# nLab global family

Contents

### Context

#### Stable Homotopy theory

stable homotopy theory

Introduction

# Contents

#### Representation theory

representation theory

geometric representation theory

# Contents

## Idea

In the context of global equivariant homotopy theory and global equivariant stable homotopy theory one considers equivariant spectra with equivariance not necessarily with respect to one fixed group, but with respect to a suitable family of groups at once. These “suitable families of groups for defining global equivariant (stable) homotopy theory” are called global families, for short.

## Definition:

###### Definition

A non-empty class of compact Lie groups is called a global family if it is closed under

(e.g. Schwede 18, def. 1.4.1)

A global family is said to be reflexive if and only if the inclusion into the category of all compact Lie groups has a left adjoint. Schwede 18, def 4.5.7.

## Examples

###### Example

The degenerate case of a global family (def. ) contains only the trivial group $1$. The (stable) global equivariant homotopy theory with respect to this choice is simply ordinary non-equivariant homotopy theory (stable homotopy theory).

###### Example

The following are examples of global families (def. ): the classes of

1. all quotient groups of closed subgroups of a fixed compact Lie group.

2. all finite groups.

3. all finite abelian groups.

4. all finite cyclic groups.

5. all finite p-primary groups.

6. all finite solvable p-groups.

## References

Last revised on June 24, 2019 at 12:57:36. See the history of this page for a list of all contributions to it.