nLab equivariant stable homotopy category

Idea

The generalization of the stable homotopy category from stable homotopy theory to equivariant stable homotopy theory.

Definition

A homomorphism $f \colon E_1 \longrightarrow E_2$ between two G-spectra, indexed on a G-universe $\mathcal{U}$, is called an equivariant weak homotopy equivalence if the following equivalent conditions hold

1. For each $V\in \mathcal{U}$ the component map $f_V \colon f$ induces ordinary weak homotopy equivalences $(E_1)_V^H \to (E_2)_V^H$ on all fixed point spaces for all closed subgroups $H \hookrightarrow G$.

2. For each $n \in \mathbb{Z}$ and each closed subgroup $H \hookrightarrow G$ the morphism $f$ induces an isomorphism of Mackey functors of equivariant homotopy groups $\pi_n^H(E_1) \stackrel{\simeq}{\longrightarrow} \pi_n^H(E_2)$.

(The equivalence of these conditions is part of the equivariant Whitehead theorem.)

The $G$-equivariant stable homotopy category is the homotopy category of G-spectra with respect to these weak equivalences.

Properties

Relation to Mackey functors

The full subcategory $\mathcal{B}_G$ of the equivariant stable homotopy category on the objects of the form

$$G/H_* \wedge \Sigma^\infty S^n$$

is, as an additive category, the domain of Mackey functors, such as the equivariant homotopy group-functors.

References

• John Greenlees, Peter May, section 2 of Equivariant stable homotopy theory, in I.M. James (ed.), Handbook of Algebraic Topology , pp. 279-325. 1995. (pdf)

• Anna Marie Bohmann, Basic notions of equivariant stable homotopy theory, (pdf)