The univalence axiom in cubical sets

In this note we show that Voevodsky’s univalence axiom holds in the model of type theory based on cubical sets as described in [2, 6]. We will also discuss Swan’s construction of the identity type in this variation of cubical sets. This proves that we have a model of type theory supporting dependent products, dependent sums, univalent universes, and identity types with the usual judgmental equality, and this model is formulated in a constructive metatheory. 1. Review of the cubical set model We give a brief overview of the cubical set model, introducing some different notations, but will otherwise assume the reader is familiar with [2, 6]. As opposed to [2, 6] let us define cubical sets as contravariant presheaves on the opposite of the category used there, that is, the category of cubes C contains as objects finite sets I = {i1, . . . , in} (n ≥ 0) of names and a morphism f : J → I is given by a set-theoretic map I → J ∪ {0, 1} which is injective when restricted to the preimage of J ; we will write compositions in applicative order. The category of cubical sets is the category [Cop,Set] of presheaves on C. A morphism f : J → I in C can be viewed as a substitution. If f(i) ∈ J , we call f defined on i. For i / ∈ I, the face morphisms are denoted by (i/0), (i/1) : I → I, i and are induced by setting i to 0 and 1, respectively; degenerating along i / ∈ I is denoted by si : I, i → I and is induced by the inclusion I ⊆ I, i. If Γ is a cubical set, we write Ty(Γ) for the collection/class of presheaves on the category of elements of Γ [2, 6]. Such a presheaf A ∈ Ty(Γ) is given by a family of sets A(I, ρ) for I ∈ C and ρ ∈ Γ(I) together with restriction functions. As ρ ∈ Γ(I) determines I we simply write Aρ for A(I, ρ). Given A ∈ Ty(Γ) and a natural transformation (substitution) σ : ∆ → Γ we get Aσ ∈ Ty(∆) defined as (Aσ)ρ = A(σρ) which extends canonically to the restrictions. For A ∈ Ty(Γ) we denote the set of sections of A by Ter(Γ, A); so a ∈ Ter(Γ, A) is given by a family aρ ∈ Aρ for ρ ∈ Γ(I) such that (aρ)f = a(ρf) for f : J → I. Substitution also extends to terms via (aσ)ρ = a(σρ). Let us recall the construction of Π-types: ΠAB ∈ Ty(Γ) for A ∈ Ty(Γ) and B ∈ Ty(Γ.A) is given by letting each element w of (ΠAB)ρ (with ρ ∈ Γ(I)) be a family of wf a ∈ B(ρf, a) for f : J → I and a ∈ Aρ satisfying (wf a)g = wfg (ag); the restriction of such a w is given by (wf)g = wfg. In the sequel we will however only have to refer to wf when f is the identity, and will thus simply write w a for wid a. We also occasionally switch between sections in Ter(Γ.A,B) and Ter(Γ,ΠAB) without warning the reader. Let A ∈ Ty(Γ), ρ ∈ Γ(I), and J ⊆ I. A J-tube in A over ρ is given by a family ~u of elements ujc ∈ Aρ(j/c) for (j, c) ∈ J × {0, 1} which is adjacent compatible, that is, ujc(k/d) = ukd(j/c) for (j, c), (k, d) ∈ J ×{0, 1}. For (i, a) ∈ (I − J)×{0, 1} we say that an element uia ∈ Aρ(i/a) is a lid of such a tube ~u if ujc(i/a) = uia(j/c) for all (j, c) ∈ J × {0, 1}. In this situation we call the pair [J 7→ ~u; (i, a) 7→ uia] an Date: October 31, 2017. 1 ar X iv :1 71 0. 10 94 1v 1 [ m at h. L O ] 3 0 O ct 2 01 7 2 MARC BEZEM, THIERRY COQUAND, AND SIMON HUBER open box in A over ρ. A filler for such an open box is an element u ∈ Aρ such that u(j/c) = ujc for (j, c) ∈ {(i, a)} ∪ (J × {0, 1}). In case J is empty, we simply write [(i, a) 7→ uia]. Given f : K → I and an open box m = [J 7→ ~u; (i, a) 7→ uia] in A over ρ we call f allowed for m if f is defined on J, i. In this case we define the open box mf in A in ρf to be [Jf 7→ ~uf ; (f(i), a) 7→ uia(f − i)] where ~uf is given by (~uf)f(j) c = ujc(f − j) with f − i : K− f(i)→ I− i being like f but skipping i, and Jf is the image of J under f . Recall from [2, Section 4] that a (uniform) Kan structure for a type A ∈ Ty(Γ) is given by an operation κ which (uniformly) fills open boxes: for any ρ ∈ Γ(I) and open box m in A over ρ we get a filler κ ρm of m subject to the uniformity condition (κ ρm)f = κ (ρf) (mf) for all f : K → I allowed for m. Any Kan structure κ defines a composition operation κ̄ which provides the missing lid of the open box, given by: κ̄ ρ [J 7→ ~u; (i, 0) 7→ ui0] = (κ ρ [J 7→ ~u; (i, 0) 7→ ui0])(i/1) κ̄ ρ [J 7→ ~u; (i, 1) 7→ ui1] = (κ ρ [J 7→ ~u; (i, 1) 7→ ui1])(i/0) We denote the set of all Kan structures on A ∈ Ty(Γ) as Fill(Γ, A). If σ : ∆→ Γ and κ is an element in Fill(Γ, A), we get an element κσ in Fill(∆, Aσ) defined by (κσ) ρ = κ (σρ). Given a cubical set Γ a Kan type is a pair (A, κ) where A ∈ Ty(Γ) and κ ∈ Fill(Γ, A). We denote the collection of all such Kan types by KTy(Γ). In [2] we showed that Kan types are closed under dependent products and sums constituting a model of type theory.