nLab
global equivariant homotopy theory

Contents

under construction (some more harmonization needed)

Context

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Representation theory

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

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 GG-infinity-actions “for all compact Lie groups GG 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 (Orb)PSh_\infty(Orb) on the global orbit category OrbOrb (Henriques-Gepner 07, section 1.3), regarded as an (∞,1)-category.

Here OrbOrb has as objects compact Lie groups and the (∞,1)-categorical hom-spaces Orb(G,H)Π[BG,BH]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 GloGlo for the (∞,1)-category whose

(Rezk 14, 2.1)

Remark

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

(Rezk 14, 2.4, 2.5)

The following is the global orbit category.

Definition

Write

OrbGlo Orb \longrightarrow Glo

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

(Rezk14, 4.5)

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

Definition

Write

Top GloPSh (Glo) Top_{Glo} \coloneqq PSh_\infty(Glo)

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

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

for the (∞,1)-Yoneda embedding.

Similarly write

Top OrbPSh (Orb) Top_{Orb} \coloneqq PSh_\infty(Orb)

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

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

for its (∞,1)-Yoneda embedding.

(Rezk 14, 3.1 and 4.5)

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

Definition

For GG a topological group, write

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

for the slice (∞,1)-topos of PSh (Orb)PSh_\infty(Orb) over the image of GG 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 (Glo)PSh_\infty(Glo) of def. is a cohesive (∞,1)-topos over the canonical base (∞,1)-topos ∞Grpd:

the global section geometric morphism

(ΔΓ):PSh (Glo)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

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

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

ΔS:GS. \Delta S \colon G \mapsto S \,.

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

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

(Rezk 14, 5.1)

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 GG a compact Lie group define an (∞,1)-functor

δ G:GTopPSh (Glo) \delta_G \;\colon\; G Top \longrightarrow PSh_\infty(Glo)

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

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

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

Δ G:GTopGTop/*PSh (Orb)/𝔹Gδ GPSh (Glo)/𝔹G. \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 \,.

(Rezk 14, 3.2)

Proposition

(ordinary quotient and homotopy quotient via equivariant cohesion)

On a GG-space XGTopX \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

|X//G||X/G|. \vert X//G \vert \longrightarrow \vert X/G \vert \,.

(Rezk 14, 5.1)

Proposition

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

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

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

The inclusion Δ G\Delta_G is that of def. .

(Rezk 14, 5.3)

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 (Orb). 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 Π\Pi such that it sends a topological action groupoid of a topological group GG acting on a topological space XX to the homotopy quotient Π(X)//Π(G)\Pi(X)//\Pi(G).)

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

Rezk-global equivariant homotopy theory:

cohesive (∞,1)-toposits (∞,1)-sitebase (∞,1)-toposits (∞,1)-site
global equivariant homotopy theory PSh (Glo)PSh_\infty(Glo)global equivariant indexing category GloGlo∞Grpd PSh (*) \simeq PSh_\infty(\ast)point
sliced over terminal orbispace: PSh (Glo) /𝒩PSh_\infty(Glo)_{/\mathcal{N}}Glo /𝒩Glo_{/\mathcal{N}}orbispaces PSh (Orb)PSh_\infty(Orb)global orbit category
sliced over BG\mathbf{B}G: PSh (Glo) /BGPSh_\infty(Glo)_{/\mathbf{B}G}Glo /BGGlo_{/\mathbf{B}G}GG-equivariant homotopy theory of G-spaces L weGTopPSh (Orb G)L_{we} G Top \simeq PSh_\infty(Orb_G)GG-orbit category Orb /BG=Orb GOrb_{/\mathbf{B}G} = Orb_G

representation theory and equivariant cohomology in terms of (∞,1)-topos theory/homotopy type theory (FSS 12 I, exmp. 4.4):

homotopy type theoryrepresentation theory
pointed connected context BG\mathbf{B}G∞-group GG
dependent type on BG\mathbf{B}GGG-∞-action/∞-representation
dependent sum along BG*\mathbf{B}G \to \astcoinvariants/homotopy quotient
context extension along BG*\mathbf{B}G \to \asttrivial representation
dependent product along BG*\mathbf{B}G \to \asthomotopy invariants/∞-group cohomology
dependent product of internal hom along BG*\mathbf{B}G \to \astequivariant cohomology
dependent sum along BGBH\mathbf{B}G \to \mathbf{B}Hinduced representation
context extension along BGBH\mathbf{B}G \to \mathbf{B}Hrestricted representation
dependent product along BGBH\mathbf{B}G \to \mathbf{B}Hcoinduced representation
spectrum object in context BG\mathbf{B}Gspectrum with G-action (naive G-spectrum)

References

The global orbit category OrbOrb 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

see also

  • Peter May, Some remarks on equivariant bundles and classifying spaces, Asterisque 191 (1990), 7, 239-253. International Conference on Homotopy Theory (Marseille-Luminy, 1988).

Discussion of the global equivariant homotopy theory as a cohesive (∞,1)-topos is in

Discussion of a model structure for global equivariance with respect to geometrically discrete simplicial groups/∞-group (globalizing the Borel model structure for ∞-actions) is in

Discussion from a perspective of homotopy type theory is in

Last revised on November 29, 2018 at 10:18:11. See the history of this page for a list of all contributions to it.