  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.
