homotopy theory, (∞,1)-category theory, homotopy type theory
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Be?linson-Bernstein localization?
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).
The concept was introduced in
following
Last revised on April 17, 2023 at 09:49:32. See the history of this page for a list of all contributions to it.