Meaning explanations at higher dimension

Abstract

Martin-Löf’s intuitionistic type theory is a widely-used framework for constructive mathematics and computer programming. In its most popular form, type theory consists of a collection of inference rules inductively defining formal proofs. These rules are justified by Martin-Löf’s meaning explanations, which extend the Brouwer–Heyting–Kolmogorov interpretation of connectives to a rich collection of types, and therefore provide a constructive realizability interpretation of formal proofs.Around 2005, researchers noticed that the rules of type theory also admit homotopy-theoretic models, and subsequently extended type theory with constructs inspired by these models: higher inductive types and Voevodsky’s univalence axiom. Although the resulting homotopy type theory has proved useful for homotopy-theoretic reasoning, it lacks a constructive interpretation. In this overview, we discuss a cubical generalization of the meaning explanations of type theory that constitutes an inherently constructive account of higher-dimensional structure in types.