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 with the following property: for every finite group
, the functor
induced on
categories of
–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.
