Abstract
AbstractWe 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.
Highlights
Cubical type theories are a family of formal systems for Homotopy Type Theory/Univalent Foundations (The Univalent Foundations Program, Institute for Advanced Study, 2013; Voevodsky, 2006)
This model validates the rules of a formal type theory and yields a a strict, rather than homotopy, canonicity result for a Cartesian cubical type theory with, path, boolean, circle, and “isovalence” types, the latter being a variant of univalence for strict isomorphisms
The formal cubical type theory that we define in this paper should interpret in this equivariant model, allowing a translation of theorems proved in it to facts about a standard notion of spaces. (The De Morgan model is not equivalent to spaces, but the question is still open for the model with connections but no reversal.) Because of these ongoing issues in proof assistant design and semantics, we believe it is worth documenting all of the variants of cubical type theory that support univalence and higher inductive types
Summary
Angiuli et al (2016, 2017a), Angiuli and Harper (2017) made progress by developing a computational Cartesian cubical type theory in the sense of Nuprl (Constable et al, 1986), in which types are construed as partial equivalence relations that specify the evaluation behavior of untyped programs This model validates the rules of a formal type theory and yields a a strict, rather than homotopy, canonicity result for a Cartesian cubical type theory with , , path, boolean, circle, and “isovalence” types, the latter being a variant of univalence for strict isomorphisms (pairs of functions that compose to the identity up to judgmental equality, not up to paths). The formal cubical type theory that we define in this paper should interpret in this equivariant model, allowing a translation of theorems proved in it to facts about a standard notion of spaces. (The De Morgan model is not equivalent to spaces, but the question is still open for the model with connections but no reversal.) Because of these ongoing issues in proof assistant design and semantics, we believe it is worth documenting all of the variants of cubical type theory that support univalence and higher inductive types
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