nLab Elmendorf's theorem

Idea

Elmendorf’s theorem states that for $G$ a topological group, the (∞,1)-category of (∞,1)-presheaves on the orbit category $Orb_G$ of $G$, naturally regarded as an (∞,1)-site, is equivalent to the classical $G$-equivariant homotopy theory, namely the localization of topological spaces with $G$-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 $G$-equivariant homotopy theory, thus identified with an (∞,1)-category of (∞,1)-presheaves, is an (∞,1)-topos and in fact (since $Orb_G$ 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 $Top^G$ for the category of compactly generated topological spaces which are equipped with a continuous $G$-action (G-spaces). Say that a $G$-equivariant continuous function $f \colon X \longrightarrow Y$ between $G$-spaces is a weak $G$-homotopy equivalence if for all closed subgroups $H \hookrightarrow G$ the induced function on $H$-fixed point spaces $f^H \colon X^H \longrightarrow Y^H$ is an ordinary weak homotopy equivalence. Write

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

for the corresponding simplicial localization.

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

$$PSh_\infty(Orb_G) \in (\infty,1)Cat \,.$$

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

$$L_{G whe} Top^G \;\simeq\; PSh_\infty(Orb_G) \,.$$

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 $G$ to be a compact Lie group, Dwyer-Kan 84 allow $G$ 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 $G$-homotopy equivalences, typically used in practice, again requires $G$ 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 $G$ any topological group; while Cordier-Porter 96, Thm. 3.11 and Guillou 06, Prop. 3.15 re-prove a (simplicial) Quillen equivalence assuming $G$ 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 $G$ 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 $G$ 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 $\mathrm{X}$ to its underlying homotopy type presented by its path $\infty$-groupoid (we follow the notation at shape via cohesive path ∞-groupoid)

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

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

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

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

In this form the statement holds for every choice of topological group $G$ and every choice of family $\big\{H_i \subset G\big\}_{i \in I}$ of subgroups: This choice affects what counts as a $G$-CW complex on the left and what the orbit category-site $Orb(G,\{H_i\}_{i \in I})$ is on the right (in both cases the coset spaces involved must be $G/H_i$ for $H_i$ 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

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

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

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

Statement

Classical version in topological spaces

Write

• $kTop$ for the category of compactly generated topological spaces;

Assumption

(general input data)
Let

• $G \,\in\, Grp(kTop)$ be any topological group,

• $\mathcal{F} \,\subset\, Sub(G)$ be any set of subgroups of $G$.

Remark

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

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

Proposition

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

• hom-objects are the singular simplicial complexes of mapping spaces:

  (1)$$[X,Y] \,\coloneqq\, Sing \big( Map(X,Y)^G \big) \;\;\; \in \; sSet$$

• weak equivalences and fibrations $X \xrightarrow{f} Y$ are $H \in \mathcal{F}$-fixed point space-wise ($[G/H, X] \xrightarrow{\;[G/H, f]\;} [G/H, Y]$, see Rem. )

  those of the classical model structure on topological spaces,

• cofibrations include the relative $G$-CW complexes which are build (only) from orbits of the form $G/H$ for $H \,\in\, \mathcal{F} \subset Sub(G)$.

(Dwyer & Kan 1984, §1.2 and Thm. 2.2)

Remark

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

1. $G$ is a compact Lie group;

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

3. $X$ is a $G$-CW complex

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

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

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

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 $G$ and $\mathcal{F}$) the plain $G$-homotopy classes:

$$\pi_0 [X,Y] \,=\, G Heq(X,\,Y) \,.$$

Definition

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

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

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

Moreover, write

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

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

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

between

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

   (Prop. )

2. the projective model structure on simplicial presheaves over the $\mathcal{F}$-type $G$-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 $\mathcal{C}$ a cofibrantly generated model category and for $G$ a discrete group (canonically regarded as a group object of $\mathcal{C}$ via its tensoring over Set) write $G \mathcal{C}$ for the category of $G$-actions in $\mathcal{C}$.

Definition

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

(Guillou 06, def. 3.7)

Example

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

• the classical model structure on topological spaces;

• the classical model structure on simplicial sets;

• the global model structure on simplicial presheaves over any small category;

• (…)

(Guillou 06, section 4)

Proposition

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

(Guillou 06, theorem 3.12)

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

Write $Orb_G$ for the orbit category of $G$.

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

Proposition

There is a pair of adjoint functors

$$(\Theta, \Phi) \;\colon\; G\mathcal{C} \stackrel{\overset{\Theta}{\longleftarrow}}{\underset{\Phi}{\longrightarrow}} \mathcal{C}^{Orb_G^{op}}$$

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

This constitutes a Quillen equivalence between the above model structures

$$(\Theta, \Phi) \;\colon\; G\mathcal{C}_{fine} \underset{Quillen}{\simeq} \mathcal{C}^{Orb_G^{op}}_{proj} \,.$$

(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 $G$ is considered (Stephan 10, also May 96).

2. There is an evident generalization of the orbit category of a fixed group $G$ 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 $G$ can also be generalized to the orbit category generated by any small category, $I$, where the $I$-orbits are $I$-diagrams in $Top$ whose strict colimit is equal to a point. If the orbits are either small or complete, then the $I$-equivariant model structure is Quillen-equivalent to a presheaf category (see pt. 11 of p. 8 of PHCTIntro and Chorny13).

Related concepts

• G-CW approximation

• equivariant Whitehead theorem

• tom Dieck splitting

• global equivariant homotopy theory

• Bredon cohomology

• equivariant rational homotopy theory

  • model structure on equivariant connective dgc-algebras

  • Quillen adjunction between equivariant simplicial sets and equivariant connective dgc-algebras

  • fundamental theorem of dg-algebraic equivariant rational homotopy theory

References

Lecture notes on Elmendorf’s theorem:

• Andrew Blumberg, Section 1.3 of: Equivariant homotopy theory, 2017 (pdf, GitHub)

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

• Glen Bredon, Equivariant cohomology theories, Springer Lecture Notes in Mathematics Vol. 34. 1967.

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

• Anthony Elmendorf, Systems of fixed point sets, Trans. Amer. Math. Soc., 277(1):275–284, 1983 (jstor:1999356)

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:

• William Dwyer, Daniel Kan, Singular functors and realization functors, Indagationes Mathematicae (Proceedings) Volume 87, Issue 2, 1984, Pages 147-153 (doi:10.1016/1385-7258(84)90016-7)

• Jean-Marc Cordier, Timothy Porter, Thm. 3.11 of: Categorical Aspects of Equivariant Homotopy, Applied Cat. Structures, 4 (1996) 195 - 212 (doi:10.1007/BF00122252) (Proceedings of the European Colloquium of Category Theory, 1994)

• Bert Guillou, A short note on models for equivariant homotopy theory, 2006 (pdf, pdf)

• Marc Stephan,

  Elmendorf’s Theorem via Model Categories, 2010 (pdf)

  Elmendorf’s Theorem for Cofibrantly Generated Model Categories, MS thesis, Zürich 2010 (pdf)

• Marc Stephan, On equivariant homotopy theory for model categories, Homology Homotopy Appl. 18(2) (2016) 183-208 (arXiv:1308.0856, doi:10.4310/HHA.2016.v18.n2.a10)

• Bertrand Guillou, Peter May, Jonathan Rubin, Sections 1 and 5.2 in: Enriched model categories in equivariant contexts, Homology, Homotopy and Applications 21 (1), 2019 (arXiv:1307.4488, doi:10.4310/HHA.2019.v21.n1.a10)

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

• Robert Piacenza, Section 6 of: Homotopy theory of diagrams and CW-complexes over a category, Can. J. Math. Vol 43 (4), 1991 (doi:10.4153/CJM-1991-046-3, pdf)

• Peter May et al., Section V.3 of: Equivariant homotopy and cohomology theory, CBMS Regional Conference Series in Mathematics Volume: 91; 1996 (ISBN:978-0-8218-0319-6, pdf, pdf)

More on the approach of Dwyer-Kan 84:

• Emmanuel Dror Farjoun, Prop. 1.3 in: Homotopy Theories for Diagrams of Spaces, Proceedings of the AMS, Vol. 101, No. 1 (Sep., 1987), pp. 181-189 (jstor:2046572)

• Boris Chorny, Homotopy theory of relative simplicial presheaves, Israel J. Math. 205 (2015), no. 1, 471–484, (arXiv:1310.2932)

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

• Jonathan Rubin, Elmendorf constructions for G-categories and G-posets (arXiv:2006.08876)

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

• André Henriques, David Gepner, Homotopy Theory of Orbispaces (arXiv:math/0701916),

relating to global equivariant homotopy theory:

• Charles Rezk, Global Homotopy Theory and Cohesion (2014)

and then to orbifolds:

• Hisham Sati, Urs Schreiber, Proper Orbifold Cohomology (arXiv:2008.01101)

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

• John Huerta, Hisham Sati, Urs Schreiber, Real ADE-equivariant (co)homotopy and Super M-branes, Comm. Math. Phys. 371: 425. (2019 (arXiv:1805.05987, doi:10.1007/s00220-019-03442-3)