Geometric Realizations of Tricategories

Abstract

Any tricategory characteristically has associated various simplicial or pseudo-simplicial objects. This paper explores the relationship amongst three of them: the pseudo-simplicial bicategory so-called Grothendieck nerve of the tricategory, the simplicial bicategory termed its Segal nerve, and the simplicial set called its Street geometric nerve, and it proves the fact that the geometric realizations of all of these possible candidate 'nerves of the tricategory' are homotopy equivalent. Our results provide coherence for all reasonable extensions to tricategories of Quillen's definition of the 'classifying space' of a category as the geometric realization of the category's Grothendieck nerve. Many properties of the classifying space construction for tricategories may be easier to establish depending on the nerve used for realizations. For instance, by using Grothendieck nerves we state and prove the precise form in which the process of taking classifying spaces transports tricategorical coherence to homotopy coherence. Segal nerves allow us to obtain an extension to bicategories of the results by Mac Lane, Stasheff, and Fiedorowicz about the relation between loop spaces and monoidal or braided monoidal categories by showing that the group completion of the classifying space of a bicategory enriched with a monoidal structure is, in a precise way, a loop space. With the use of geometric nerves, we obtain genuine simplicial sets whose simplices have a pleasing geometrical description in terms of the cells of the tricategory and, furthermore, we obtain an extension of the results by Joyal, Street, and Tierney about the classification of braided categorical groups and their relationship with connected, simply connected homotopy 3-types, by showing that, via the classifying space construction, bicategorical groups are a convenient algebraic model for connected homotopy 3-types.