Algebraic & Geometric Topology

Nerves and classifying spaces for bicategories

Pilar Carrasco, Antonio M Cegarra, and Antonio R Garzón

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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.

Article information

Algebr. Geom. Topol., Volume 10, Number 1 (2010), 219-274.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 18D05: Double categories, 2-categories, bicategories and generalizations
Secondary: 55U40: Topological categories, foundations of homotopy theory

category bicategory monoidal category pseudosimplicial category nerve classifying space homotopy type simplicial set


Carrasco, Pilar; Cegarra, Antonio M; Garzón, Antonio R. Nerves and classifying spaces for bicategories. Algebr. Geom. Topol. 10 (2010), no. 1, 219--274. doi:10.2140/agt.2010.10.219.

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