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nLab > Latest Changes: equivariant suspension spectrum

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeDec 4th 2018

    Is the equivariant suspension spectrum functor still strong monoidal, homotopically?

    diff, v10, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeDec 4th 2018
    • (edited Dec 4th 2018)

    [ duplicate deleted ]

    diff, v10, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeDec 29th 2018

    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.

    • CommentRowNumber4.
    • CommentAuthorGuest
    • CommentTimeDec 30th 2018
    If you construct the stable equivariant homotopy category by inverting representation spheres, then this holds by construction ; see the paper of Marco Robalo, "K-Theory and the bridge from Motives to non-commutative Motives", for the universal property of monoidal inversion of objects in symmetric monoidal infinity-categories.
    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeDec 30th 2018

    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?

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeDec 30th 2018

    While I have your attention (if I do), here is another simple question of this kind:

    for E 1,E 2E_1, E_2 two genuine GG-spectra, let [E 1,E 2][E_1, E_2] denote their internal hom, i.e. the genuine G-spectrum of maps, and let Hom(E 1,E 2)Hom(E_1, E_2) denote their external hom, i.e. the plain mapping spectrum.

    Then, I suppose, the latter is the GG-fixed point spectrum of the former

    Hom(E 1,E 2)≃[E 1,E 2] G Hom(E_1, E_2) \;\simeq\; [E_1, E_2]^G

    ?

    • CommentRowNumber7.
    • CommentAuthorGuest
    • CommentTimeJan 1st 2019

    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,−)Hom(X, -) of both sides, for a spectrum XX. You have Hom(X,Y G)=Hom(X⊗S G,Y)Hom(X, Y^G) = Hom(X \otimes S_G, Y) where S GS_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.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeJan 2nd 2019

    Thanks! Okay, I have tried to sum this up in the entry (here) as follows:

    [[monoidalness of Σ G ∞\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.

    diff, v11, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeJan 4th 2019
    • (edited Jan 4th 2019)

    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.

    diff, v13, current

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