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.Homology, Homotopy and Applications, Vol. 9 (2007), No. 1, pp.399-438.