Cartesian Cubical Type Theory

We present a cubical type theory based on the Cartesian cube category (faces, degeneracies, symmetries, diagonals, but no connections or reversal) with univalent universes, each containing Π, Σ, path, identity, natural number, boolean, pushout, and glue (equivalence extension) types. The type theory includes a syntactic description of a uniform Kan operation, along with judgemental equality rules defining the Kan operation on each type. The Kan operation uses both a different set of trivial cofibrations and a different set of cofibrations than the Cohen, Coquand, Huber, and Mörtberg (CCHM) model. Next, we describe a constructive model in Cartesian cubical sets; the syntactic type theory is inspired by this model, though we have not yet given a formal interpretation. We describe a mechanized proof, using the internal language of cubical sets in the style introduced by Orton and Pitts, that glue, Π, Σ, path, identity, boolean, natural number, and pushout types are Kan in this model; we also sketch a proof that this internal construction implies univalent universes externally. An advantage of this formal approach is that our construction can also be interpreted in cubical sets on the connections cube category, and on the de Morgan cube category used in the CCHM model. As a first step towards comparing these approaches, we show that the two Kan operations are interderivable in a setting where both exist (presheaves on the de Morgan cube category, with the additional cofibration required by our construction).