nLab global equivariant homotopy theory

Contents

under construction (some more harmonization needed)

Context

Representation theory

representation theory

geometric representation theory

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)

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