No abstract

Abstract. This paper proves three different coherence theorems for symmetric monoidal bicategories. First, we show that in a free symmetric monoidal bicategory every diagram of 2-cells commutes. Second, we show that this implies that the free symmetric monoidal bicategory on one object is equivalent, as a symmetric monoidal bicategory, to the discrete symmetric monoidal bicategory given by the disjoint union of the symmetric groups. Third, we show that every symmetric monoidal bicategory is equivalent to a strict one.We give two topological applications of these coherence results. First, we show that the classifying space of a symmetric monoidal bicategory can be equipped with an E∞ structure. Second, we show that the fundamental 2-groupoid of an En space, n ≥ 4, has a symmetric monoidal structure. These calculations also show that the fundamental 2-groupoid of an E 3 space has a sylleptic monoidal structure.

ABSTRACT. We establish an equivalence of homotopy theories between symmetric monoidal bicategories and connective spectra. For this, we develop the theory of Γ-objects in 2-categories. In the course of the proof we establish strictification results of independent interest for symmetric monoidal bicategories and for diagrams of 2-categories.

In a previous paper, we showed that a discrete version of the S•-construction gives an equivalence of categories between unital 2-Segal sets and augmented stable double categories. Here, we generalize this result to the homotopical setting, by showing that there is a Quillen equivalence between a model category for unital 2-Segal objects and a model category for augmented stable double Segal objects which is given by an S•-construction. We show that this equivalence fits together with the result in the discrete case and briefly discuss how it encompasses other known S•-constructions.

We define a model structure on the category GCat of small categories with an action by a discrete group G by lifting the Thomason model structure on Cat. We show there is a Quillen equivalence between GCat with this model structure and GTop with the standard model structure.