Naturality for higher-dimensional path types

Abstract

We define a naturality construction for the operations of weak omega-categories, as a meta-operation in a dependent type theory. Our construction has a geometrical motivation as a local tensor product with a directed interval, and behaves logically as a globular analogue of Reynolds parametricity. Our construction operates as a ``power tool'' to support construction of terms with geometrical structure, and we use it to define composition operations for cylinders and cones in omega-categories. The machinery can generate terms of high complexity, and we have implemented our construction in a proof assistant, which verifies that the generated terms have the correct type. All our results can be exported to homotopy type theory, allowing the explicit computation of complex path type inhabitants.