Cubical Type Theory: A Constructive Interpretation of the Univalence Axiom

Canonicity and homotopy canonicity for cubical type theory

“…Thus, every element of Elem(Γ, Path(A, a 0 , a 1 )) is uniquely of the form (u) with u in i:I Elem(Γ, Ai) such that u 0 = a 0 and u 1 = a 1 . Using path types, we define isContr c (A) in Type(Γ) for A in Type(Γ) as well as isEquiv c in Type(Γ.A → B) and Equiv c in Type(Γ) for A, B in Type(Γ) as in [CCHM18]. (We use a subscript here and for some other notions to distinguish them from analogous notions defined later in a different setting in Subsection 2.2.)…”

“…Assume that I has a connection algebra structure and that F forms a sublattice of Ω 0 that contains the interval endpoint inclusions. As in [CCHM18], it is then possible in the above context of the glue type former to construct an element of Elem(Γ, isEquiv c [unglue]). From this, one derives an element of Elem(Γ, iUnivalence n ) where…”

“…In order to simplify the notations, we assume here that the interval I also has a reversal operation, like in [CCHM18,CHM18]. This assumption is not necessary (for instance, as noted in [CCHM18], and indeed as we did in Subsection 1.3, we can avoid the reverse operation at the cost of carrying around an external boolean parameter) but it simplifies the presentation slightly.…”

“…It is however best expressed in an algebraic setting. We first define a general notion of model, called cubical category with families, defined as a category with families [Dyb96] with certain special operations internal to presheaves over a category C (such as a cube category) with respect to the parameters of an interval I and a cofibration classifier F. In this article, we will work with models of the cubical type theory described by [CCHM18,OP16]. However, our methods apply equally well to other versions of cubical type theories that can be presented in a similar setting, for example [ABC + 21].…”

Typal Heterogeneous Equality Types

“…Equality types in the intensional version of Martin-Löf Type Theory (see, for example, Reference [12,Section 8.1]) are traditionally formulated in terms of an introduction rule (reflexivity) together with a rule for eliminating proofs of equality and a rule describing how elimination computes when it meets a reflexivity proof. Some recent work [4,7] on models of Homotopy Type Theory [17] uses a formulation of equality types that differs from this in two respects. First, the elimination operation is replaced by the combination of a simple operation for transporting elements along proofs of equality, together with an axiom asserting contractibility of singleton types.…”