There is a theorem of Schwede and Shipley which classifies categories of modules over an A_{∞} ring spectrum as those stable presentable (∞,1)categories with a compact generator. Suppose I allow my A_{∞} rings to "have many objects", that is, I consider categories of the form Fun_{Sp}(I^{op}, Sp) where Sp is the category of spectra, I is a small Spenriched category (in some appropriate sense) and Fun_{sp} denotes the category of Spenriched functors. Is there a classification of which stable presentable categories can be obtained in this way? Is it possible that all stable presentable categories are of this form?

According to the abstract of http://arxiv.org/abs/math/0108143 (Schwede & Shipley, Classification of Stable Model Categories), they deal with the case of stable model categories (=stable presentable (∞,1)categories, I suppose) which have a set of compact generators, and show they are the same as model categories of functors from spectralenriched categories. (Edited, in the light of Reid's comment, to include the hypothesis of compact generators.) 

