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
–types.
Keywords
monoidal bicategory, tricategory, nerve, classifying space,
homotopy type, loop space
