Comparing geometric realizations of tricategories

Abstract

This paper contains some contributions to the study of classifying spaces for tricategories, with applications to the homotopy theory of monoidal categories, bicategories, braided monoidal categories and monoidal bicategories. Any small tricategory has various associated simplicial or pseudosimplicial objects and we explore the relationship between three of them: the pseudosimplicial bicategory (so-called Grothendieck nerve) of the tricategory, the simplicial bicategory termed its Segal nerve and the simplicial set called its Street geometric nerve. We prove that the geometric realizations of all of these 'nerves of the tricategory' are homotopy equivalent. By using Grothendieck nerves we state the precise form in which the process of taking classifying spaces transports tricategorical coherence to homotopy coherence. Segal nerves allow us to prove that, under natural requirements, the classifying space of a monoidal bicategory is, in a precise way, a loop space. With the use of geometric nerves, we obtain simplicial sets whose simplices have a pleasing geometrical description in terms of the cells of the tricategory and we prove that, via the classifying space construction, bicategorical groups are a convenient algebraic model for connected homotopy 3-types.

The Catalan simplicial set

“…The possibility of taking natural transformations as 2-cells makes Cat not just a monoidal category, but a monoidal bicategory in the sense of [7]. Just as one can form a nerve of a monoidal category by viewing it as a one-object bicategory, so one can form a nerve of a monoidal bicategory by viewing it as a one-object tricategory (= weak 3-category), and in fact, various nerve constructions are possible-see [4]. The following definitions are specialisations of some of these nerves to the case of Cat.…”