Contents

# 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 a rigid group action.

Orbispaces are to topological spaces what orbifolds are to manifolds.

## 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.

### 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):

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

Rezk-global equivariant homotopy theory:

cohesive (∞,1)-toposits (∞,1)-sitebase (∞,1)-toposits (∞,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 $Orb_{/\mathbf{B}G} = Orb_G$

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 (arXiv:1711.06019)

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