nLab Elmendorf's theorem

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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

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Representation theory

Cohomology

cohomology

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Special notions

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Contents

Idea

Elmendorf’s theorem states that for GG a topological group, the (∞,1)-category of (∞,1)-presheaves on the orbit category Orb GOrb_G of GG, naturally regarded as an (∞,1)-site, is equivalent to the classical GG-equivariant homotopy theory, namely the localization of topological spaces with GG-action (G-spaces) at the weak homotopy equivalences on all fixed point spaces of closed subgroups (the equivariant weak homotopy equivalences).

This is due to Elmendorf 83 and Dwyer-Kan 84, Sec. 1.2, 1.7 & Thm. 3.1; with alternative proofs in various stages of refinement (see Remark below) given by Piacenza 91, Sec. 6, May 96, Sec. V.3, Cordier-Porter 96, Thm. 3.11, Guillou 06, Prop. 3.15, Stephan 13, Cor. 3.20, Guillou-May-Rubin 13, Thm. 1.8 & 5.6. Exposition is in Blumberg 17, Sec. 1.3.

In particular this means that the GG-equivariant homotopy theory, thus identified with an (∞,1)-category of (∞,1)-presheaves, is an (∞,1)-topos and in fact (since Orb GOrb_G has finite products) a cohesive (∞,1)-topos; a point expanded on in Rezk 14, Sati-Schreiber 20 (for more on this see at orbifold cohomology).

More in detail, write Top GTop^G for the category of compactly generated topological spaces which are equipped with a continuous GG-action (G-spaces). Say that a GG-equivariant continuous function f:XYf \colon X \longrightarrow Y between GG-spaces is a weak GG-homotopy equivalence if for all closed subgroups HGH \hookrightarrow G the induced function on HH-fixed point spaces f H:X HY Hf^H \colon X^H \longrightarrow Y^H is an ordinary weak homotopy equivalence. Write

L GwheTop G(,1)Cat L_{G whe} Top^G \;\in\; (\infty,1)Cat

for the corresponding simplicial localization.

Next, write Orb GOrb_G for the full subcategory of Top GTop^G on the GG-homogeneous spaces of the form G/HG/H, but regarded as an (∞,1)-category by regarding each hom-space as its homotopy type. Write moreover Top Orb GTop^{Orb_G} for the category of continuous functors Orb G opTopOrb_G^{op} \longrightarrow Top. Write finally

PSh (Orb G)(,1)Cat. PSh_\infty(Orb_G) \in (\infty,1)Cat \,.

Then Elmendorf’s theorem asserts that there is an equivalence of (∞,1)-categories

L GwheTop GPSh (Orb G). L_{G whe} Top^G \;\simeq\; PSh_\infty(Orb_G) \,.

Remark

(survey of available proofs.)

Elmendorf’s theorem is stated in Elmendorf 83 as an equivalence of homotopy categories; and is enhanced in Dwyer-Kan 84, Sec. 1.2, 1.7 & Thm. 3.1 to a simplicial Quillen equivalence of model categories presenting (in hindsight) the equivalence of (∞,1)-categories stated above (e.g. Blumberg 17. Thm. 1.3.8).

Moreover, while Elmendorf 83 assumes GG to be a compact Lie group, Dwyer-Kan 84 allow GG to be any topological group. But beware that invoking the equivariant Whitehead theorem to identify the fixed-locus-wise weak homotopy equivalences, used in the theorem, with the GG-homotopy equivalences, typically used in practice, again requires GG to be a compact Lie group.

Later Piacenza 91, Sec 6 and May 96, Sec. V.3 re-prove the equivalence of homotopy categories for GG any topological group; while Cordier-Porter 96, Thm. 3.11 and Guillou 06, Prop. 3.15 re-prove a (simplicial) Quillen equivalence assuming GG to be a discrete or even finite group, respectively; and Stephan 13, Cor. 3.20, Guillou-May-Rubin 13, Thm. 1.8 re-prove the Quillen equivalence for GG again any topological group.

(Stephan 13 credits Piacenza 91 with proving a Quillen equivalence. While this is not what Piacenza 91, Thm. 6.3 actually states, the conclusion follows with hindsight.)

These results are all based on the classical model structure on topological spaces. The analogous Quillen equivalence based on the classical model structure on simplicial sets is proven in Guillou-May-Rubin 13, Thm. 5.6, assuming GG to be a discrete group.

Remark

(as a statement about concordance of equivariant maps)
One way to understand Elmendorf’s theorem (in its full \infty-category theoretic version) is as translating concordance of equivariant maps into actual homotopies in another \infty -category. Namely, if we think of sending a topological space X\mathrm{X} to its underlying homotopy type presented by its path \infty -groupoid (we follow the notation at shape via cohesive path ∞-groupoid)

Top Grpd X ʃX \array{ Top &\longrightarrow& Grpd_\infty \\ \mathrm{X} &\overset{\phantom{--}}{\mapsto}& \esh \mathrm{X} }

then it says that got each G G -CW complex XX we have:

ʃ(Maps(X,Y) G)GGrpd (ʃ(X ()),ʃ(Y ())), \esh \Big( Maps \big( \mathrm{X} ,\, \mathrm{Y} \big)^G \Big) \;\; \simeq \;\; G Grpd_\infty \Big( \esh \big(\mathrm{X}^{(-)}\big) ,\, \esh \big(\mathrm{Y}^{(-)}\big) \Big) \,,

where Map(,)Map(-,-) denotes the mapping space.

In this form the statement holds for every choice of topological group GG and every choice of family {H iG} iI\big\{H_i \subset G\big\}_{i \in I} of subgroups: This choice affects what counts as a GG-CW complex on the left and what the orbit category-site Orb(G,{H i} iI)Orb(G,\{H_i\}_{i \in I}) is on the right (in both cases the coset spaces involved must be G/H iG/H_i for H iH_i in the given family) – but with that understood, the above equivalence holds, by Dwyer-Kan 84, Sec. 1.2, 1.7 & Thm. 3.1.

To bring this out, write

  • Orb(G,)GTopOrb(G,\mathcal{F}) \hookrightarrow G Top for the full subcategory on the coset spaces G/HG/H for HH \in \mathcal{F} in the family of subgroups,

  • (G,)CWCplxGTop(G, \mathcal{F})CWCplx \hookrightarrow G Top for the full subcategory of G-CW complexes build via these coset spaces, then

X(G,)CWCplxʃ(Maps(X,Y) G)Psh (Orb(G,))(ʃ(X ()),ʃ(Y ())). X \,\in\, (G,\mathcal{F})CWCplx \;\;\;\; \vdash \;\;\;\; \esh \Big( Maps \big( \mathrm{X} ,\, \mathrm{Y} \big)^G \Big) \;\; \simeq \;\; Psh_\infty\big(Orb(G,\mathcal{F})\big) \Big( \esh \big(\mathrm{X}^{(-)}\big) ,\, \esh \big(\mathrm{Y}^{(-)}\big) \Big) \,.

Statement

Classical version in topological spaces

Write

Assumption

(general input data)
Let

Remark

(fixed loci as equivariant mapping spaces)
For HGH \subset G any subgroup, notice the usual identification of HH-fixed loci X HXX^H \subset X inside topological G G -spaces XX with the GG-equivariant mapping space out of the orbit/coset space G/HG/H:

Map(G/H,)() H. Map(G/H,\, -) \;\simeq\; (-)^H \,.

Proposition

(\mathcal{F}-projective model structure on GG-spaces)
In the generality of Assumption , the category of topological G G -spaces carries the structure of a simplicial model category GAct(kTop) projG Act(kTop)_{\mathcal{F} proj} whose

(Dwyer & Kan 1984, §1.2 and Thm. 2.2)

Remark

(relation to classical GG-equivariant homotopy theory)
In the special case of Assumption that

  1. GG is a compact Lie group;

  2. ClsSub(G)\mathcal{F} \coloneqq ClsSub(G) is the set of all closed subgroups

  3. XX is a G G -CW complex

the equivariant Whitehead theorem says that weak equivalence XYX \longrightarrow{Y} in GAct(kTop) ClsSub(G)G Act(kTop)_{ClsSub(G)} (from Prop. ) are exactly the GG-equivariant homotopy equivalences.

More generally there may be more weak equivalences in GAct(KTop) G Act(KTop)_{\mathcal{F}} than there are GG-homotopy equivalences:

GHeqW . G Heq \hookrightarrow W_{\mathcal{F}} \,.

However, observe that the simplicial hom-complex (1) is always the classical one, in particular its connected components are always (no matter the choice of GG and \mathcal{F}) the plain GG-homotopy classes:

π 0[X,Y]=GHeq(X,Y). \pi_0 [X,Y] \,=\, G Heq(X,\,Y) \,.

Definition

(simplicial presheaves over \mathcal{F}-type GG-orbits)
Given data as in Assumption , write

(2)Orb(G,)GAct(kTop) Orb(G,\mathcal{F}) \hookrightarrow G Act(kTop)

for the category of \mathcal{F} -type G G orbits, namely the full sub-sSet-enriched category of topological G-spaces (1) on the coset spaces G/HG/H for HSub(G)H \,\in\, \mathcal{F} \subset Sub(G).

Moreover, write

sPSh(Orb(G,)) proj sPSh \big( Orb(G,\mathcal{F}) \big)_{proj}

for the projective model structure on sSet-presheaves over (2), hence that whose weak equivalences and fibrations are object-wise those in the classical model structure on simplicial sets.

Proposition

(Dwyer-Kan theorem)
In the generality of Assumption , there is a Quillen equivalence

GAct(kTop) QuX(G/H[G/H,X])sPSh(Orb(G,)) G Act(kTop) \underoverset { \underset{ X \mapsto \big( G/H \mapsto [G/H, X] \big) }{\longrightarrow} } {\longleftarrow} {\simeq_{\mathrlap{Qu}}} sPSh( Orb(G,\mathcal{F}) )

between

  1. the \mathcal{F}-projective model structure on topological GG-spaces

    (Prop. )

  2. the projective model structure on simplicial presheaves over the \mathcal{F}-type GG-orbits

    (Def. )

This is Dwyer & Kan 1984, Thm. 3.1, formulated here as a simplicial Quillen adjunction in view of the recognition principle for simplicial Quillen adjunctions (from this Prop.).

General model category presentation

A version of the theorem that applies fairly generally for (discrete) group objects in suitable cofibrantly generated model categories is in (Guillou 06, Stephan 10, Stephan 13).

Definition

For 𝒞\mathcal{C} a cofibrantly generated model category and for GG a discrete group (canonically regarded as a group object of 𝒞\mathcal{C} via its tensoring over Set) write G𝒞G \mathcal{C} for the category of GG-actions in 𝒞\mathcal{C}.

Definition

A cellular fixed point functor on 𝒞\mathcal{C} is …

(Guillou 06, def. 3.7)

Example

The fixed point spaces-functors on the following kinds of model categories are cellular

(Guillou 06, section 4)

Proposition

For 𝒞\mathcal{C} a cofibrantly generated model category with cellular fixed point functor, def. , then the category G𝒞G \mathcal{C} of GG-actions in 𝒞\mathcal{C} carries a cofibrantly generated model category structure G𝒞 fineG \mathcal{C}_{fine} whose weak equivalences and fibrations are those morphisms whose underlying maps of HH-fixed loci, for all subgroups HH of GG, are equivalences or fibrations in 𝒞\mathcal{C}, respectively.

(Guillou 06, theorem 3.12)

For the case that 𝒞\mathcal{C} is the classical model structure on topological spaces this yields the fine model structure on topological G-spaces.

Write Orb GOrb_G for the orbit category of GG.

Write (𝒞 Orb G op) proj(\mathcal{C}^{Orb_G^{op}})_{proj} for the projective global model structure on functors from the GG-orbit category to 𝒞\mathcal{C}.

Proposition

There is a pair of adjoint functors

(Θ,Φ):G𝒞ΦΘ𝒞 Orb G op (\Theta, \Phi) \;\colon\; G\mathcal{C} \stackrel{\overset{\Theta}{\longleftarrow}}{\underset{\Phi}{\longrightarrow}} \mathcal{C}^{Orb_G^{op}}

where ΦX:G/HX H\Phi X \colon G/H \mapsto X^H assigns fixed-point objects and where ΘS\Theta S has as underlying object S(G/1)S(G/1).

This constitutes a Quillen equivalence between the above model structures

(Θ,Φ):G𝒞 fineQuillen𝒞 proj Orb G op. (\Theta, \Phi) \;\colon\; G\mathcal{C}_{fine} \underset{Quillen}{\simeq} \mathcal{C}^{Orb_G^{op}}_{proj} \,.

(Guillou 06, prop. 3.15, Stephan 13, Theorem 3.17)

Generalizations

  1. Elmendorf’s theorem may be generalized to the case where only a sub-family \mathcal{H} of the closed subgroups of GG is considered (Stephan 10, also May 96).

  2. There is an evident generalization of the orbit category of a fixed group GG to the global orbit category. Under this generalization an analog of Elmendorf’s theorem plays a central role in global equivariant homotopy theory (Rezk 14).

  3. The orbit category for GG can also be generalized to the orbit category generated by any small category, II, where the II-orbits are II-diagrams in TopTop whose strict colimit is equal to a point. If the orbits are either small or complete, then the II-equivariant model structure is Quillen-equivalent to a presheaf category (see pt. 11 of p. 8 of PHCTIntro and Chorny13).

References

Lecture notes on Elmendorf’s theorem:

The “fine” homotopical structure on G-spaces (with fixed-point-wise weak equivalences) is originally due to

The equivalence of its homotopy category to that of presheaves over the orbit category is then due to:

Enhancement of Elmendorf’s equivalence of homotopy categories to an sSet-enriched adjunction and/or to a Quillen equivalence of model categories based on the classical model structure on topological spaces and/or the classical model structure on simplicial sets:

Other discussions generalizing Elmendorf 83 (i.e. the equivalence of homotopy categories) to general topological equivariance groups:

More on the approach of Dwyer-Kan 84:

These Elmendorf-theorem Quillen equivalences (as in Stephan 13) apply to other model categories, and yield Elmendorf-like equivalences in other contexts:

See also related discussion of “orbispaces”, starting with:

relating to global equivariant homotopy theory:

and then to orbifolds:

An n-cat café discussion initiated by John Huerta and probing some of its uses in Mathematical Physics, can be found here, following

Last revised on July 4, 2022 at 17:07:47. See the history of this page for a list of all contributions to it.