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Let $G$ be a compact Lie group; I will write $SH_G$ for the (equivariant) stable homotopy category of $G$-spectra (say, with respect to a complete universe; does its choice affect the homotopy category?). Now, in the well-known announcement paper "Ordinary $RO(G)$-graded cohomology" (by G. Lewis, J.P. May, and J. McClure; https://projecteuclid.org/euclid.bams/1183548004) to any G-CW spectrum $Y\in SH_G$ a complex $C_*Y$ whose terms are Mackey functors (i.e., contravariant functors from the orbit category) was associated. Using this functor one easily defines an (equivariant) cohomology functor $H(-,M)$ with values in abelian groups corresponding to any Mackey functor $M$.

My question is: where can I find details on these functors? In particular, does the association $Y\mapsto C_*(Y)$ gives an exact functor from $SH_G$ into the derived category of Mackey functors?

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    $\begingroup$ The derived category of Mackey functors is equivalent to the category of $HA$-modules, where $A$ is the Burnside Mackey functor (the unit for the tensor product of Mackey functors). Then the functor you want is simply $HA\wedge -$. Would you be interested in a reference for the above statement? $\endgroup$ – Denis Nardin Aug 24 '18 at 12:21
  • $\begingroup$ Dear Denis: yes, a reference of this sort would certainly be very interesting. $\endgroup$ – Mikhail Bondarko Aug 24 '18 at 12:41
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The book "Equivariant Homotopy and Cohomology Theory" by May et al probably counts as the standard reference. I don't think that it answers your last question, however. That is probably best addressed in the way suggested by Denis Nardin.

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  • $\begingroup$ Sorry; may I ask you to help me in locating (co)homology with coefficients in Mackey functors in this book? Mackey functors are mentioned several times in it; yet I wasn't able to find (co)homology similar to that mentioned in "Ordinary RO(G)-graded cohomology". $\endgroup$ – Mikhail Bondarko Aug 28 '18 at 8:32

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