∞-Lie theory

# 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.

## 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 orbit category $Orb$ (Henriques-Gepner 07, section 1.3), regarded as an (∞,1)-category.

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

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

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

(As such that global equivariant homotopy theory should be similar to ETop∞Grpd. Observe that this is a cohesive (∞,1)-topos with $\Pi$ such that it sends a topological action groupoid of a topological group $G$ acting on a topological space $X$ to the homotopy quotient $\Pi(X)//\Pi(G)$.)

The central theorem of (Rezk 14) (using a slightly different definition than Henriques-Gepner 07) is that $PSh_\infty(Orb)$ is a cohesive (∞,1)-topos with $\Gamma$ producing homotopy quotients.

## References

A detailed but elementary approach via atlases can be found in

and another approach is discussed in

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Revised on March 7, 2014 02:05:51 by Urs Schreiber (89.204.138.194)