Categories and orbispaces. (arXiv:1810.06632v1 [math.AT])

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 equi 查看全文>>