Differential graded categories or dg-categories are linear analogues of spectral categories. In other words they are linear stable (infinity,1)-categories. It is common and useful to view them as enhanced triangulated categories.
A dg-category over a commutative ring is an (infinity,1)-category enriched in the (infinity,1)-category of chain complexes of -modules. Equivalently, it is an ordinary category strictly enriched in chain complexes (see Haugseng 13).
Hence a dg-category is a category with mapping complexes of morphisms between any two objects. By taking the homologies of these chain complexes in degree zero, one gets an ordinary category, called the homotopy category of a dg-category. Notice that a dg-category with a single object is the same thing as a dg-algebra.
By the stable Dold-Kan correspondence, the (infinity,1)-category of dg-categories is equivalent to the (infinity,1)-category of (infinity,1)-categories enriched in the symmetric monoidal (infinity,1)-category of modules over the Eilenberg-Mac Lane spectrum . The latter is equivalent, at least morally, to the (infinity,1)-category of -linear stable (infinity,1)-categories.
Historically, dg-categories were introduced in
whilst their modern development can be traced to
For concise reviews of the theory, see section 1 of
as well as the introduction and appendices to
For longer surveys, see
Gonçalo Tabuada, Une structure de catégorie de modèles de Quillen sur la catégorie des dg-catégories, C. R. Math. Acad. Sci. Paris 340 (2005), no. 1, 15–19.
(Note that the proof works over any ring, even though it is stated there for characteristic zero.)
In the following it is shown that the homotopy theory of (infinity,1)-categories enriched in the (infinity,1)-category of chain complexes is equivalent to the homotopy theory of ordinary categories strictly enriched in chain complexes.
Gonçalo Tabuada, Invariants additifs de DG-catégories, Int. Math. Res. Not. 2005, no. 53, 3309–3339; Addendum in Int. Math. Res. Not. 2006, Art. ID 75853, 3 pp. ; Erratum in Int. Math. Res. Not. IMRN 2007, no. 24, Art. ID rnm149, 17 pp.
S. Mahanta, Noncommutative geometry in the framework of differential graded categories, (arXiv:0805.1628)
Dmitry Tamarkin, What do dg-categories form?, Compos. Math. 143 (2007), no. 5, 1335–1358.