Categories and orbispaces

TL;DR

This paper extends the classical correspondence between categories and spaces to orbispaces, establishing a model structure on small categories that captures their refined homotopy types and relates to orbispaces via a Quillen equivalence.

Contribution

It introduces a global model structure on small categories that models the homotopy theory of orbispaces, generalizing the classical space-category correspondence.

Findings

Global equivalences induce weak equivalences of nerves of G-objects for all finite groups G.

The model structure on small categories is Quillen equivalent to the homotopy theory of orbispaces.

Every cofibrant category is opposite to a complex of groups as defined by Haefliger.

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 between small categories that induces weak equivalences of nerves of the categories of $G$-objects, for all finite groups $G$. 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.