nLab
Elmendorf's theorem

Contents

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

Elmendorf’s theorem states that for GG a topological group, then 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 localization of topological spaces with GG-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 GG a topological group, write Top GTop^G for the category of compactly generated topological spaces which are equipped with a continuous GG-action. Say that a continuous map f:XYf \colon X \longrightarrow Y between GG-spaces is a weak GG-homotopy equivalence if for any closed subgroup 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

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

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

In particular the theorem hence asserts that the GG-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, 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, def. 3.7)

Example

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

(Guillou, 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 maps which induce weak equivalences or fibrations in 𝒞\mathcal{C}, respectively, on objects of HH-fixed points, for all subgroups HH of GG.

(Guillou, theorem 3.12)

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, prop. 3.15)

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 include

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

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.

which considered all closed subgroups of GG. 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 (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

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

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

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 II-orbits for a small category II is in

Last revised on September 17, 2018 at 06:37:24. See the history of this page for a list of all contributions to it.