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2010 Nerves and classifying spaces for bicategories
Pilar Carrasco, Antonio M Cegarra, Antonio R Garzón
Algebr. Geom. Topol. 10(1): 219-274 (2010). DOI: 10.2140/agt.2010.10.219

Abstract

This paper explores the relationship amongst the various simplicial and pseudosimplicial 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.

Citation

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Pilar Carrasco. Antonio M Cegarra. Antonio R Garzón. "Nerves and classifying spaces for bicategories." Algebr. Geom. Topol. 10 (1) 219 - 274, 2010. https://doi.org/10.2140/agt.2010.10.219

Information

Received: 30 March 2009; Accepted: 8 November 2009; Published: 2010
First available in Project Euclid: 21 December 2017

zbMATH: 1183.18005
MathSciNet: MR2602835
Digital Object Identifier: 10.2140/agt.2010.10.219

Subjects:
Primary: 18D05
Secondary: 55U40

Rights: Copyright © 2010 Mathematical Sciences Publishers

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Vol.10 • No. 1 • 2010
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