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equivariant homotopy theory

Idea

Equivariant homotopy theory is homotopy theory for the case that a group $G$ acts on all the topological spaces or other objects involved.

In topological spaces

Let $G$ be a discrete group.

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

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

A $G$-homotopy $\eta$ between $G$-maps, $f,g:X\to Y$, is a left homotopy with respect to this $I$

$$\begin{array}{c}X\times *=X\\ {}^{\mathrm{Id}\times 0}\downarrow & {\searrow }^f\\ X\times I& \stackrel{\eta }{\to }& Y\\ {}^1\uparrow & {\nearrow }_g\\ X\times *=X\end{array}.$$\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 } \,.

Homotopical categories of $G$-equivariant spaces

Definition

(models for $G$-equivariant spaces)

Consider the following three homotopical categories that model $G$-spaces:

1. Write

   $$G{\mathrm{Top}}_{\mathrm{cof}}\subset G\mathrm{Top}$$G Top_{cof} \subset G Top

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

2. Write

   $$G{\mathrm{Top}}_{\mathrm{loc}}$$G Top_{loc}

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

3. Write

   $$[O_G^{\mathrm{op}},\mathrm{Top}{]}_{\mathrm{proj}}$$[O_G^{op}, Top]_{proj}

   for the projective global model structure on functors from the opposite category of the orbit category $O_G$ of $G$ to Top.

Theorem

(Elmendorf’s theorem)

The homotopy categories of all three models are equivalent:

$$\mathrm{Ho}(G{\mathrm{Top}}_{\mathrm{loc}})\simeq \mathrm{Ho}(G{\mathrm{Top}}_{\mathrm{cof}})\stackrel{\simeq }{\to }\mathrm{Ho}([O_G^{\mathrm{op}},\mathrm{Top}]),$$Ho(G Top_{loc}) \simeq Ho(G Top_{cof}) \stackrel{\simeq}{\to} Ho([O_G^{op}, Top]) \,,

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

Proof

Stated as theorem VI.6.3 in EqHoCo.

$(\mathrm{\infty },1)$-category of $G$-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 ${\mathrm{Sh}}_{(\mathrm{\infty },1)}(\mathrm{Top})$ of (∞,1)-sheaves on $\mathrm{Top}$, at the collection of morphisms of the form $\{X\times I\to X\}$ with $I$ the real line.

The analogous statement is true for $G$-spaces: the equivariant homotopy category is the homotopy localization of the category of $\mathrm{\infty }$-stacks on $G\mathrm{Top}$.

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

Write

$$\mathrm{sSh}(G\mathrm{Top}{)}_{\mathrm{loc}}$$sSh(G Top)_{loc}

for the corresponding hypercomplete local model structure on simplicial sheaves.

Let $I$ be the unit interval, the standard interval object in Top, equipped with the trivial $G$-action, regarded as an object of $G\mathrm{Top}$ and hence in $\mathrm{sSh}(G\mathrm{Top})$.

Write

$$\mathrm{sSh}(G\mathrm{Top}{)}_{\mathrm{loc}}^I\overleftarrow{\to }\mathrm{sSh}(G\mathrm{Top}{)}_{\mathrm{loc}}$$sSh(G Top)_{loc}^I \stackrel{\leftarrow}{\to} sSh(G Top)_{loc}

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

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

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

This is exaple 3, page 50 of

• Morel, Voevosky, $A^1$-homtopy theory of schemes (pdf)

In more general model categories

Let $G$ be a finite group as above. We describe the generalizaton of the above story as Top is replaced by a more general model category $C$.

Definition and proposition

1. Let $C$ be a cofibrantly generated model category with generating cofibrations $I$ and generating acyclic cofibrations $J$.

   There is a cofibrantly generated model category

   $$[O_G^{\mathrm{op}},C{]}_{\mathrm{loc}}$$[O_G^{op}, C]_{loc}

   on the functor category from the orbit category of $G$ to $C$ by taking the generating cofibrations to be

   $$I_{O_G}:=\{G/H\times i{\}}_{i\in I,H\subset G}$$I_{O_G} := \{G/H \times i\}_{i \in I, H \subset G}

   and the generating acyclic cofibrations to be

   $$J_{O_G}:=\{G/H\times j{\}}_{j\in I,H\subset G}.$$J_{O_G} := \{G/H \times j\}_{j \in I, H \subset G} \,.

2. Let $BG$ be the delooping groupoid of $G$ and let

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

   be the functor category from $BG$ to $C$ – the category of objects in $C$ equipped with a $G$-action equipped with a set of generatinc (acyclic) cofibrations

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

   and the generating acyclic cofibrations to be

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

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

Definition and proposition

(generalized Elmendorf’s theorem)

There is a Quillen adjunction

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

and a Quillen equivalence

$$\Theta :[O_G^{\mathrm{op}},C{]}_{\mathrm{loc}}\overleftarrow{\to }[B{G}^{\mathrm{op}},C{]}_{\mathrm{loc}}:\Phi .$$\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 $\mathrm{\infty }$-stack $(\mathrm{\infty },1)$-toposes

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

$$C:=L_A\mathrm{SPSh}(D)$$C := L_A SPSh(D)

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

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

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

• For instance motivic cohomology is the cohomology of the ∞-stack (∞,1)-topos on the Nisnevich site, presented by $C:=L_{\mathrm{Cech}}\mathrm{SPSh}(\mathrm{Nis})$ . Its $G$-equivariant version as above should be the right context for the Bredon $G$-equivariant cohomology refinement of motivic cohomology.

  This is example 4.5 in Guillou.

  (Actually here one localizes moreover at hypercovers and at A1-homotopies.)

Related concepts

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

References

• 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)

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

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