Nerves and classifying spaces for bicategories

TL;DR

This paper establishes that various nerve constructions of bicategories have homotopy equivalent realizations, defining their classifying space, and extends Thomason's Homotopy Colimit Theorem to bicategories, linking diagram colimits to bicategorical structures.

Contribution

It proves the homotopy equivalence of different nerve realizations of bicategories and extends Thomason's theorem to bicategories, providing a unified framework for classifying spaces.

Findings

All nerve realizations are homotopy equivalent.

Classifying space of a bicategory can be chosen from any nerve realization.

Extension of Thomason's Homotopy Colimit Theorem to bicategories.

Abstract

This paper explores the relationship amongst the various simplicial and pseudo-simplicial objects characteristically associated to any bicategory C. It proves the fact that the geometric realizations of all of these possible candidate `nerves of C' are homotopy equivalent. Any one of these realizations could therefore be taken as the classifying space BC 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.