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An orbispace is a space, specifically a topological stack, that is locally modeled on the homotopy quotient/action groupoid of a locally compact topological space by a rigid group action.

Orbispaces are to topological spaces what orbifolds are to manifolds.


Write OrbOrb for the global orbit category. Then its (∞,1)-presheaf (∞,1)-category PSh (Orb)PSh_\infty(Orb) is the (∞,1)-category of orbispaces. (Henriques-Gepner 07, Rezk 14, remark 1.5.1)


Relation of global equivariant homotopy theory

By the main theorem of (Henriques-Gepner 07, (4)) the (∞,1)-presheaves on the global orbit category are equivalently “cellular” topological stacks/topological groupoids (“orbispaces”), we might write this as

TopGrpd cellPSh (Orb). TopGrpd^{cell} \simeq PSh_\infty(Orb) \,.

Hence cellular topological stacks are equivalently the objects of equivariant homotopy theory.

See also (Rezk 14, p. 4 and section 7)

Relation to GG-equivariant homotopy theory

Fixing a compact topological group GG and writing BG*//G\mathbf{B}G \simeq \ast // G for its delooping stack (the moduli stack of GG-principal bundles), then the slice homotopy theory of topological stacks over BG\mathbf{B}G on the representable morphisms (those inducing closed monos on isotropy groups) is equivalently that of topological G-spaces (with their G-equivariant homotoy theoretical structure, see at equivariant Whitehead theorem):

TopGrpd /BG reprs.GSpaces TopGrpd_{/\mathbf{B}G}^{reprs.} \simeq G Spaces

(Henriques-Gepner 07, p.7)

Rezk-global equivariant homotopy theory:

cohesive (∞,1)-toposits (∞,1)-sitebase (∞,1)-toposits (∞,1)-site
global equivariant homotopy theory PSh (Glo)PSh_\infty(Glo)global equivariant indexing category GloGlo∞Grpd PSh (*) \simeq PSh_\infty(\ast)point
sliced over terminal orbispace: PSh (Glo) /𝒩PSh_\infty(Glo)_{/\mathcal{N}}Glo /𝒩Glo_{/\mathcal{N}}orbispaces PSh (Orb)PSh_\infty(Orb)global orbit category
sliced over BG\mathbf{B}G: PSh (Glo) /BGPSh_\infty(Glo)_{/\mathbf{B}G}Glo /BGGlo_{/\mathbf{B}G}GG-equivariant homotopy theory of G-spaces L weGTopPSh (Orb G)L_{we} G Top \simeq PSh_\infty(Orb_G)GG-orbit category Orb /BG=Orb GOrb_{/\mathbf{B}G} = Orb_G


A detailed but elementary approach via atlases can be found in

and another approach is discussed in

Yet another description is due to

  • Alexander Körschgen, A Comparison of two Models of Orbispaces, Homology, Homotopy and Applications, vol. 20(1), 2018, pp.329–358 (arXiv:1612.04267)

  • Stefan Schwede, Orbispaces, orthogonal spaces, and the universal compact Lie group, Mathematische Zeitschrift 294 (2020), 71-107 (arXiv:1711.06019)

The relation between the two versions (arbitrary group homomorphisms or subgroup inclusions) is discussed in

See also

  • Jacob Lurie, Section 3 of Elliptic cohomology III: Tempered Cohomology (pdf)

Last revised on June 3, 2020 at 09:43:29. See the history of this page for a list of all contributions to it.