Computational Higher Type Theory IV: Inductive Types

This is the fourth in a series of papers extending Martin-Löf’s meaning explanation of dependent type theory to higher-dimensional types. In this installment, we show how to define cubical type systems supporting a general schema of cubical inductive types, inductive types whose constructors may take dimension parameters and may have specified boundaries. Using this schema, we are able to specify and implement many of the higher inductive types which have been postulated in homotopy type theory, including homotopy pushouts, the torus, W-quotients, truncations, and arbitrary localizations. We also construct one indexed inductive type, the fiber family of a term. Using the fiber family, it is possible to define an identity type whose eliminator satisfies an exact computation rule on the reflexivity constructor. We believe that the techniques used to construct the fiber family could be straightforwardly combined with our schema for inductive types in order to give a schema for indexed cubical inductive types. The addition of higher inductive types and identity types makes computational higher type theory a model of homotopy type theory, capable of interpreting almost all of the constructions in the HoTT Book [41] (with the exception of general indexed inductive types and inductive-inductive types). This is the first such model with an explicit canonicity theorem stating that all closed terms of boolean type evaluate either to true or to false.