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I just read some parts of the book "Equivariant orthogonal spectra and S-modules" by Mandell and May. I wonder whether there is any description of the relation between the categories of orthogonal G-spectra when G varies, other than the change of group functor and fixed point functor in Chapter V.

Thanks.

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    $\begingroup$ What kind of reationship are you looking for? There are a bunch of functors between them: genuine fixed points, geometric fixed points, induction, coinduction, HHR Norm,... $\endgroup$
    – Denis Nardin
    Commented Feb 4, 2018 at 16:19

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Megan, I imagine that what you have in mind is comparisons of $G$-spectra as $G$ varies, not restricting attention just to subgroups of a given $G$ as in the references cited so far. There are some things that hold universally in broad classes of groups and others that are specific to a given group (and its subgroups). You might be interested in ``Global orthogonal spectra'' as studied by Anna Marie Bohmann and, very extensively, Stefan Schwede. See

http://intlpress.com/site/pub/pages/journals/items/hha/content/vols/0016/0001/a017/index.html

http://www.math.uni-bonn.de/people/schwede/global.pdf

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In the paper you mention, Chapter V, section 2 is all about "change of group," and tells you about the various adjoint functors involved. See also Mandell's paper Equivariant Symmetric Spectra for a summary of what's known about change of group.

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