nLab equivariant stable homotopy category

Contents

Context

Representation theory

representation theory

geometric representation theory

Contents

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)

Last revised on December 14, 2020 at 15:05:11. See the history of this page for a list of all contributions to it.