Abstract
The goal of this paper is to provide the last equivalence needed in order to identify all known models for (Formula presented.) -categories. We do this by showing that Verity's model of saturated 2-trivial complicial sets is equivalent to Lurie's model of (Formula presented.) -bicategories, which, in turn, has been shown to be equivalent to all other known models for (Formula presented.) -categories.
A key technical input is given by identifying the notion of (Formula presented.) -bicategories with that of weak (Formula presented.) -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 (Formula presented.) -bicategory retains enough information to detect thin 2-simplices.