2019/11/05 by Gagna, Andrea, Harpaz, Yonatan, Lanari, Edoardo
#18N40 #18N50 #18N65 #55U10 #55U35 #Algebraic Topology (math.AT) #Category Theory (math.CT) #FOS: Mathematics
paper · doi:10.48550/arxiv.1911.01905
The goal of this paper is to provide the last equivalence needed in order to identify all known models for (∞,2)-categories. We do this by showing that Verity's model of saturated 2-trivial complicial sets is equivalent to Lurie's model of ∞-bicategories, which, in turn, has been shown to be equivalent to all other known models for (∞,2)-categories. A key technical input is given by identifying the notion of ∞-bicategories with that of weak ∞-bicategories, a step which allows us to understand Lurie's model structure in terms of Cisinski--Olschok's theory. This description of ∞-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 ∞-bicategory retains enough information to detect thin 2-simplices.