of type class inference to emulate fibrancy. We then propose a refinement of the notion of fibrancy—justified in the cubical model—by distinguishing between degenerate and regular fibrant families. In this system, a fibrant replacement is admissible (which is an open issue in the community) and gives rise to a model structure on the universe of all types. 1998 ACM Subject Classification F.4.1 Mathematical logic