A Type Theory for Strictly Unital ∞-Categories

Abstract

We use type-theoretic techniques to present an algebraic theory of ∞-categories with strict units. Starting with a known type-theoretic presentation of fully weak ∞-categories, in which terms denote valid operations, we extend the theory with a non-trivial definitional equality. This forces some operations to coincide strictly in any model, yielding the strict unit behaviour. We make a detailed investigation of the meta-theoretic properties of this theory. We give a reduction relation that generates definitional equality, and prove that it is confluent and terminating, thus yielding the first decision procedure for equality in a strictly-unital setting. Moreover, we show that our definitional equality relation identifies all terms in a disc context, providing a point comparison with a previously proposed definition of strictly unital ∞-category. We also prove a conservativity result, showing that every operation of the strictly unital theory indeed arises from a valid operation in the fully weak theory. From this, we infer that strict unitality is a property of an ∞-category rather than additional structure.