Homotopy Theory of Orbispaces

2007/01/31 by André Henriques, David Gepner, Henriques, Andre +1 · 4 citations
Mathematics · #Homotopy and Cohomology in Algebraic Topology #Algebraic structures and combinatorial models #Advanced Topics in Algebra

paper · pdf · doi:10.48550/arxiv.math/0701916

Abstract

Given a topological group G, its orbit category Orb_G has the transitive G-spaces G/H as objects and the G-equivariant maps between them as morphisms. A well known theorem of Elmendorf then states that the category of G-spaces and the category of contravariant functors Func(Orb_G,Spaces) have equivalent homotopy theories. We extend this result to the context of orbispaces, with the role of Orb_G now played by a category whose objects are topological groups and whose morphisms are given by Hom(H,G) = Mono(H,G) x_G EG. On our way, we endow the category of topological groupoids with notions of weak equivalence, fibrant objects, and cofibrant objects, and show that it then shares many of the properties of a Quillen model category.

Cited by

• The homotopy type of a topological stack
• Quasi-elliptic cohomology and its Spectrum
• Sheaf Theory for Étale Geometric Stacks
• Topological Mathieu Moonshine

Related

• Orbispaces and their Mapping Spaces via Groupoids: A Categorical Approach
• Orbispaces and Orbifolds from the Point of View of the Borel Construction, a new Definition
• G-category versus orbifold category
• Algebraic models of homotopy types and the homotopy hypothesis
• Homotopical Algebra in Categories with Enough Projectives