Syntax and models of Cartesian cubical type theory

Abstract

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, suspension, and glue (equivalence extension) types. The type theory includes a syntactic description of a uniform Kan operation, along with judgmental equality rules defining the Kan operation on each type. The Kan operation uses both a different set of generating trivial cofibrations and a different set of generating cofibrations than the Cohen, Coquand, Huber, and Mörtberg (CCHM) model. Next, we describe a constructive model of this type theory in Cartesian cubical sets. We give a mechanized proof, using Agda as the internal language of cubical sets in the style introduced by Orton and Pitts, that glue, Π, Σ, path, identity, boolean, natural number, suspension types, and the universe itself are Kan in this model, and that the universe is univalent. An advantage of this formal approach is that our construction can also be interpreted in a range of other models, including cubical sets on the connections cube category and the De Morgan cube category, as used in the CCHM model, and bicubical sets, as used in directed type theory.

Internalizing Representation Independence with Univalence

“…In Cubical Agda, the interval I is a primitive type which, in addition to the elements i0 : I and i1 : I, is equipped with three operations-minimum (_∧_ : I → I → I), maximum (_∨_ : I → I → I), and reversal (∼_ : I → I)-which satisfy the laws of a De Morgan algebra, i.e., a bounded distributive lattice (i0, i1, _∧_, _∨_) with a De Morgan involution ∼_. Other formulations of Cubical Type Theory requiring less structure on I are also possible [Angiuli et al 2019[Angiuli et al , 2018.…”

Canonicity and homotopy canonicity for cubical type theory

“…In order to later model universes à la Russell, we define them in a stratified manner where instead of a single presheaf of types, we specify a filtration of presheaves of "small" types. 2 The length of the filtration is not essential: we have chosen 1 + ω so that we may specify constructions just at the top level.…”

“…For the rest of the paper, we fix a category C in the lowest Grothendieck universe. As in [2,20,18], we will use the language of extensional type theory (with subtypes) to describe constructions in the presheaf topos over C.…”

“…This paper is a contribution to the analysis of the computational content of the univalence axiom [28] (and higher inductive types). In previous work [2,3,4,5,20], various presheaf models of this axiom have been described in a constructive meta theory. In this formalism, the notion of fibrant type is stated as a refinement of the path lifting operation where one not only provides one of the endpoints but also a partial lift (for a suitable notion of partiality).…”

“…The axiom of univalence is then captured by a suitable equivalence extension operation (the "glueing" operation), which expresses that we can extend a partially defined equivalence of a given total codomain to a total equivalence. These presheaf models suggest possible extensions of type theory where we manipulate higher dimensional objects [2,4]. One can define a notion of reduction and prove canonicity for this extension [13]: any closed term of type N (natural number) is convertible to a numeral.…”