Derived categories and their frame, triangulated categories after their invention by Grothendieck and Verdier in the early sixties of the last century, have been immediately an important tool in several branches of mathematics. The theory of differential graded categories, briefly dg-categories, together with one of \(A_\infty\)-categories are an approach to overcome certain main problems, difficulties appearing in the investigation of both derived and triangulated categories.
In this survey the author reviews the foundations of dg categories and summerizes recent results of Drinfeld, Dugger-Shipley…on this field. The author gives for example, the derived category of a dg-category, structure theorems for algebraic triangulated categories, a topological Morita equivalence for dg-categories, and most important invariance for \(K\)-theory, Hochschild
(co-)homology and the derived Hall algebra…As the author mentiones, the main aim is an attempt to answer Drinfeld’s question: what do differential graded categories form?
For the entire collection see [Zbl 1095.00005].