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nLab > Latest Changes: spectrum with G-action

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeApr 14th 2014

    added to spectrum with G-action brief paragraphs “Relation to genuine G-spectra”, and “relation to equivariant cohomology”.

    Both would deserve to be expanded much more, but it’s a start.

    • CommentRowNumber2.
    • CommentAuthoradeelkh
    • CommentTimeDec 19th 2017
    • (edited Dec 19th 2017)

    The obvious definition of spectrum with G-action is just that: a spectrum with a G-action. This is even more naive than the notion discussed at spectrum with the G-action (which are sometimes called naive G-spectra), so these are sometimes called “doubly naive” G-spectra; they’re what you get if you start with the naive version of G-spaces, i.e. just spaces with G-action, and then naively stabilize. These doubly naive G-spectra can also be identified with a certain full reflective subcategory of genuine G-spectra whose objects are sometimes called “Borel-complete” G-spectra: they have the property that homotopy fixed points coincide with genuine fixed points with respect to all subgroups of GG (at least if GG is finite).

    Anyway, I’m not an expert here, but maybe we should use a different terminology for spectrum with G-action.

    • CommentRowNumber3.
    • CommentAuthoradeelkh
    • CommentTimeDec 19th 2017
    • (edited Dec 19th 2017)

    Let me also mention a further argument for reserving some terminology for “a spectrum with a G-action”. In the special case when G is cyclic of prime order, we have simple models of naive and genuine G-spectra in terms of spectra with G-action:

    • A naive G-spectrum is a spectrum EE with GG-action, together with another spectrum E GE^G (called the genuine fixed point spectrum) and a map E G→E hGE^G \to E^{hG} (E hGE^{hG} = homotopy fixed points).

    • A genuine G-spectra is a spectrum EE with GG-action, together with another spectrum E GE^G and morphisms E hG→E G→E hGE_{hG} \to E^G \to E^{hG} (E hGE_{hG} = homotopy orbits). (Alternatively, you can specify a geometric fixed point spectrum Φ G(E)\Phi^G(E) and a map to the Tate construction.)

    This is proven in:

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeDec 19th 2017

    Sure, let’s change the terminology in the entry.

    • CommentRowNumber5.
    • CommentAuthoradeelkh
    • CommentTimeDec 19th 2017

    Ok, done.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeDec 19th 2017
    • (edited Dec 19th 2017)

    Thanks. You should add the content of your comment #3 to some entry, too!

    • CommentRowNumber7.
    • CommentAuthorDylan Wilson
    • CommentTimeDec 19th 2017

    Does “sometimes referred to as” mean “once, by Jacob, during the Thursday seminar, as a joke”?

    I think ’spectrum with a G-action’ is fine, and maybe also ’Borel G-spectrum’. All terminology with the word “naive” should probably be removed from the equivariant literature… (Especially because the ’official’ usage of ’naive G-spectrum’ from LMS- equivalent to presheaves of spectra on the orbit category- seems to be mostly unused in practice and also isn’t what people think it means.)

    • CommentRowNumber8.
    • CommentAuthoradeelkh
    • CommentTimeDec 19th 2017
    • (edited Dec 19th 2017)

    Does “sometimes referred to as” mean “once, by Jacob, during the Thursday seminar, as a joke”?

    Yes.

    I think ’spectrum with a G-action’ is fine, and maybe also ’Borel G-spectrum’.

    Well, those are the two suggestions I gave as well. I realize the above discussion doesn’t make sense anymore, after I renamed the pages. The page spectrum with G-action used to refer to what is now at naive G-spectrum.

    All terminology with the word “naive” should probably be removed from the equivariant literature… (Especially because the ’official’ usage of ’naive G-spectrum’ from LMS- equivalent to presheaves of spectra on the orbit category- seems to be mostly unused in practice and also isn’t what people think it means.)

    “Presheaves of spectra on the orbit category” is the definition I would give of naive G-spectrum as well. I would agree “naive” is not the best terminology, but is there a more standard option?

    • CommentRowNumber9.
    • CommentAuthoradeelkh
    • CommentTimeDec 19th 2017

    Just to be clear, as an outsider to equivariant homotopy theory, I thought the following terminology was close to standard these days:

    • Genuine G-spectra: Invert all representation spheres in G-spaces. Modeled by spectral Mackey functors, for example.

    • Naive G-spectra: Invert S^1 in G-spaces. Modeled by presheaves of spectra on the orbit category.

    • Spectra with G-action: Functors BG -> Spectra, or Borel-complete genuine G-spectra.

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