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 $\Theta_2$. We prove that the nerve of a bicategory is a $2$-quasi-category (a model for $(\infty,2)$-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 $(2,2)$-$\Theta$-spaces on the category of simplicial presheaves over $\Theta_2$. 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-category-enriched 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.