On the equivalence of all models for $(\infty,2)$-categories
Résumé
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 Cisinski--Olschok'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.