Nerves and classifying spaces for bicategories

Abstract

This paper explores the relationship amongst the various simplicial and pseudosimplicial objects characteristically associated to any bicategory $\mathcal{C}$. It proves the fact that the geometric realizations of all of these possible candidate “nerves of $\mathcal{C}$” are homotopy equivalent. Any one of these realizations could therefore be taken as the classifying space $B\mathcal{C}$ of the bicategory. Its other major result proves a direct extension of Thomason’s “Homotopy Colimit Theorem” to bicategories: When the homotopy colimit construction is carried out on a diagram of spaces obtained by applying the classifying space functor to a diagram of bicategories, the resulting space has the homotopy type of a certain bicategory, called the “Grothendieck construction on the diagram”. Our results provide coherence for all reasonable extensions to bicategories of Quillen’s definition of the “classifying space” of a category as the geometric realization of the category’s Grothendieck nerve, and they are applied to monoidal (tensor) categories through the elemental “delooping” construction.