nLab Spectral Mackey functor theorem

Idea

Let $G$ be a finite group. The Wirthmüller isomorphism implies that the suspension G-spectrum of a transitive G-set is $\mathbb{S}_G$-linearly self-dual; in particular, given a map $[G/K] \rightarrow [G/H]$, self-duality yields a transfer map $\Sigma_{G,+}^\infty [G/H] \rightarrow \Sigma_{G,+}^{\infty} [G/K]$. This underlies a functor from the Burnside category into $\mathrm{Sp}_G$; taking mapping spectra out of $[G/K]$ then yields a functor from $G$-spectra to Mackey functors valued in spectra:

$$F\colon \mathrm{Sp}_G \rightarrow \mathrm{Fun}^{\times}\left(\mathrm{Span}(\mathbb{F}_G), \mathrm{Sp}\right).$$

The Spectral Mackey functor theorem states that $F$ is an equivalence.

Precise statements

The first proof of the Spectral Mackey functor appeared in Guillou-May 11. To state it, we write $\mathrm{Sp}_G^{O}$ for the model category of orthogonal $G$-spectra, and $\mathrm{Sp}^O$ for orthogonal spectra. $\mathrm{Sp}_G^O$ comes enriched over $\mathrm{Sp}^O$, as does the projective model structure on the spectral presheaf category $\mathrm{Fun}_{\mathrm{Sp}^{\mathcal{O}}}(\mathrm{Span}^+(\mathbb{F}_G)^{\op}, \mathrm{Sp}^O)$.

Theorem

There is a zigzag of $\mathrm{Sp}^O$-enriched quillen equivalences connecting $\mathrm{Sp}_G^O$ and $\mathrm{Fun}_{\mathrm{Sp}^{\mathcal{O}}}(\mathrm{Span}^+(\mathbb{F}_G)^{\op}, \mathrm{Sp}^O)$.

This presents an $\infty$-categorical equivalence, who was later directly proved using $\infty$-categorical means in Nardin 16. The strategy is to recognize the G-∞-category of orthogonal G-spectra? as the $G$-stabilization of G-spaces; then, one may recognize $G$-stability as equivalent to naive stability and $G$-semiadditivity. Since each are presented by smashing localizations of the $\infty$-category of presentable $G$-$\infty$-categories, these processes preserve each other; $G$-semiadditivization of a $G$-category of coefficient systems is computed by (semi)-Mackey functors, and naive stabilization by stabilization of the value category, so $G$-stabilization of coefficient systems in $\mathcal{C}$ is equivalent to Mackey functors valued in spectrum objects in $\mathcal{C}$. Said altogether, Nardin proved the following.

Theorem

The forgetful $G$-functor $\mathrm{Sp}_G^O[\mathrm{weq}^{-1}] \rightarrow \mathcal{S}_G$ lifts through an equivalence

$$\mathrm{Sp}_G^O[\mathrm{weq}^{-1}] \simeq \mathrm{Fun}^\times \left( \mathrm{Span}(\mathbb{F}_G), \mathrm{Sp} \right).$$

References

• Bertrand Guillou, Peter May, Models of $G$-spectra as presheaves of spectra (arXiv:1110.3571)

• Denis Nardin, Parametrized higher category theory and higher algebra: Exposé IV – Stability with respect to an orbital \infty-category (arXiv:1608.07704v4)