Comparison of the $n$-categorical nerves

Abstract

Our aim is to compare three nerve functors for strict $n$-categories: the Street nerve, the cellular nerve and the multi-simplicial nerve. We show that these three functors are equivalent in some appropriate sense. In particular, the classes of $n$-categorical weak equivalences that they define coincide: they are the Thomason equivalences. We give two applications of this result: the first one states that a Dyer-Kan-type equivalence for Thomason equivalences is a Thomason equivalence; the second one, fundamental, is the stability of the class of Thomason equivalences under the dualities of the category of strict $n$-categories.