# nLab global equivariant homotopy theory

Contents

under construction (some more harmonization needed)

### Context

#### Representation theory

representation theory

geometric representation theory

cohomology

# Contents

## Idea

What is called global equivariant homotopy theory is a variant of equivariant cohomology in homotopy theory where pointed topological spaces/homotopy types are equipped with $G$-infinity-actions “for all compact Lie groups $G$ at once”, or more generally for a global family.

Sometimes this is referred to just as “global homotopy theory”, leaving the equivariance implicit. There is also a stable version involving spectra equipped with infinity-actions, see at global equivariant stable homotopy theory.

More precisely, the global equivariant homotopy category is the (∞,1)-category (or else its homotopy category) of (∞,1)-presheaves $PSh_\infty(Orb)$ 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.

## Definition

We follow (Rezk 14). Beware that the terminology there differs slightly but crucially in some places from (Henriques-Gepner 07). Whatever terminology one uses, the following are the key definitions.

The following is the global equivariant indexing category.

###### Definition

Write $Glo$ for the (∞,1)-category whose

###### Remark

Equivalent models for the global indexing category, def. include the category “$O_{gl}$” of (May 90). Another variant is $\mathbf{O}_{gl}$ of (Schwede 13).

The following is the global orbit category.

###### Definition

Write

$Orb \longrightarrow Glo$

for the non-full sub-(∞,1)-category of the global indexing category, def. , on the injective group homomorphisms.

The following defines the global equivariant homotopy theory $PSh_\infty(Glo)$.

###### Definition

Write

$Top_{Glo} \coloneqq PSh_\infty(Glo)$

for the (∞,1)-category of (∞,1)-presheaves (an (∞,1)-topos) on the global indexing category $Glo$ of def. , and write

$\mathbb{B} \;\colon\; Glo \longrightarrow PSh_\infty(Glo)$

for the (∞,1)-Yoneda embedding.

Similarly write

$Top_{Orb} \coloneqq PSh_\infty(Orb)$

for the (∞,1)-category of (∞,1)-presheaves on the global orbit category $Orb$ of def. , and write again

$\mathbb{B} \;\colon\; Orb \longrightarrow PSh_\infty(Orb)$

for its (∞,1)-Yoneda embedding.

The following recovers the ordinary (“local”) equivariant homotopy theory of a given topological group $G$ (“of $G$-spaces”).

###### Definition

For $G$ a topological group, write

$G Top \coloneqq PSh_\infty(Orb)/\mathbb{B}G$

for the slice (∞,1)-topos of $PSh_\infty(Orb)$ over the image of $G$ under the (∞,1)-Yoneda embedding, as in def. .

This is (Rezk 14, 1.5). Depending on axiomatization this is either a definition or Elmendorf's theorem, see at equivariant homotopy theory for more on this.

## Properties

### Cohesion

###### Proposition

The global equivariant homotopy theory $PSh_\infty(Glo)$ of def. is a cohesive (∞,1)-topos over the canonical base (∞,1)-topos ∞Grpd:

$(\Delta \dashv \Gamma) \;\colon\; PSh_\infty(Glo) \longrightarrow \infty Grpd$

is given (as for all (∞,1)-presheaf (∞,1)-toposes) by the direct image/global section functor being the homotopy limit over the opposite (∞,1)-site

$\Gamma X \simeq \underset{\leftarrow}{\lim}(Glo^{op}\stackrel{X}{\to} \infty Grpd)$

and the inverse image/constant ∞-stack functor literally assigning constant presheaves:

$\Delta S \colon G \mapsto S \,.$

This is a full and faithful (∞,1)-functor.

Moreover, $\Delta$ has a further left adjoint $\Pi$ which preserves finite products, and $\Gamma$ has a further right adjoint $\nabla$.

More in detail, the shape modality, flat modality and sharp modality of this cohesion of the global equivariant homotopy theory has the following description.

### Relation between global and local equivariant homotopy theory

###### Definition

For $G$ a compact Lie group define an (∞,1)-functor

$\delta_G \;\colon\; G Top \longrightarrow PSh_\infty(Glo)$

sending a topological G-space to the he presheaf which sends a group $H$ to the geometric realization of the topological groupoid of maps from $\mathbf{B}H$ to the action groupoid $X//G$:

$\delta_G(X)\;\colon\; H \mapsto \Pi( [\mathbf{B}H, X//G] ) \,.$

Observe that by def. this gives $\delta_G(\ast) \simeq \mathbb{B}G$ and so $\delta_G$ induces a functor

$\Delta_G \;\colon\; G Top \simeq G Top/\ast \simeq PSh_\infty(Orb)/\mathbb{B}G \stackrel{\delta_G}{\longrightarrow} PSh_\infty(Glo)/\mathbb{B}G \,.$
###### Proposition

(ordinary quotient and homotopy quotient via equivariant cohesion)

On a $G$-space $X \in G Top$ included via def. into the global equivariant homotopy theory,

In particular then the points-to-pieces transform of general cohesion yields the comparison map

$\vert X//G \vert \longrightarrow \vert X/G \vert \,.$
###### Proposition

For $G$ any compact Lie group, the cohesion of the global equivariant homotopy theory, prop. , descends to the slice (∞,1)-toposes

$PSh_\infty(Glo)/\mathbb{B}G \longrightarrow PSh_\infty(Orb)/\mathbb{B}G \simeq G Top \,,$

hence to cohesion over the “local” $G$-equivariant homotopy theory.

The inclusion $\Delta_G$ is that of def. .

### Relation to topological stacks and orbispaces

under construction

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 the 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.

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 $Orb_{/\mathbf{B}G} = Orb_G$
homotopy type theoryrepresentation theory
pointed connected context $\mathbf{B}G$∞-group $G$
dependent type on $\mathbf{B}G$$G$-∞-action/∞-representation
dependent sum along $\mathbf{B}G \to \ast$coinvariants/homotopy quotient
context extension along $\mathbf{B}G \to \ast$trivial representation
dependent product along $\mathbf{B}G \to \ast$homotopy invariants/∞-group cohomology
dependent product of internal hom along $\mathbf{B}G \to \ast$equivariant cohomology
dependent sum along $\mathbf{B}G \to \mathbf{B}H$induced representation
context extension along $\mathbf{B}G \to \mathbf{B}H$restricted representation
dependent product along $\mathbf{B}G \to \mathbf{B}H$coinduced representation
spectrum object in context $\mathbf{B}G$spectrum with G-action (naive G-spectrum)

## References

The global orbit category $Orb$ is considered in

Global unstable equivariant homotopy theory is discussed as a localization of the category of “orthogonal spaces” (the unstable version of orthogonal spectra) in