nLab proper equivariant homotopy theory



Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology



Paths and cylinders

Homotopy groups

Basic facts


Representation theory



Proper GG-equivariant homotopy theory (DHLPS 19) is the variant of GG-equivariant homotopy theory where the topological group GG 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 GG.

Hence in the presentation by topological G-spaces, proper GG-equivariant homotopy theory is obtained by localizing at those equivariant continuous functions f:XYf \;\colon\; X \to Y which induce weak homotopy equivalences f H:X HY Hf^H \;\colon\; X^H \to Y^H for all compact subgroups HGH \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 GG-orbit category on those orbits corresponding to compact subgroups (e.g. finite subgroups for GG discrete, details in DHLPS 19, Remark 3.1.12).


The concept was introduced in


  • 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 April 17, 2023 at 09:49:32. See the history of this page for a list of all contributions to it.