Is there a relationship between norms/transfers in equivariant homotopy theory and norms in the Tate construction / ambidexterity?

Question

In homotopy theory, the word "norm" is commonly used in two different ways (well, surely there are other ways, but these two have a particular familial resemblence).

1. Let $G$ be a finite group. A $G$-spectrum $E$ can be restricted to an $H$-spectrum for any subgroup $H \subseteq G$, and there is an "inclusion of fixed points" map $E^G \to E^H$. If $E$ is "genuine" then there is also a "wrong-way" map in the other direction, called the transfer. There is a similar multiplicative story: if $E$ is a $G$-ring spectrum, then for any map of finite $G$-sets $T \to S$, there is an "inclusion of fixed points" map $(\wedge^S E)^G \to (\wedge^T E)^G$. If the multiplicative structure on $E$ is "genuine" then there is also a "wrong-way" map in the other direction, called the norm.

2. Let $G$ be a finite group acting on a spectrum $E$. Then there is a natural composite map $E^{hG} \to E \to E_{hG}$, but there is also a "wrong-way" map in the other direction, called the norm. If the norm is invertible, the action is ambidextrous, and the obstruction to ambidexterity (i.e. the cofiber $E^{tG}$ of the norm) is the Tate construction.

Question: Is there a relationship between these two uses of the word "norm" (or perhaps between "norm" in the second sense and "transfer")?

I think this is more than a coincidence of terminology partly because Nikolaus and Scholze use norm data in the second sense to describe cyclotomic spectra, a sort of equivariant spectrum, which data is normally encoded in terms of transfers. But I haven't studied Nikolaus-Scholze closely enough to extract what's going on.

Answer 1

Here's a bit more of an organized answer:

First you have to decide how you'd like to model the notion of a genuine $G$-spectrum. There are at least ways that it's currently in vogue to do this, each of which is convenient for different purposes. (My groups are finite below, and I won't write down functors that aren't homotopically meaningful- so everything is 'derived'.)

1. One could begin with the homotopy theory of $G$-spaces, where weak equivalences are detected by checking on all fixed points and then 'invert' $\Omega^{\rho}$ or $\Sigma^{\rho}$, where $\rho$ is the regular representation. There are lots of ways of doing this, but no matter how you do it you will be able to present any $G$-spectrum as a homotopy colimit $X = \mathrm{hocolim}\, S^{-n\rho}\wedge\Sigma^{\infty}X_{n\rho}$ where the $X_{n\rho}$ are pointed $G$-spaces. This way of thinking about things makes it easy to define symmetric monoidal, accessible functors $\mathrm{Sp}^G \to \mathcal{C}$ since they'll be determined by what they do to suspension spectra. For example, from this point of view you can define geometric fixed points by doing what you expect: commuting with the formation of suspension spectra, commuting with homotopy colimits, and being symmetric monoidal. The other type of fixed points, genuine fixed points, are essentially uniquely determined by the property that the functor $(-)^G: \mathrm{Sp}^G \to \mathrm{Sp}$ is exact and satisfies $\Omega^{\infty}(X^G) = (\Omega^{\infty}X)^G$. (This functor is not symmetric monoidal- for example, it doesn't even send the unit to the right place, by the tom Dieck splitting). Alternatively, one can use that every stable homotopy theory is canonically enriched in spectra, and take the spectrum of maps from $S^0$ to $X$. Now, in this setting I sketched an answer to your question about transfers (the classical answer) which came from the map $\mathrm{E}G_+ \to S^0$. This induces a map $(\mathrm{EG}_+ \wedge X)^G \to X^G$. Now, why is it the case that $(\mathrm{EG}_+ \wedge X)^G$ is the same as the homotopy orbits? (A case of the 'Adams isomorphism'). You can prove this inductively using the standard bar filtration of $\mathrm{EG}_+$, i.e. inductively define a map $(\mathrm{sk}_j\mathrm{E}G_+ \wedge X)_{hG} \to (\mathrm{sk}_j\mathrm{E}G_+ \wedge X)^G$ and show it's an equivalence. The base case and inductive case reduce to the statement that there's a natural equivalence $(T_+ \wedge X)_{hG} \to (T_+ \wedge X)^G$ where $T$ is a finite free $G$-set. In other words, $(T/G)_+ \wedge X = \mathrm{map}_{\mathrm{Sp}}((T/G)_+, X)$ (using the self-duality of finite sets in $\mathrm{Sp}$) is naturally equivalent to $\mathrm{map}_{\mathrm{Sp}^G}(S^0, T_+ \wedge X)$. This, finally, follows because every finite $G$-set is self dual in $\mathrm{Sp}^G$: embed into some $G$-representation and produce a collapse map as in the proof of Atiyah duality to define the duality datum.
2. Another choice is to define the homotopy theory of genuine G-spectra as 'spectral Mackey functors', following Guillou-May and Barwick-Dotto-Glasman-Nardin-Shah, just take the homotopy theory of product preserving functors $\mathrm{Span}(G) \to \mathrm{Sp}$ where the former denotes the $(2,1)$-category of finite $G$-sets with mapping groupoids given by the groupoid of spans. And here I'm working $\infty$-categorically, so these are homotopy coherent functors. The value of the functor on the orbit [G/H] is the H-fixed points. In particular, there are natural inclusions of both the orbit category and its opposite into the category of spans. The map $[G] \to [*]$ and its 'opposite' automatically, by functoriality, gives the maps $X_{hG} \to X^G$ and $X^G \to X^{hG}$. The identification of the composite as the usual norm is then a kind of `double-coset formula', i.e. it follows from the formula for composition of spans coming from pullbacks.
3. Finally, there is the paper of Glasman which codifies and makes precise ideas of Greenlees and Greenlees-May: G-spectra can be described by their collection of geometric fixed points, and gluing data having something to do with Tate spectra. From this point of view it's easiest to say what the transfer maps here are in the case of something like $G=C_p$ (so I don't have to talk about generalized Tate constructions and so on). From this point of view, a $C_p$-spectrum is a triple $(X, X^{\Phi C_p}, X^{\Phi C_p} \to X^{tC_p})$ where $X$ is a Borel $C_p$-spectrum. The genuine fixed points are then defined as the pullback of $X^{hC_p} \to X^{tC_p} \leftarrow X^{\Phi C_p}$. This automatically produces a map $X_{hC_p} \to X^{C_p}$ factoring the 'norm/trace' since the Tate spectrum is the cofiber of the norm/trace.

All of the above is well-documented, with references enough to suit any taste. There's Lewis-May-Steinberger, the 'Alaska notes', an online book of Schwede, the first few sections of Hill-Hopkins-Ravenel and their appendices, that paper of Glasman's, etc. etc.