Algebraic & Geometric Topology
- Algebr. Geom. Topol.
- Volume 10, Number 1 (2010), 219-274.
Nerves and classifying spaces for bicategories
This paper explores the relationship amongst the various simplicial and pseudosimplicial objects characteristically associated to any bicategory . It proves the fact that the geometric realizations of all of these possible candidate “nerves of ” are homotopy equivalent. Any one of these realizations could therefore be taken as the classifying space 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.
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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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. https://projecteuclid.org/euclid.agt/1513882312