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equivariant homotopy theory

Context

Homotopy theory

Representation theory

Contents

Idea

Equivariant homotopy theory is homotopy theory for the case that a group GG acts on all the topological spaces or other objects involved, hence the homotopy theory of topological G-spaces.

(Beware that this is crucially different from (namely “finer” and “more geometric” than) the homotopy theory of the ∞-actions of the underlying homotopy type ∞-group of GG, and this is so even when GG is a discrete group, see below).

The union of GG-equivariant homotopy theories as GG is allowed to vary is global equivariant homotopy theory.

The direct stabilization of equivariant homotopy theory is the theory of spectra with G-action. More generally there is a concept of G-spectra and they are the subject of equivariant stable homotopy theory.

The concept of cohomology of equivariant homotopy theory is equivariant cohomology:

cohomology in the presence of ∞-group GG ∞-action:

Borel equivariant cohomology\leftarrowgeneral (Bredon) equivariant cohomology\rightarrownon-equivariant cohomology with homotopy fixed point coefficients
H(X G,A)\mathbf{H}(X_G, A)trivial action on coefficients AA[X,A] G[X,A]^Gtrivial action on domain space XXH(X,A G)\mathbf{H}(X, A^G)

In topological spaces

Let GG be a discrete group.

GG-Homotopy

A G-space is a topological space equipped with a GG-action.

Let I=I = \mathbb{R} be the interval object (*0I1*)({*} \stackrel{0}{\to} I \stackrel{1}{\leftarrow} {*}) regarded as a GG-space by equipping it with the trivial GG-action.

A GG-homotopy η\eta between GG-maps, f,g:XYf, g : X \to Y, is a left homotopy with respect to this II

X×*=X Id×0 f X×I η Y 1 g X×*=X. \array{ X \times {*} = X \\ {}^{\mathllap{Id \times 0}}\downarrow & \searrow^{f} \\ X \times I &\stackrel{\eta}{\to}& Y \\ {}^{\mathllap{1}}\uparrow & \nearrow_{g} \\ X\times {*} = X } \,.

Homotopy theory of GG-spaces

Definition

(models for GG-equivariant spaces)

Consider the following three homotopical categories that model GG-spaces:

  1. Write

    GTop cofGTop G Top_{cof} \subset G Top

    for the full subcategory of GG-CW-complexes, regarded as equipped with the structure of a category with weak equivalences by taking the weak equivalences to be the GG- homotopy equivalences with the above definition.

  2. Write

    GTop loc G Top_{loc}

    for all of GTopG Top equipped with weak equivalences given by those morphisms (f:XY)GTop(f : X \to Y) \in G Top that induce on for all subgroups HGH \subset G weak equivalences f H:X HY Hf^H : X^H \to Y^H on the HH-fixed point spaces, in the standard model structure on topological spaces (i.e. inducing isomorphism on homotopy groups).

  3. Write

    [Orb G op,Top] proj [Orb_G^{op}, Top]_{proj}

    for the projective global model structure on functors from the opposite category of the orbit category O GO_G of GG to Top.

The following theorem (Elmendorf's theorem) says that these models all present the same homotopy theory.

Theorem

(Elmendorf’s theorem)

The homotopy categories of all three models are equivalent:

Ho(GTop loc)Ho(GTop cof)Ho([Orb G op,Top]), Ho(G Top_{loc}) \simeq Ho(G Top_{cof}) \stackrel{\simeq}{\to} Ho([Orb_G^{op}, Top]) \,,

where the equivalence is induced by the functor that sends GG-space to the presheaf that it represents is an equivalence of categories.

This is stated as (May 96,theorem VI.6.3).

(,1)(\infty,1)-category of GG-equivariant spaces

At topological ∞-groupoid it is discussed that the category Top of topological spaces may be understood as the localization of an (∞,1)-category Sh (,1)(Top)Sh_{(\infty,1)}(Top) of (∞,1)-sheaves on TopTop, at the collection of morphisms of the form {X×IX}\{X \times I \to X\} with II the real line.

The analogous statement is true for GG-spaces: the equivariant homotopy category is the homotopy localization of the category of \infty-stacks on GTopG Top.

More in detail: let GTopG Top be the site whose objects are GG-spaces that admit GG-equivariant open covers, morphisms are GG-equivariant maps and morphism YXY \to X is in the coverage if it admits a GG-equivariant splitting over such GG-equivariant open covers.

Write

sSh(GTop) loc sSh(G Top)_{loc}

for the corresponding hypercomplete local model structure on simplicial sheaves.

Let II be the unit interval, the standard interval object in Top, equipped with the trivial GG-action, regarded as an object of GTopG Top and hence in sSh(GTop)sSh(G Top).

Write

sSh(GTop) loc IsSh(GTop) loc sSh(G Top)_{loc}^I \stackrel{\leftarrow}{\to} sSh(G Top)_{loc}

for the left Bousfield localization at thecollection of morphisms {XId×0X×I}\{X \stackrel{Id \times 0}{\to} X \times I\}.

Then the homotopy category of sSh(GTop) loc IsSh(G Top)_{loc}^I is the equivariant homotopy category described above

Ho(sSh(GTop) loc I)GTop loc. Ho(sSh(G Top)_{loc}^{I}) \simeq G Top_{loc} \,.

This is (Morel-Voevodsky 03, example 3, p. 50).

Global equivariant homotopy theory

The above constructions may be unified to apply “for all groups at once”, this is the content of global equivariant homotopy theory.

In more general model categories

Let GG be a finite group as above. We describe the generalizaton of the above story as Top is replaced by a more general model category CC (Guillou).

Definition and proposition
  1. Let CC be a cofibrantly generated model category with generating cofibrations II and generating acyclic cofibrations JJ.

    There is a cofibrantly generated model category

    [O G op,C] loc [O_G^{op}, C]_{loc}

    on the functor category from the orbit category of GG to CC by taking the generating cofibrations to be

    I O G:={G/H×i} iI,HG I_{O_G} := \{G/H \times i\}_{i \in I, H \subset G}

    and the generating acyclic cofibrations to be

    J O G:={G/H×j} jI,HG. J_{O_G} := \{G/H \times j\}_{j \in I, H \subset G} \,.
  2. Let BG\mathbf{B}G be the delooping groupoid of GG and let

    [BG op,C] loc [\mathbf{B}G^{op}, C]_{loc}

    be the functor category from BG\mathbf{B}G to CC – the category of objects in CC equipped with a GG-action equipped with a set of generatinc (acyclic) cofibrations

    I BG:={G/H×i} iI,HG I_{\mathbf{B}G} := \{G/H \times i\}_{i \in I, H \subset G}

    and the generating acyclic cofibrations to be

    J BG:={G/H×j} jI,HG. J_{\mathbf{B}G} := \{G/H \times j\}_{j \in I, H \subset G} \,.

    This defines a cofibrantly generated model category if [BG op,C][\mathbf{B}G^{op}, C] has a cellular fixed point functor (see…).

Definition and proposition

(generalized Elmendorf’s theorem)

There is a Quillen adjunction

G/e×():C[BG op,C] loc:() e G/e \times (-) : C \stackrel{\leftarrow}{\to} [\mathbf{B}G^{op},C]_{loc} : (-)^e

and a Quillen equivalence

Θ:[O G op,C] loc[BG op,C] loc:Φ. \Theta : [O_G^{op}, C]_{loc} \stackrel{\leftarrow}{\to} [\mathbf{B}G^{op},C]_{loc} : \Phi \,.
Proof

This is proposition 3.1.5 in Guillou.

In \infty-stack (,1)(\infty,1)-toposes

The assumption on the model category CC entering the generalized Elmendorf theorem above is satisfied in particular by every left Bousfield localization

C:=L ASPSh(D) C := L_A SPSh(D)

of the global projective model structure on simplicial presheaves onany small category CC at any set AA of morphisms, i.e. for every combinatorial model category CC. This is example 4.4 in Guillou.

For A={C({U i})X}A = \{C(\{U_i\}) \to X\} the collection of Cech covers for all covering families of a Grothendieck topology on DD, this are the standard models for ∞-stack (∞,1)-toposes H\mathbf{H}.

This way the above theorem provides a model for GG-equivariant refinements of ∞-stack (∞,1)-toposes.

Properties

(,1)(\infty,1)-topos

By Elmendorf's theorem the GG-equivariant homotopy theory is an (∞,1)-topos.

By (Rezk 14) GTopG Top is also the base (∞,1)-topos of the cohesion of the global equivariant homotopy theory sliced over BG\mathbf{B}G.

Stabilization

The stabilization of the (∞,1)-topos GTopPSh (Orb G)G Top \simeq PSh_\infty(Orb_G) is the equivariant stable homotopy theory of spectra with G-action (“naive G-spectra”).

Relation to ∞-Actions (fine and coarse equivariance)

For GG a discrete group (geometrically discrete) the homotopy theory of G-spaces which enters Elmendorf's theorem is different (finer) than the standard homotopy theory of GG-∞-actions, which is presented by the Borel model structure (see there for more, and see (Guillou)).

Examples

S 1S^1-Equivariance

circle group-equivariant homotopy theory may be presented by cyclic sets.

Equivariant homotopy theory is to equivariant stable homotopy theory as homotopy theory is to stable homotopy theory.

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

References

A standard text is

  • Peter May, Equivariant homotopy and cohomology theory CBMS Regional Conference Series in Mathematics, vol. 91, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1996. With contributions by M. Cole, G. Comeza˜na, S. Costenoble, A. D. Elmenddorf, J. P. C. Greenlees, L. G. Lewis, Jr., R. J. Piacenza, G. Triantafillou, and S. Waner. (pdf)

For a brief modern surves see also the first three sections of

The generalization of the homotopy theory of GG-spaces and of Elmendorf’s theorem to that of GG-objects in more general model categories is in

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

and further discussed in

  • Marc Stephan, Elmendorf’s theorem for cofibrantly generated model categories, arXiv:1308.0856, also Masters Thesis E.T.H., 2010.

See also

Discussion in the context of global equivariant homotopy theory is in

Revised on April 15, 2014 08:26:59 by Urs Schreiber (88.128.80.29)