Stable infinity categories vs dg-categories What is the relation between dg-categories and stable $\infty$-categories?
Given a dg-category one can form its dg-nerve and get a $\infty$-category
(which will be stable if the dg-category is?). 
Can one turn a stable $\infty$-category into a dg-category or $A_\infty$-category somehow?
I have heard the statement that at least over a field of characteristic zero the theories
of stable $\infty$-categories and dg-categories are "equivalent".
What would be a precise formulation of this statement and what would be a reference?
 A: See the recent paper


*

*Lee Cohn, Differential graded categories are k-linear stable infinity categories, arXiv:1308.2587
where a proof has been written down.  The precise statement is that the underlying $(\infty,1)$-category associated to the Morita model structure on dg-categories over $k$ (where fibrant objects are karoubian pretriangulated dg-categories) is equivalent to the $(\infty,1)$-category of karoubian stable $k$-linear $(\infty,1)$-categories.
Update: Though Cohn works over characteristic zero, Bertrand Toen told me his arguments in fact hold in arbitrary characteristic.
Also, see the new paper 


*

*Giovanni Faonte, Simplicial nerve of an A-infinity category, arXiv:1312.2127
where it is proved that the dg-nerve functor takes pretriangulated dg-categories to stable $(\infty,1)$-categories.
A: Here are a few observations...


*

*I think there exist stable infinity categories that are not the dg-nerve (resp. $A_\infty$-nerve) of a dg-category (resp. $A_\infty$ category). In particular, the category of spectra should not arise in this way. I think Keller has a paper on differential graded categories that answers this question; he notes at some point that the homotopy category of spectra is not "algebraic" but that homotopy categories of differential graded categories are (and in fact sort of encompass all such algebraic categories.) Basically it comes down to something like the existence of Hopf maps. Now- could one define somehow the "closest dg-category approximation" to a given stable infty category? Probably. I don't know how. Or maybe I could come up with how, but I'm not sure how useful this would be if the functor wasn't an equivalence? 

*To answer Fernando's question, see DAG X.5 or DAG VII.6.2. That is, a stable $\infty$-category over a field $k$ is a presentable, stable $\infty$-category "equipped with an action of the monoidal $\infty$-category of $k$-module spectra". Unless I'm mistaken I think this basically implies that it is enriched and tensored over k-module spectra.

*Here would be a precise formulation of the statement about categories over a field of characteristic zero: The dg-nerve functor induces an equivalence of $\infty$-categories between the $\infty$-category underlying the model category of dg-categories over k and the $\infty$-category of stable, k-linear $\infty$-categories. (I don't mean to overwhelm with "infinities", I would state this in the perhaps friendlier world of model categories, but I'm not sure what precisely the model category is that corresponds to stable, k-linear $\infty$-categories.) I don't know of a reference for a proof, though Lurie alludes to this a lot. It would be great if someone wrote this down!

