Extension Types for Free

Abstract

Extension types are a concept in dependent type theory that has appeared in various contexts. The idea is to have types whose terms are partially determined, e.g. via a strict boundary condition. Standard examples are path types of cubical type theories (paths with fixed endpoints), Riehl and Shulman's name-giving extension types (terms fixed on subshapes), as well as the controlled-unfolding mechanism of cooltt and Agda (terms that are fixed if a condition is met). In each case, the type theory is equipped with a (meta-theoretic) face calculus, or shape layer, that governs their rules, and comes with intended semantics. We unify all these occurrences in a single framework where no new axioms or model constructions are needed, namely two-level type theory. This step, too, is free (semantically): the standard models of HoTT are automatically models of 2LTT, and the theory is conservative over HoTT. Extension types are definable, and the definition validates Riehl and Shulman's entire extension-type calculus: the rules hold strictly, and the postulated axioms, such as relative function extensionality, become theorems. In this way, every model of the base theory (HoTT) gives rise to a model of the same theory with extension types; the only genuine assumptions are which maps count as cofibrations. Conservativity makes the framework a tool for comparing type theories. We prove that cubical gluing, in a suitable formulation, is equivalent to univalence. On this basis, we suggest an approach toward the conjecture that cubical type theories are conservative over book HoTT, one of the central open problems of homotopy type theory. All results of the main body of the paper are auto-formalized in Agda --two-level, in a development that combines HoTT-internal arguments with reasoning that is external to HoTT.