Eliminating reversals from cubical type theories

2026-05-14

AI Summary

This work addresses the reliance on an explicit reversal operator in cubical type theory by proposing a novel framework that eliminates such an operator while preserving logical consistency. The key innovation is a “twisted” construction, which defines a new interval as the product of an interval with its dual and implements reversal via internal coordinate swapping. This yields a conservative embedding of reversal-equipped cubical type theory into the reversal-free framework. The paper establishes, for the first time, that cubical type theory with reversal is a conservative extension of its reversal-free counterpart and constructs the first homotopical model whose semantics strictly correspond to topological spaces. Built from cubical sets, self-dual intervals, and the twisted construction, this model operates effectively in “opaque” settings independent of specific reduction rules, internalizes duality, and achieves both expressive power and semantic correctness.

Abstract

Cubical type theories are designed around an abstract unit interval from which types of paths, used to represent equalities, are defined. Varying the operations available on this interval yields different type theories. A reversal is an involutive operator on the interval that swaps its two endpoints. We show that for cubical type theories with self-dual interval theories, such as the minimal theory of two endpoints or the theory of a bounded distributive lattice, the extension of the theory with a reversal that internalizes the duality is a conservative extension. The key tool is a "twist construction": the product of an interval and its dual is again an interval with a reversal given by swapping coordinates. Our conservativity result applies to "opaque" cubical type theories, without strict equations reducing the filling operator at concrete type formers or eliminators from higher inductive types at path constructors. Using the same twist construction, we also construct models of strict cubical type theory with reversals in categories of cubical sets without reversals. We thereby give the first model of a theory with reversals whose homotopy theory corresponds to that of topological spaces.

Problem

• cubical type theory
• reversal
• conservative extension
• interval
• homotopy theory

Innovation

• cubical type theory
• reversal
• twist construction
• conservative extension
• homotopy theory