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2019 Categories and orbispaces
Stefan Schwede
Algebr. Geom. Topol. 19(6): 3171-3215 (2019). DOI: 10.2140/agt.2019.19.3171

Abstract

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 Φ:CD between small categories with the following property: for every finite group G, the functor GΦ:GCGD induced on categories of G–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.

Citation

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Stefan Schwede. "Categories and orbispaces." Algebr. Geom. Topol. 19 (6) 3171 - 3215, 2019. https://doi.org/10.2140/agt.2019.19.3171

Information

Received: 28 October 2018; Revised: 4 February 2019; Accepted: 23 February 2019; Published: 2019
First available in Project Euclid: 29 October 2019

zbMATH: 07142628
MathSciNet: MR4023338
Digital Object Identifier: 10.2140/agt.2019.19.3171

Subjects:
Primary: 55P91

Keywords: category , global homotopy theory , orbispace

Rights: Copyright © 2019 Mathematical Sciences Publishers

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Vol.19 • No. 6 • 2019
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