nLab global family

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

1. isomorphism;

2. closedsubgroups;

3. quotient groups

(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 compact Lie groups;

2. all abelian compact Lie groups;

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

4. all finite groups.

5. all finite abelian groups.

6. all topologically cyclic groups.

7. all finite cyclic groups.

8. all finite p-primary groups.

9. all finite solvable p-groups.

References

• Stefan Schwede, section 1.4 of Global homotopy theory