A homotopy coherent cellular nerve for bicategories

Abstract

The subject of this paper is a nerve construction for bicategories introduced by Leinster, which defines a fully faithful functor from the category of bicategories and normal pseudofunctors to the category of presheaves over Joyal's category Θ2. We prove that the nerve of a bicategory is a 2-quasi-category (a model for (∞, 2)-categories due to Ara), and moreover that the nerve functor restricts to the right part of a Quillen equivalence between Lack's model structure for bicategories and a Bousfield localisation of Ara's model structure for 2-quasi-categories. We deduce that Lack's model structure for bicategories is Quillen equivalent to Rezk's model structure for (2, 2)-Θ-spaces on the category of simplicial presheaves over Θ2.To this end, we construct the homotopy bicategory of a 2-quasi-category, and prove that a morphism of 2-quasi-categories is an equivalence if and only if it is essentially surjective on objects and fully faithful. We also prove a Quillen equivalence between Ara's model structure for 2-quasi-categories and the Hirschowitz-Simpson-Pellissier model structure for quasi-categoryenriched Segal categories, from which we deduce a few more results about 2-quasi-categories, including a conjecture of Ara concerning weak equivalences of 2-categories.

Nerves of 2-categories and 2-categorification of $(\infty,2)$-categories

“…This formalizes the idea that (∞, 2)-categories generalize 2-categories, and that the homotopy theory of 2-categories completely embeds in that of (∞, 2)-categories. Similar ideas working with (∞, 2)-categories in the form of Θ 2 -sets are pursued in [Cam19].…”

Nerves and cones of free loop-free ω-categories

“…The realization of an ω-category D as a weak higher category is traditionally implemented by some kind of nerve construction N D. Efforts towards defining homotopical nerve constructions include work by different groups of authors and for different models of weak higher categories, e.g. [Ver08a,OR19,Cam20,GHL19,Mos20,Lou21].…”

On $\infty$-cosmoi of bicategories

“…Thus N B(x, y) ∼ = Hom N B (x, y), proving the essential commutativity of (6.1). Campbell proves that a 2-quasi-category is equivalent to the homotopy coherent cellular nerve of a bicategory if and only if each of its hom-quasi-categories Hom X (x, y) are equivalent to nerves of categories [Ca20,7.28].…”

“…both adjoints preserve binary products[Ca20, 6.27,6.29]. It follows that the right adjoint preserves exponentials -for A, B ∈ Bicat, N (B A ) ∼ = N B N A -and more generally that N (B HoX ) ∼ = N B X for any bicategory B and 2-quasicategory X.The essential image of the homotopy coherent cellular nerve functor can be characterized using an essentially commutative diagram of right adjoint functors (6.1)…”