Up to weak homotopy equivalence, every space is the classifying space of a category. But much more is true: in view of e.g., [R. W. Thomason, Cah. Topologie Géom. Différ. Catégoriques 21, 305–324 (1980; Zbl 0473.18012)] the entire homotopy theory of topological spaces and continuous maps can be modeled by categories and functors.
The author establishes a vast generalization of the equivalence of the homotopy theories of categories and spaces: small categories represent refined homotopy types of orbispaces introduced by A. Gepner and D. Henriques [“Homotopy theory of orbispaces”, Preprint, arXiv:math/0701916] whose underlying coarse moduli space is the traditional homotopy type hitherto considered.
A global equivalence is a functor \(\Phi : \mathcal{C}\to \mathcal{D}\) between small categories with the following property: for every finite group \(G\), the functor \(G\Phi : G\mathcal{C}\to G\mathcal{D}\) induced on categories of \(G\)-objects is a weak equivalence. The author shows that the global equivalences are part of a model structure on the category of small categories, which is moreover Quillen equivalent [D. G. Quillen, Homotopical algebra. Berlin-Heidelberg-New York: Springer-Verlag (1967; Zbl 0168.20903)] to the homotopy theory of orbispaces.