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Is the equivariant suspension spectrum functor still strong monoidal, homotopically?
Of course it ought to be, but what I’d like to have is a convenient citation for a proof. Checking on MO’s homotopy chat here it looks like that citation may not exist.
Thanks.
I gather the relevant section in Rabolo’s “bridge” publication (doi:10.1016/j.aim.2014.10.011) is equivalently section 4 in his preprint “Noncommutative Motives I” (arXiv:1206.3645). And here we need the last clause of Prop. 4.1 combined with the last clause of Prop. 4.10 (1).
Okay, this is for inverting a single object, right? Propbably the idea is that the same conclusion goes through for inversion at a set of objects?
While I have your attention (if I do), here is another simple question of this kind:
for $E_1, E_2$ two genuine $G$-spectra, let $[E_1, E_2]$ denote their internal hom, i.e. the genuine G-spectrum of maps, and let $Hom(E_1, E_2)$ denote their external hom, i.e. the plain mapping spectrum.
Then, I suppose, the latter is the $G$-fixed point spectrum of the former
$Hom(E_1, E_2) \;\simeq\; [E_1, E_2]^G$?
First question : exactly. This works for any finite set of course (same as inverting the tensor product of all the objects) and in general you can write it is a filtered colimit of finite inversions. A reference for this general consideration is in the paper of Marc HOyois, http://www-bcf.usc.edu/~hoyois/papers/equivariant.pdf (6.1).
Second question : actually I am not an expert on equivariant homotopy, sorry to give you a wrong impression. But it seems to me that this statement is formal: consider $Hom(X, -)$ of both sides, for a spectrum $X$. You have $Hom(X, Y^G) = Hom(X \otimes S_G, Y)$ where $S_G$ is the equivariant sphere spectrum. (To prove it depends on your foundations I guess but for example I think this is obvious with the spectral Mackey functors definition). So I think you get the same thing on both sides.
Thanks! Okay, I have tried to sum this up in the entry (here) as follows:
$[$monoidalness of $\Sigma^\infty_G$ $]$ follows from general properties of stabilization when regarding equivariant stable homotopy theory as the result of inverting smash product with all representation spheres, via Robalo 12, last clause of Prop. 4.1 with last clause of Prop. 4.10 (1) , generalized to sets of objects as in Hoyois 15, section 6.1, see also Hoyois 15, Def. 6.1.
added this pointer:
Alternatively, under the equivalence of genuine G-spectra with spectral Mackey functors on the Burnside category, it follows as in Nardin 12, Remark A.12.
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