On the geometry of 2-categories and their classifying spaces
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 \(y'\in {\mathcal C}'\).
scientific article; zbMATH DE number 2037061
scientific article