We extend the theory of equivariant orthogonal spectra from finite groups to profinite groups, and more generally from compact Lie groups to compact Hausdorff groups. The G -homotopy theory is "pieced together" from the G=U -homotopy theories for suitable quotient groups G=U of G ; a motivation is the way continuous group cohomology of a profinite group is built out of the cohomology of its finite quotient groups. In the model category of equivariant spectra Postnikov towers are studied from a general perspective. We introduce pro-G -spectra and construct various model structures on them. A key property of the model structures is that pro-spectra are weakly equivalent to their Postnikov towers. We discuss two versions of a model structure with "underlying weak equivalences". One of the versions only makes sense for pro-spectra. In the end we use the theory to study homotopy fixed points of pro-G -spectra. 55P91; 18G55

For every stable model category M with a certain extra structure, we produce an associated model structure on the pro-category pro-M and a spectral sequence, analogous to the Atiyah-Hirzebruch spectral sequence, with reasonably good convergence properties for computing in the homotopy category of pro-M. Our motivating example is the category of pro-spectra.The extra structure referred to above is a t-model structure. This is a rigidification of the usual notion of a t-structure on a triangulated category. A t-model structure is a proper simplicial stable model category M with a t-structure on its homotopy category together with an additional factorization axiom.

We introduce a notion of a filtered model structure and use this notion to produce various model structures on procategories. We give several examples, including a homotopy theory for G-spaces, where G is a profinite group. The class of weak equivalences are an approximation to the class of underlying weak equivalences.

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