# nLab proper equivariant homotopy theory

Contents

### Context

#### Representation theory

representation theory

geometric representation theory

# Contents

## Idea

Proper $G$-equivariant homotopy theory (DHLPS 19) is the variant of $G$-equivariant homotopy theory where the topological group $G$ is allowed to be non-compact (for instance a non-finite discrete group) but whose weak equivalences are still detected only on fixed point spaces of compact subgroups of $G$.

Hence in the presentation by topological G-spaces, proper $G$-equivariant homotopy theory is obtained by localizing at those equivariant continuous functions $f \;\colon\; X \to Y$ which induce weak homotopy equivalences $f^H \;\colon\; X^H \to Y^H$ for all compact subgroups $H \subset G$ (DHLPS 19, Def. 1.1.2).

Equivalently, under Elmendorf's theorem (DHLPS 19, p. 87) this is the homotopy theory of (∞,1)-presheaves on the full sub-(∞,1)-category of the $G$-orbit category on those orbits corresponding to compact subgroups (e.g. finite subgroups for $G$ discrete, details in DHLPS 19, Remark 3.1.12).

## References

The concept was introduced in

following

• James Davis, Wolfgang Lück, Spaces over a Category and Assembly Maps in Isomorphism Conjectures in K- and L-Theory, K-Theory 15:201–252, 1998 (pdf)

Last revised on October 30, 2020 at 18:13:12. See the history of this page for a list of all contributions to it.