Abstract
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.
Citation
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
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