geometric representation theory
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Be?linson-Bernstein localization?
For $G$ a topological group, a naive $G$-spectrum is a spectrum object in the category of G-spaces (e.g. Carlsson 92, def. 3.1). The stable homotopy theory of spectra with $G$-action is part of the subject of equivariant stable homotopy theory.
There one typically considers a richer concept of G-spectra. In that context a spectrum with $G$-action is a G-spectrum for the “trivial universe” and is called a naive G-spectrum. Naive G-spectra are obtained by stabilizing G-spaces with respect to the sphere $S^1$, while (genuine) G-spectra are obtained by inverting all representation spheres.
For $U$ a universe of $G$-representations in the sense of genuine G-spectra, there is the inclusion
of the fixed points. A genuine G-spectrum modeled on $U^G$ is a spectrum with G-action (“naive G-spectrum”). The induced adjunction
has unit and counit which are equivalences on the underlying bare spectra.
(e.g. Carlsson 92, p. 14, Greenlees May, p. 16)
Write
for the fixed point (invariants) functor and
for the quotient (coinvariants) functor.
Write $E G$, as usual, for a contractible topological G-space whose $G$-action is free. Write $E G_+$ for this regarded as a pointed $G$-space with a point freely adjoined.
Then there are two functor
given by forming smash product with $E G_+$ and forming the internal hom out of $E G_+$.
Then the homotopy fixed point functor is
and the homotopy quotient functor is
Spectra with $G$-action represent $\mathbb{Z}$-graded equivariant cohomology on G-spaces.
For $X$ a G-space and $E$ a spectrum with $G$-action, then the corresponding cohomology is
where on the right we have the homotopy fixed points of the mapping spectrum, which inherits a conjugation action by $G$ from the $G$-action on $X$ and $E$.
More abstractly, in terms of the tangent cohesive (∞,1)-topos $T PSh_\infty(Orb)_{/\mathbf{B}G}$ of the slice (∞,1)-topos of orbispaces over $\mathbf{B}G$, this means that
is the dependent product over $\mathbf{B}G$ of the intrinsic cohomology of the tangent slice topos. See at ∞-action for more on this.
Notice here Elmendorf's theorem which identifies G-spaces with (∞,1)-presheaves over the orbit category $Orb_G$. It is via this equivalence that spectra with $G$-action represent equivariant cohomology in the form of Bredon cohomology.
Hence exhibiting a spectrum $E$ with $G$-action as a spectrum-valued presheaf on the orbit category means to assign to any coset space $G/H$ of $G$ the $H$-homotopy fixed points of $E$:
Gunnar Carlsson, A survey of equivariant stable homotopy theory, Topology, Vol 31, No. 1, pp. 1-27, 1992 (pdf)
John Greenlees, Peter May, Equivariant stable homotopy theory (pdf)
On model category structure on naive $G$-spectra:
Last revised on February 1, 2024 at 13:20:57. See the history of this page for a list of all contributions to it.