Strictly Associative and Unital $\infty$-Categories as a Generalized Algebraic Theory

Abstract: We present the first definition of strictly associative and unital $\infty$-category. Our proposal takes the form of a generalized algebraic theory, with operations that give the composition and coherence laws, and equations encoding the strict associative and unital structure. The key technical idea of the paper is an equality generator called insertion, which can ``insert'' an argument context into the head context, simplifying the syntax of a term. The equational theory is defined by a reduction relation, and we study its properties in detail, showing that it yields a decision procedure for equality. Expressed as a type theory, our model is well-adapted for generating and verifying efficient proofs of higher categorical statements. We illustrate this via an OCaml implementation, and give a number of examples, including a short encoding of the syllepsis, a 5-dimensional homotopy that plays an important role in the homotopy groups of spheres.