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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.
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.
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 $E^G$ (called the genuine fixed point spectrum) and a map $E^G \to E^{hG}$ ($E^{hG}$ = homotopy fixed points).
A genuine G-spectra is a spectrum $E$ with $G$-action, together with another spectrum $E^G$ and morphisms $E_{hG} \to E^G \to E^{hG}$ ($E_{hG}$ = homotopy orbits). (Alternatively, you can specify a geometric fixed point spectrum $\Phi^G(E)$ and a map to the Tate construction.)
This is proven in:
Sure, let’s change the terminology in the entry.
Ok, done.
Thanks. You should add the content of your comment #3 to some entry, too!
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.)
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?
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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