On Higher Inductive Types in Cubical Type Theory

TL;DR

This paper develops a constructive semantics for higher inductive types within cubical type theory, extending its syntax to include various complex types and ensuring they have computational rules and universe closure.

Contribution

It introduces a semantics for higher inductive types in cubical type theory and extends the syntax to include spheres, tori, suspensions, truncations, and pushouts.

Findings

Constructive semantics for higher inductive types in cubical type theory.

Extended syntax with new higher inductive types and judgmental computation rules.

Universes closed under these new type formers.

Abstract

Cubical type theory provides a constructive justification to certain aspects of homotopy type theory such as Voevodsky's univalence axiom. This makes many extensionality principles, like function and propositional extensionality, directly provable in the theory. This paper describes a constructive semantics, expressed in a presheaf topos with suitable structure inspired by cubical sets, of some higher inductive types. It also extends cubical type theory by a syntax for the higher inductive types of spheres, torus, suspensions,truncations, and pushouts. All of these types are justified by the semantics and have judgmental computation rules for all constructors, including the higher dimensional ones, and the universes are closed under these type formers.