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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 G (at least if G 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 E with G-action, together with another spectrum EG (called the genuine fixed point spectrum) and a map EGEhG (EhG = homotopy fixed points).

    • A genuine G-spectra is a spectrum E with G-action, together with another spectrum EG and morphisms EhGEGEhG (EhG = homotopy orbits). (Alternatively, you can specify a geometric fixed point spectrum ΦG(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.