#### Volume 10, issue 1 (2010)

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Nerves and classifying spaces for bicategories

### Pilar Carrasco, Antonio M Cegarra and Antonio R Garzón

Algebraic & Geometric Topology 10 (2010) 219–274
##### Abstract

This paper explores the relationship amongst the various simplicial and pseudosimplicial objects characteristically associated to any bicategory $\mathsc{C}$. It proves the fact that the geometric realizations of all of these possible candidate “nerves of $\mathsc{C}$” are homotopy equivalent. Any one of these realizations could therefore be taken as the classifying space $B\mathsc{C}$ 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.

##### Keywords
category, bicategory, monoidal category, pseudosimplicial category, nerve, classifying space, homotopy type, simplicial set
Primary: 18D05
Secondary: 55U40
##### Publication
Received: 30 March 2009
Accepted: 8 November 2009
Published: 12 February 2010