ω-weak equivalences between weak ω-categories

Abstract

We study ω-weak equivalences between weak ω-categories in the sense of Batanin–Leinster. Our ω-weak equivalences are strict ω-functors satisfying essential surjectivity in every dimension, and when restricted to those between strict ω-categories, they coincide with the weak equivalences in the model category of strict ω-categories defined by Lafont, Métayer, and Worytkiewicz. We show that the class of ω-weak equivalences has the 2-out-of-3 property. We also consider a generalisation of ω-weak equivalences, defined as weak ω-functors (in the sense of Garner) satisfying essential surjectivity, and show that this class also has the 2-out-of-3 property.