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Elmendorf's theorem

Idea

Elmendorf’s theorem states that for $G$ a topological group, then the (∞,1)-category of (∞,1)-presheaves on the orbit category $Orb_G$ of $G$, naturally regarded as an (∞,1)-site, is equivalent to the localization of topological spaces with $G$-action (G-spaces) at the weak homotopy equivalences on fixed point spaces (also presented by G-CW complexes, see at equivariant homotopy theory for more).

More in detail, for $G$ a topological group, write $Top^G$ for the category of compactly generated topological spaces which are equipped with a continuous $G$-action. Say that a continuous map $f \colon X \longrightarrow Y$ between $G$-spaces is a weak $G$-homotopy equivalence if for any closed subgroup $H \hookrightarrow G$, the induced function on $H$-fixed point spaces $f^H \colon X^H \longrightarrow Y^H$ is an ordinary weak homotopy equivalence. Write

$$Top^G[\{weak\,G-homotopy\;equivalences\}^{-1}] \in (\infty,1)Cat$$

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

$$Top^G[\{weak\;G-homotopy\;equivalences\}^{-1}] \simeq PSh_\infty(Orb_G) \,.$$

Remark

In particular the theorem hence asserts that the $G$-equivariant homotopy theory is an (∞,1)-topos.

Model category presentation / Quillen equivalence

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 maps which induce weak equivalences or fibrations in $\mathcal{C}$, respectively, on objects of $H$-fixed points, for all subgroups $H$ of $G$.

(Guillou 06, theorem 3.12)

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

• 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 dgc-algebraic equivariant rational homotopy theory?

References

Lecture notes include

• Andrew Blumberg, 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 the homotopy theory (homotopy category) of that to 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)

which considered all closed subgroups of $G$. Also

• Robert Piacenza, section 6 of Homotopy theory of diagrams and CW-complexes over a category, Can. J. Math. Vol 43 (4), 1991 (pdf)

  also chapter VI of Peter May et al, Equivariant homotopy and cohomology theory, 1996 (pdf)

The generalization of the proof to other choices of families of subgroups is due to

• Peter May, Equivariant homotopy and cohomology theory With contributions by M. Cole, G. Comezaa, S. Costenoble, A. D. Elmendorf, J. P. C. Greenlees, L. G. Lewis, Jr., R. J. Piacenza, G. Triantafillou, and S. Waner. Number 91 in CBMS Regional Conference Series in

  Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1996

Discussion in terms of Quillen equivalence of model categories is due to

• 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, Zurich 2010 (pdf)

• Marc Stephan, On equivariant homotopy theory for model categories, Homology Homotopy Appl. 18(2) (2016) 183-208 (arXiv:1308.0856)

A generalization to orbispaces is discussed in

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

Discussion in the broader context of global equivariant homotopy theory is in

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

Some of the categorical aspects of Elmendorf’s theorem are examined in

• Jean-Marc Cordier, Timothy Porter, 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)

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

Generalization to $I$-orbits for a small category $I$ is in

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