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This is a very naive question but 1)given a compact Lie group G, is there a good notion of a sheaf of equivariant spectra on a G-space X analogous to the model structure that Brown develops in his paper on ordinary sheaves of spectra?

2)Is there a homotopy theory which allows you to take cohomology with coefficients in a sheaf of spectra and in which it makes sense to take for a G-space X the constant sheaf Z and recover Borel cohomology or "the constant sheaf in the equivariant K-theory spectrum" and recover K_G(X)?

3) Assuming answers to the previous questions, does the picture simplify in a reasonable way when one works rationally by analogy with the usual rational homotopy theory?

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This sounds somewhat close to twisted cohomology, where one takes the sheaf of sections of a bundle of spectra. You might look at some of the literature on twisted K-theory (and in particular, twisted K-theory of stacks, which is essentially an equivariant version). –  Jeffrey Giansiracusa May 19 '10 at 10:16

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