# nLab orbispace

Contents

### Context

Ingredients

Concepts

Constructions

Examples

Theorems

#### $\infty$-Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Related topics

Examples

$\infty$-Lie groupoids

$\infty$-Lie groups

$\infty$-Lie algebroids

$\infty$-Lie algebras

# 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:

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

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.

### 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 $Orb_{/\mathbf{B}G} = Orb_G$

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