A Type Theory for Strictly Unital $\infty$Categories
Abstract
We use typetheoretic techniques to present an algebraic theory of $\infty$categories with strict units. Starting with a known typetheoretic presentation of fully weak $\infty$categories, in which terms denote valid operations, we extend the theory with a nontrivial definitional equality. This forces some operations to coincide strictly in any model, yielding the strict unit behaviour. We make a detailed investigation of the metatheoretic 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 strictlyunital 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 $\infty$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 $\infty$category rather than additional structure.
 Publication:

arXiv eprints
 Pub Date:
 July 2020
 DOI:
 10.48550/arXiv.2007.08307
 arXiv:
 arXiv:2007.08307
 Bibcode:
 2020arXiv200708307F
 Keywords:

 Computer Science  Logic in Computer Science
 EPrint:
 46 pages