Model independence of $(\infty,2)$-categorical nerves

Abstract

For most models of (∞, 2)-categories an embedding of the ∞-category of 2-categories into that of (∞, 2)-categories has been constructed in the form of a nerve construction of some flavor. We prove that all those nerve embeddings induce equivalent functors, modulo change of model. We also show that all the nerve embeddings realize the ∞-category of 2-categories as the sub-∞-category of (∞, 2)-categories that are local with respect to a certain class of maps.

What is an equivalence in a higher category?

“…can be realized as a nerve-categorification Quillen reflection in most models of (∞, 𝑛)-categories presented by model categories. For 𝑛 = 0, this can be easily implemented in Kan complexes, and for 𝑛 = 1, this can be done in quasi-categories, naturally marked quasi-categories, and complete Segal spaces (see, for example, [88,Section 4.2]). For 𝑛 = 2, this was done in saturated 2-complicial sets [92], in 2-quasi-categories [23], in 2-fold complete Segal spaces [86] Consider the functor…”