# Contents

## 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 $I = \mathbb{R}$ 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 $I$

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

### Homotopical categories of $G$-equivariant spaces

###### Definition

(models for $G$-equivariant spaces)

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

1. Write

$G Top_{cof} \subset G Top$

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

$G Top_{loc}$

for all of $G Top$ equipped with weak equivalences given by those morphisms $(f : X \to Y) \in G Top$ 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

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

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:

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

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.

### $(\infty,1)$-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 $Sh_{(\infty,1)}(Top)$ of (∞,1)-sheaves on $Top$, 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 $\infty$-stacks on $G Top$.

More in detail: let $G Top$ 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

$sSh(G Top)_{loc}$

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 $G Top$ and hence in $sSh(G Top)$.

Write

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

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

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

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

This is example 3, page 50 of

## 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

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

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

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

and the generating acyclic cofibrations to be

$J_{O_G} := \{G/H \times j\}_{j \in I, H \subset G} \,.$
2. Let $\mathbf{B}G$ be the delooping groupoid of $G$ and let

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

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

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

and the generating acyclic cofibrations to be

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

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

###### Definition and proposition

(generalized Elmendorf’s theorem)

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

and a Quillen equivalence

$\Theta : [O_G^{op}, C]_{loc} \stackrel{\leftarrow}{\to} [\mathbf{B}G^{op},C]_{loc} : \Phi \,.$
###### Proof

This is proposition 3.1.5 in Guillou.

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

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

$C := L_A SPSh(D)$

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 $\mathbf{H}$.

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

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

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