On the equivalence of all models for (∞,2)$(\infty,2)$‐categories

Abstract

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. 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 2category of an ∞-bicategory retains enough information to detect thin 2-simplices.

Models for $(\infty,n)$-categories with discreteness conditions, I

“…Equivalences between some of these different models have been given by the author and Rezk [10], [11], Haugseng [17], and Doherty, Kapulkin, and Maehara [14], and, using an axiomatic approach, Barwick and Schommer-Pries [3]. In the special case when n = 2, there are further results by the author with Ozornova and Rovelli [9], and by Gagna, Harpaz, and Lanari [16]. Results when n = 1 are now well-established, and include the comparisons of Barwick and Kan [2], the author [8], Dugger and Spivak [15], Joyal [21], Joyal and Tierney [22], Lurie [23], and the axiomatic approach of Toën [34].…”

Lax monoidal adjunctions, two‐variable fibrations and the calculus of mates

“…Remark An explicit description of the fibrant objects in $$\mathrm{sSet}^{\mathrm{sc}}$$ in terms of lifting properties has been obtained by Gagna, Harpaz and Lanari in [6]. …”

Model independence of $(\infty,2)$-categorical nerves

“…was considered by Harpaz-Nuiten-Prasma in [HNP19, §2] and shown to be a right Quillen embedding by Gagna-Harpaz-Lanari in [GHL22,Prop. 8.2,8.3].…”

“…Ara's 2-quasi-categories [Ara14], and 2-comical sets [CKM20,DKM21], as well as categories strictly enriched over a model of (∞, 1)-categories [Lur09b,BR13,BR20]. In the past few years, the proof that all models are equivalent was completed, combining work by Lurie [Lur09b], Bergner-Rezk [BR13,BR20], Ara [Ara14], Gagna-Harpaz-Lanari [GHL22], Campion-Doherty-Kapulkin-Maehara [CKM20,DKM21].…”

“…In the recent years a well-behaved embedding has also been constructed in most models of (∞, 2)-categories presented by model categories in the form of a homotopical functor that is homotopically fully faithful, which is in addition either right Quillen or fully faithful (but generally not both). This was achieved by the second and third author [OR21] for 2-complicial sets, by Campbell [Cam20] for 2quasi-categories, by Gagna-Harpaz-Lanari for scaled simplicial sets [GHL22], and by the first author [Mos20] for 2-fold complete Segal spaces. For (∞, 2)-categories presented by categories enriched over a model of (∞, 1)-categories, this can be done by base-change along a suitable 1-dimensional nerve.…”