On the geometry of 2-categories and their classifying spaces
type: Item
instance of: scholarly article
title: On the geometry of 2-categories and their classifying spaces
zbMATH Open document ID: 1037.18003
DOI: 10.1023/B:KTHE.0000006921.50151.00
published in: \(K\)-Theory
publication date: 2004-02-03T00:00:00Z
review text: Quillen's definition of the classifying space of a category \({\mathcal C}\) is \(| \text{Ner}({\mathcal C})|\), the geometric realization of Grothendieck's nerve of \({\mathcal C}\). For any 2-category \({\mathcal C}\) there are two different simplicial objects ``nerves of \({\mathcal C}\)'' which can be functorially associated to \({\mathcal C}\) and the proposal of this interesting paper is to explore the relationship between the geometric realizations of these simplicial objects. Segal's definition of the classifying space of a topological category provides, as an instance, a definition of classifying space of \({\mathcal C}\), \(B{\mathcal C}\), but the geometric realization of the Duskin-Street geometric nerve of the 2-category \({\mathcal C}\), \(|\Delta{\mathcal C}|\), also provides such a definition of classifying space of \({\mathcal C}\). The main result of the paper proves that there is a natural homotopy equivalence \(B{\mathcal C}\rightarrow |\Delta{\mathcal C}|\). Any one of these realizations can thus be taken as the classifying space \(B{\mathcal C}\) of the 2-category. This result then extends to 2-categories Quillen's definition of the classifying space of a category and, since monoidal categories may be identified with 2-categories with only one object, the paper will quite possibly be of interest to researches in K-theory and higher-dimensional category theory. As an application of this result and using Gray's notion of homotopy fiber 2-category, in the paper it is also proved a direct extension of Quillen's ``Theorem A'' to 2-categories: A 2-functor \(F:{\mathcal C}\rightarrow {\mathcal C}'\) induces a homotopy equivalence \(BF:B{\mathcal C}\rightarrow B{\mathcal C}'\) if Gray's homotopy fiber \(y'// F\) is contractible for every object \(y'\in {\mathcal C}'\).
reviewed by: Antonio R. Garzón
Mathematics Subject Classification ID: 18D05
Mathematics Subject Classification ID: 18D10
Mathematics Subject Classification ID: 55U40
Mathematics Subject Classification ID: 55P15
Mathematics Subject Classification ID: 55U10
Mathematics Subject Classification ID: 18G30
zbMATH DE Number: 2037061
zbMATH Keywords: 2-category
zbMATH Keywords: monoidal category
zbMATH Keywords: simplicial space
zbMATH Keywords: classifying space
zbMATH Keywords: nerve
zbMATH Keywords: homotopy equivalence
author: Manuel Bullejos
author: Antonio M. Cegarra
MaRDI profile type: Publication
full work available at URL: b:kthe.0000006921.50151.00
OpenAlex ID: W2164858042
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description: scientific article; zbMATH DE number 2037061