2-nerves for bicategories

We describe a Cat-valued nerve of bicategories, which associates to every bicategory a simplicial object in Cat, called the 2-nerve. This becomes the object part of a 2-functor N : NHom → [∆,Cat], where NHom is a 2-category whose objects are bicategories and whose 1-cells are normal homomorphisms of bicategories. The 2-functor N is fully faithful and has a left biadjoint, and we characterize its image. The 2-nerve of a bicategory is always a weak 2-category in the sense of Tamsamani, and we show that NHom is biequivalent to a certain 2-category whose objects are Tamsamani weak 2-categories. This paper concerns a notion of “2-nerve”, or Cat-valued nerve, of bicategories. To every category, one can associate its nerve; this is the simplicial set whose 0-simplices are the objects, whose 1-simplices are the morphisms, and whose n-simplices are the composable n-tuples of morphisms. The face maps encode the domains and codomains of morphisms, the composition law, and the associativity property, while the degeneracies record information about the identities. This construction is the object part of a functor N : Cat1 → [∆op,Set] from the category of categories and functors, to the category of simplicial sets. This functor is fully faithful and has a left adjoint. It arises in a natural way, as the “singular functor” (see Section 1 below) of the inclusion J : ∆→ Cat1 in Cat1 of the full subcategory ∆ consisting of the non-empty finite ordinals. One can characterize the simplicial sets which lie in the image of the nerve functor as those for which certain diagrams are pullbacks. As observed by Street, one may define the nerve of a bicategory as the simplicial set whose 0-simplices are the objects, whose 1-simplices are the morphisms, whose 2-simplices consist of a composable pair f and g and a 2-cell gf → h, and so on. In this way, the category Bicat1 of bicategories and normal lax functors becomes a full subcategory of [∆op,Set]; here a normal lax functor preserves the identities strictly, but preserves composition only up to coherent, but not necessarily invertible, comparison maps. In the important special case of invertible comparison maps, one speaks rather of normal homomorphisms. Once again this nerve functor Bicat1 → [∆op,Set] is a singular functor, this time of the inclusion ∆→ Bicat1, where the non-empty finite ordinals are now seen as locally discrete (no non-identity 2-cells) bicategories. The image of this nerve functor was characterized explicitly in [4]. ∗The hospitality of Macquarie University and the support of the Australian Research Council are gratefully acknowledged. †The support of the Australian Research Council is gratefully acknowledged.