nLab
equivariant stable homotopy category
Contents
Context
Stable Homotopy theory
Representation theory
representation theory

geometric representation theory

Ingredients
Definitions
representation , 2-representation , ∞-representation

group , ∞-group

group algebra , algebraic group , Lie algebra

vector space , n-vector space

affine space , symplectic vector space

action , ∞-action

module , equivariant object

bimodule , Morita equivalence

induced representation , Frobenius reciprocity

Hilbert space , Banach space , Fourier transform , functional analysis

orbit , coadjoint orbit , Killing form

unitary representation

geometric quantization , coherent state

socle , quiver

module algebra , comodule algebra , Hopf action , measuring

Geometric representation theory
D-module , perverse sheaf ,

Grothendieck group , lambda-ring , symmetric function , formal group

principal bundle , torsor , vector bundle , Atiyah Lie algebroid

geometric function theory , groupoidification

Eilenberg-Moore category , algebra over an operad , actegory , crossed module

reconstruction theorems

Theorems
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

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 $H \hookrightarrow G$ .

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
Last revised on December 14, 2020 at 20:05:11.
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