$\ddot{I}‰$-weak equivalences between weak $\ddot{I}‰$-categories

Abstract

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

33 pages. Major revision. Comments welcome!