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?
1 Answer
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.)

$\begingroup$ I think that "has a set of generators" is implied by "presentable", right? $\endgroup$ Nov 9, 2009 at 0:24

$\begingroup$ Ah. I should have said "compact generators", as that appears the condition SchwedeShipley require. So "presentable" implies "set of generators", but "compact generators" is much more special. $\endgroup$ Nov 9, 2009 at 0:33

$\begingroup$ I don't have a ready example in the case where you drop the hypothesis of compactness. If you form the localization of spectra with respect to a homology theory E, the localization L_E(S) of the sphere spectrum is a generator for this stable homotopy theory, but in general L_E(S) is not compact. But this doesn't preclude the existence of some other compact generator, and that actually happens in some cases, e.g., E= a Morava Ktheory. $\endgroup$ Nov 9, 2009 at 0:44

2$\begingroup$ Corollary B.13 of the reference [HS99] gives examples of what I presume are stable presentable categories without any nonzero compact objects. $\endgroup$ Nov 9, 2009 at 0:51