On the equivalence of all models for $(\infty,2)$categories
Abstract
The goal of this paper is to provide the last equivalence needed in order to identify all known models for $(\infty,2)$categories. We do this by showing that Verity's model of saturated $2$trivial complicial sets is equivalent to Lurie's model of $\infty$bicategories, which, in turn, has been shown to be equivalent to all other known models for $(\infty,2)$categories. A key technical input is given by identifying the notion of $\infty$bicategories with that of weak $\infty$bicategories, a step which allows us to understand Lurie's model structure in terms of CisinskiOlschok's theory. This description of $\infty$bicategories, which may be of independent interest, is proved using tools coming from a new theory of outer (co)cartesian fibrations, further developed in a companion paper. In the last part of the paper we construct a homotopically fully faithful scaled simplicial nerve functor for $2$categories, we give two equivalent descriptions of it, and we show that the homotopy $2$category of an $\infty$bicategory retains enough information to detect thin $2$simplices.
 Publication:

arXiv eprints
 Pub Date:
 November 2019
 DOI:
 10.48550/arXiv.1911.01905
 arXiv:
 arXiv:1911.01905
 Bibcode:
 2019arXiv191101905G
 Keywords:

 Mathematics  Algebraic Topology;
 Mathematics  Category Theory;
 18N65;
 18N50;
 18N40;
 55U10;
 55U35
 EPrint:
 Many typos fixed, some details added and exposition clarified. To appear on the JLMS