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orbispace

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∞-Lie theory (higher geometry)

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Contents

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:

smooth manifold topological manifold orbifold orbispace Lie groupoid topological groupoid differentiable stack topological stack \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:

TopologicalGroupoids Orbispaces 𝒳 (GMaps(BG,𝒳)) \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 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)

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 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

References

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

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

developed further in

See also

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

Last revised on July 19, 2020 at 13:43:20. See the history of this page for a list of all contributions to it.