triangulated vs. dg/A-infinity

The analogy that Nadler and I like is that if (oo,1)-categories are like vector spaces, then model categories are like vector spaces with a fixed basis and homotopy categories are like vector spaces mod isomorphism - i.e., dimensions of vector spaces. Which of the three would you rather work with?

In more details I'll take the low road and quote my paper with Francis and Nadler:

The theory of triangulated categories is inadequate to handle many basic algebraic and geometric operations. Examples include the absence of a good theory of gluing or of descent, of functor categories, or of generators and relations. The essential problem is that passing to homotopy categories discards essential information (in particular, homotopy coherent structures, homotopy limits and homotopy colimits).

This information can be captured in many alternative ways, the most common of which is the theory of model categories. Model structures keep weakly equivalent objects distinct but retain the extra structure of resolutions which enables the formulation of homotopy coherence. This extra structure can be very useful for calculations but makes some functorial operations difficult. In particular, it can be hard to construct certain derived functors because the given resolutions are inadequate. There are also fundamental difficulties with the consideration of functor categories between model categories.

Anyway, in characteristic zero, the theories of dg categories, Aoo categories and stable oo,1 categories are all the same, and provide a "promised land" between the two extremes. (The problems with localization of the Fukaya category are very serious, but are issues in geometry - the existence of instanton corrections - and not in the theory of A_oo categories). Perhaps the most useful result in this context that doesn't have analogs for dg or Aoo categories is Jacob Lurie's oo-Barr-Beck theorem, which has lots and lots of consequences - such as descent theory or the generation result that Ben mentioned. Another is having a good theory of tensor products of categories and internal hom of categories, neither of which works in the two extreme theories.

I don't really think that triangulated categories are abominable, but they certainly have their problems which are a result of having forgotten the higher homotopies. For instance, non-functoriality of mapping cones can be fixed via dg-enhancement.

Another problem has to do with localization for triangulated categories. The fact that one can put a model structure on a suitable category of dg-categories and take homotopy limits for instance is a very useful thing and allows one to talk more meaningfully about gluing and working locally. One can do this in the derived category but only in quite simple situations and it requires extra structure for it to work well (e.g. the structure of a rigid tensor category compatible with the triangulation).

At least one good place to read in more detail about these problems and how to fix them using dg-categories is To¨en's Lectures on DG-categories which can be found here.

I haven't really said anything about the A-infinity point of view, but I don't know it so well yet - hopefully someone else can say something. But I do know that at least one of the same problems rears its head namely that of wanting to work locally. For instance in homological mirror symmetry one would like to glue Fukaya categories of complicated things together from those of easier things. To do this one would certainly need to use the A-infinity point of view before taking derived categories and if it works it should be because one can take the appropriate homotopy limit (does the model structure to do this actually exist by the way?).

I know other people have longer answers, but I have a short one: a dg-category can be reconstructed by knowing the Ext-algebra of any generating object as a dg-algebra, in the same way that an abelian category can be reconstucted from the endomorphisms of a projective generator. This isn't true if you only remember the Ext-algebra as a graded algebra; you might have needed those higher homotopies.