Higher Inductive Types in Cubical Computational Type Theory

In homotopy type theory (HoTT), higher inductive types provide a means of defining and reasoning about higher-dimensional objects such as circles and tori. The formulation of a schema for such types remains a matter of current research. We investigate the question in the context of cubical type theory, where the homotopical structure implicit in HoTT is made explicit in the judgmental apparatus. Within the computational cubical type system framework of Angiuli et al., we implement a class we call cubical inductive types, which includes n-truncations,W-quotients, and localizations. We suggest an extension to indexed inductive types by defining an example, a homotopy fiber type. From this we derive an identity type, making our theory a model of Martin-Löf type theory. Using Angiuli et al.’s implementation of univalence, we obtain a computational interpretation of HoTT with a general class of higher inductive types. This interpretation admits a canonicity theorem: any zero-dimensional element of a cubical inductive type evaluates to a constructor.