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I was recently in the situation of having access to the category of $G$-modules (for some group $G$ which I had forgotten), as just a category, i.e. no monoidal structure, together with the forgetful functor to $\mathbb{Z}$-modules, and wanting to reconstruct $G$.

Alas, I could only reconstruct $\mathbb{Z}[G]$ (as endomorphisms of the forgetful functor), and apparently, there are groups $G$ for which $\mathbb{Z}[G]$ does not know $G$.

In principle, I could learn what a spectrum is and work with those instead --- would it help? I.e.,

Does the category of $G$-spectra (together with the forgetful functor to spectra) know $G$?

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    $\begingroup$ A spectrum is a topologist's abelian group, so the result is similar: you can only learn the stabilization $\Sigma^\infty G$. If you need $G$ itself, then a better choice would be the unstable equivariant category of $G$-spaces, but even then I am unsure if you can reconstruct $G$ itself and not just its homotopy type. You can probably do it for compact groups, but I don't have a reference. $\endgroup$
    – Anton Fetisov
    Commented Apr 16, 2016 at 9:17
  • $\begingroup$ What kind of $G$-spectra are talking about here? I know at least three types.. $\endgroup$
    – Denis Nardin
    Commented Apr 16, 2016 at 11:28
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    $\begingroup$ I think that the question might have an affermative answer for genuine $G$-spectra (a.k.a. spectral Mackey functors) although the lack of a monoidal structure worries me a little (also I'm assuming that $G$ is a (pro)finite group here, or at least some kind of compact Lie group because otherwise I don't know how genuine $G$-spectra are defined) $\endgroup$
    – Denis Nardin
    Commented Apr 16, 2016 at 11:56
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    $\begingroup$ @Vivek: explicitly it's the suspension spectrum, and heuristically it's the "group algebra over the sphere spectrum." No idea how to show it doesn't know $G$ (as an $E_1$ space; that's the best we can hope for), but it at least doesn't obviously know $G$. $\endgroup$
    – Qiaochu Yuan
    Commented Apr 16, 2016 at 18:47
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    $\begingroup$ Re. the question of "why not use some nonabelian coefficients like all $G$-spaces", the point for me is that my access to the category of $G$-modules comes via constructible sheaves on a space with fundamental group $G$. Of course, one can take constructible sheaves with nonabelian coefficients. However, I need to use microlocal techniques in sheaf theory, which fundamentally depend on the ability to take cones, shifts, etc. Thus, I need to use coefficients which allow this; ergo, spectra rather than spaces. $\endgroup$
    – Vivek Shende
    Commented Apr 17, 2016 at 20:08

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