$\omega$-weak equivalences between weak $\omega$-categories

TL;DR

This paper investigates $oldsymbol{ extomega}$-weak equivalences between weak $oldsymbol{ extomega}$-categories, establishing properties like the 2-out-of-3 rule and connecting to existing models of strict $oldsymbol{ extomega}$-categories.

Contribution

It defines and analyzes $oldsymbol{ extomega}$-weak equivalences in the context of weak $oldsymbol{ extomega}$-categories, linking them to known models and extending the concept to a broader class of functors.

Findings

$oldsymbol{ extomega}$-weak equivalences satisfy the 2-out-of-3 property.

The class of $oldsymbol{ extomega}$-weak equivalences coincides with known weak equivalences for strict $oldsymbol{ extomega}$-categories.

A generalized class of weak $oldsymbol{ extomega}$-functors also satisfies the 2-out-of-3 property.

Abstract

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