On the equivalence of all models for (∞,2)$(\infty,2)$‐categories
Andrea Gagna, Yonatan Harpaz, Edoardo Lanari
Abstract
The goal of this paper is to provide the last equivalence needed in order to identify all known models for ( ∞ , 2 ) $(\infty,2)$ -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 ( ∞ , 2 ) $(\infty,2)$ -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. Several of our arguments use 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, give two equivalent descriptions of it, and show that the homotopy 2-category of an ∞ $\infty$ -bicategory retains enough information to detect thin 2-simplices.