We define a naturality construction for the operations of weak ω-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 ω-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.

We present the proof assistant homotopy.io for working with finitely-presented semistrict higher categories. The tool runs in the browser with a point-and-click interface, allowing direct manipulation of proof objects via a graphical representation. We describe the user interface and explain how the tool can be used in practice. We also describe the essential subsystems of the tool, including collapse, contraction, expansion, typechecking, and layout, as well as key implementation details including data structure encoding, memoisation, and rendering. These technical innovations have been essential for achieving good performance in a resource-constrained setting.