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 Sp-enriched category (in some appropriate sense) and Fun_sp denotes the category of Sp-enriched 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?