Comparison of the $n$-categorical nerves
Abstract
Our aim is to compare three nerve functors for strict -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 -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 -categories.
Keywords
Cite
@article{arxiv.2010.00266,
title = {Comparison of the $n$-categorical nerves},
author = {Dimitri Ara and Georges Maltsiniotis},
journal= {arXiv preprint arXiv:2010.00266},
year = {2022}
}
Comments
42 pages, in French, v2: minor modifications, numbering has changed in section 2 due to addition of Remark 2.10, journal version