Categories and orbispaces

Abstract

Constructing and manipulating homotopy types from categorical input data has been an important theme in algebraic topology for decades. Every category gives rise to a “classifying space”, the geometric realization of the nerve. Up to weak homotopy equivalence, every space is the classifying space of a small category. More is true: the entire homotopy theory of topological spaces and continuous maps can be modeled by categories and functors. We establish a vast generalization of the equivalence of the homotopy theories of categories and spaces: small categories represent refined homotopy types of orbispaces whose underlying coarse moduli space is the traditional homotopy type hitherto considered.

A global equivalence is a functor $\mathrm{\Phi }:\mathcal{C}\to \mathcal{D}$ between small categories with the following property: for every finite group $G$, the functor $G\mathrm{\Phi }:G\mathcal{C}\to G\mathcal{D}$ induced on categories of $G$–objects is a weak equivalence. We show that the global equivalences are part of a model structure on the category of small categories, which is moreover Quillen equivalent to the homotopy theory of orbispaces in the sense of Gepner and Henriques. Every cofibrant category in this global model structure is opposite to a complex of groups in the sense of Haefliger.