We describe a Cat-valued nerve of bicategories, which associates to every bicategory a simplicial object in Cat, called the 2-nerve. We define a 2-category NHom whose objects are bicategories and whose 1-cells are normal homomorphisms of bicategories, in such a way that the 2-nerve construction becomes a full embedding of NHom in the 2-category of simplicial objects in Cat. This embedding 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.Comment: 23 page

In this paper we obtain several model structures on DblCat, the category of small double categories. Our model structures have three sources. We first transfer across a categorification-nerve adjunction. Secondly, we view double categories as internal categories in Cat and take as our weak equivalences various internal equivalences defined via Grothendieck topologies. Thirdly, DblCat inherits a model structure as a category of algebras over a 2-monad. Some of these model structures coincide and the different points of view give us further results about cofibrant replacements and cofibrant objects. As part of this program we give explicit descriptions for and discuss properties of free double categories, quotient double categories, colimits of double categories, horizontal nerve and horizontal categorification.18D05, 18G55; 55P99, 55U10

Abstract. For each n ≥ 1, we introduce two new Segal-type models of ntypes of topological spaces: weakly globular n-fold groupoids, and a lax version of these. We show that any n-type can be represented up to homotopy by such models via an explicit algebraic fundamental n-fold groupoid functor. We compare these models to Tamsamani's weak n-groupoids, and extract from them a model for (k − 1)-connected n-types.

We introduce a new model of connected (n + 1)-types which consists of a subcategory of cat n -groups. We study the homotopical properties of this model; this includes an algebraic description of the Postnikov decomposition and of the homotopy groups of its objects. Further, we use this model to build a comparison functor from cat n -groups to Tamsamani weak (n + 1)-groupoids which preserves the homotopy type. As an application, we obtain a homotopical semistrictification result for those Tamsamani weak (n + 1)-groupoids whose classifying space is path-connected.