nLab naive G-spectrum

Idea

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.

Properties

Relation to $G$-spectra

For $U$ a universe of $G$-representations in the sense of genuine G-spectra, there is the inclusion

$$i \;\colon\; U^G \longrightarrow U$$

of the fixed points. A genuine G-spectrum modeled on $U^G$ is a spectrum with G-action (“naive G-spectrum”). The induced adjunction

$$(i_\ast \dashv i^\ast) \;\colon\; G Spectra \longrightarrow Spectra(PSh_\infty(Orb_G))$$

has unit and counit which are equivalences on the underlying bare spectra.

(e.g. Carlsson 92, p. 14, Greenlees May, p. 16)

Homotopy fixed points and homotopy quotients

Write

$$(-)^G \;\colon\; Spectra(G Top) \longrightarrow Spectra \,.$$

for the fixed point (invariants) functor and

$$(-)/Q \;\colon\; Spectra(G Top) \longrightarrow Spectra$$

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

$$EG_+ \wedge (-) , [EG_+,-] \;\colon\; Spectra(G Top) \to Spectra(G Top)$$

given by forming smash product with $E G_+$ and forming the internal hom out of $E G_+$.

Then the homotopy fixed point functor is

$$(-)^{h G} \coloneqq [E G_+, -]^G \;\colon\; Spectra(G Top) \longrightarrow Spectra$$

and the homotopy quotient functor is

$$(-)//G \coloneqq (-)_{h G} \coloneqq (E G_+ \wedge (-))/G \;\colon\; Spectra(G Top) \longrightarrow Spectra$$

Relation to equivariant cohomology

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

$$E^\bullet(X) \coloneqq \pi_\bullet(Hom(X,E)^G) \,,$$

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

$$E(X) = \underset{\mathbf{B}G}{\prod} [X,E]$$

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 $E$:

$$E//G \;\colon\; G/H \mapsto E^{H} \,.$$

References

• 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:

• Stefan Schwede, Spectra in model categories and applications to the algebraic cotangent complex, Journal of Pure and Applied Algebra 120 (1997) 77-104 (pdf)