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orbispace

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Geometry

\infty-Lie theory

∞-Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Contents

Idea

An orbispace is a space, particularly 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.

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)

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

ETopGrpd cell=PSh (Orb). ETopGrpd^{cell} = PSh_\infty(Orb) \,.

Properties

Relation to global equivariant homotopy theory

The global equivariant homotopy theory is the (∞,1)-category (or else its homotopy category) of (∞,1)-presheaves on the global equivariant indexing category GloGlo

Here GloGlo has as objects compact Lie groups and the (∞,1)-categorical hom-spaces Glo(G,H)Π[BG,BH]Glo(G,H) \coloneqq \Pi [\mathbf{B}G, \mathbf{B}H] , where on the right we have the fundamental (∞,1)-groupoid of the topological groupoid of group homomorphisms and conjugations.

The global orbit category is the non-full subcategory of the global equivariant indexing category on the faithful maps BGBH\mathbf{B}G\to \mathbf{B}H.

The central theorem of (Rezk 14) is that PSh (Orb)PSh_\infty(Orb) is the base (∞,1)-topos over the cohesion of the slice of the global equivariant homotopy theory PSh (Glo)PSh_\infty(Glo) over the terminal orbispace 𝒩\mathcal{N} (Rezk 14, p. 4 and section 7)

(ΠΔΓ):PSh (Glo)/𝒩PSh (Orb). (\Pi \dashv \Delta \dashv \Gamma \dashv \nabla) \;\colon\; PSh_\infty(Glo)/\mathcal{N} \longrightarrow PSh_\infty(Orb) \,.

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

A detailed but elementary approach via atlases can be found in

and another approach is discussed in

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Revised on April 13, 2014 23:36:44 by Urs Schreiber (185.37.147.12)