∞-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.

## 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)

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

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

## 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 $Glo$

Here $Glo$ has as objects compact Lie groups and the (∞,1)-categorical hom-spaces $Glo(G,H) \coloneqq \Pi [\mathbf{B}G, \mathbf{B}H]$, 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 $\mathbf{B}G\to \mathbf{B}H$.

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

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

Rezk-global equivariant homotopy theory:

cohesive (∞,1)-toposits (∞,1)-sitebase (∞,1)-toposits (∞,1)-site
global equivariant homotopy theory $PSh_\infty(Glo)$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 $Orb_{/\mathbf{B}G} = Orb_G$

## 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)