nLab orbispace

Idea

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 the action of a compact topological group. Thus orbispaces are to topological spaces roughly what orbifolds are to manifolds.

As to what this means precisely, there is a good deal of variance in the literature:

Following Haefliger 90, early references on orbispaces essentially mean just the topological version of orbifold (i.e. without considering smooth structure), which, in the language of étale stacks, means to consider topological groupoids/topological stacks instead of Lie groupoids/differentiable stacks (Haefliger 84, Haefliger 91, 5, Chen 01, Henriques 01, Henriques 05). Hence this use of the term “orbispace” is such as to complete the following patter:

$$\array{ \text{smooth manifold} & \text{topological manifold} \\ \text{orbifold} & {\color{blue}\text{orbispace}} \\ \text{Lie groupoid} & \text{topological groupoid} \\ \text{differentiable stack} & \text{topological stack} }$$

However, in Henriques-Gepner 07 it was suggested that orbispaces should be these topological groupoids but regarded in global homotopy theory as the (∞,1)-presheaves on a global orbit category which they represent:

$$\array{ TopologicalGroupoids &\overset{\prec}{\longrightarrow}& Orbispaces \\ \mathcal{X} &\mapsto& (G \mapsto Maps(\mathbf{B}G, \mathcal{X})) }$$

This use of the term has become adopted among (equivariant) homotopy theorists (e.g. Rezk 14, Koerschgen 16, Schwede 17, Lurie EllIII, Juran 20). But even here there are at least two variants of the definition to be distinguished, depending on whether the morphisms in the global orbit category are taken to be general or only injective group homomorphisms.

Definition

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

Properties

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

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

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

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

Relation to $G$-equivariant homotopy theory

Fixing a compact topological group $G$ and writing $\mathbf{B}G \simeq \ast // G$ for its delooping stack (the moduli stack of $G$-principal bundles), then the slice homotopy theory of topological stacks over $\mathbf{B}G$ 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):

$$TopGrpd_{/\mathbf{B}G}^{reprs.} \simeq G Spaces$$

(Henriques-Gepner 07, p.7)

Related concepts

• Elmendorf's theorem

• orbispace cohomology

Rezk-global equivariant homotopy theory:

cohesive (∞,1)-topos	its (∞,1)-site	base (∞,1)-topos	its (∞,1)-site
global equivariant homotopy theory $PSh_\infty(Glo)$	global equivariant indexing category $Glo$	∞Grpd $\simeq PSh_\infty(\ast)$	point
… sliced over terminal orbispace: $PSh_\infty(Glo)_{/\mathcal{N}}$	$Glo_{/\mathcal{N}}$	orbispaces $PSh_\infty(Orb)$	global orbit category
… sliced over $\mathbf{B}G$: $PSh_\infty(Glo)_{/\mathbf{B}G}$	$Glo_{/\mathbf{B}G}$	$G$-equivariant homotopy theory of G-spaces $L_{we} G Top \simeq PSh_\infty(Orb_G)$	$G$-orbit category $Orb_{/\mathbf{B}G} = Orb_G$

References

Orbispaces as the topological version of orbifolds (in particular as topological groupoids/topological stacks):

• André Haefliger, Groupoides d’holonomie et classifiants, Astérisque no. 116 (1984), p. 70-97 (numdam:AST_1984__116__70_0)

• André Haefliger, Orbi-Espaces, In: E. Ghys, P. de la Harpe (eds.), Sur les Groupes Hyperboliques d’après Mikhael Gromov, Progress in Mathematics, vol 83. Birkhäuser 1990 (doi:10.1007/978-1-4684-9167-8_11)

  (apparently the term is first used here?)

• André Haefliger, Complexes of Groups and Orbihedra, in: E. Ghys, A. Haefliger, A Verjovsky (eds.), Proceedings of the Group Theory from a Geometrical Viewpoint, ICTP, Trieste, Italy , 26 March – 6 April 1990_, World Scientific 1991 (doi:10.1142/1235)

• Weimin Chen, A homotopy theory of orbispaces (arXiv:math/0102020)

• André Henriques, Orbispaces and orbifolds from the point of view of the Borel construction, a new definition (arXiv:0112006)

• André Henriques, Vector bundles on orbispaces (2005) (pdf, pdf)

The idea to regard these topological groupoids as in global homotopy theory via the (infinity,1)-presheaves on a global orbit category which they represent is due to

• André Henriques, David Gepner, Homotopy Theory of Orbispaces (arXiv:math/0701916)

developed further in

• Charles Rezk, Global Homotopy Theory and Cohesion (2014)

• 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)

• Branko Juran, Orbifolds, Orbispaces and Global Homotopy Theory [arXiv:2006.12374]

See also

• Jacob Lurie, Section 3 of Elliptic cohomology III: Tempered Cohomology, 2019 (pdf)