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