(also nonabelian homological algebra)
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 $k$ is an (infinity,1)-category enriched in the (infinity,1)-category of chain complexes of $k$-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.
The Dwyer-Kan model structure on dg-categories presents the (infinity,1)-category of dg-categories.
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 $H k$. The latter is equivalent, at least morally, to the (infinity,1)-category of $k$-linear stable (infinity,1)-categories.
More precisely, it is shown in Cohn 13 that the Morita model structure on dg-categories presents the (infinity,1)-category of idempotent complete 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
and
The homotopy theory of dg-categories is studied in
Gonçalo Tabuada, Homotopy theory of DG categories, Thesis, Paris, 2007, pdf.
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.
The equivalence with the homotopy theory of stable (infinity,1)-categories is discussed in
(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.
The following references discuss the use of dg-categories in derived noncommutative algebraic geometry and noncommutative motives.
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.
Marco Robalo, Théorie homotopique motivique des espaces noncommutatifs, pdf.
S. Mahanta, Noncommutative geometry in the framework of differential graded categories, (arXiv:0805.1628)
D. Orlov, Smooth and proper noncommutative schemes and gluing of DG categories, Adv. Math. 302 (2014) doi
Bernhard Keller, Deriving DG categories, Ann. Sci. École Norm. Sup. (4) 27 (1994), no. 1, 63–102 (numdam)
Dmitry Tamarkin, What do dg-categories form?, Compos. Math. 143 (2007), no. 5, 1335–1358.
M. Batanin, What do dg-categories form (after Tamarkin), talks at Paris 7 and Australian category seminar (abstract), math.CT/0606553
Oren Ben-Bassat, Jonathan Block, Cohesive DG categories I: Milnor descent, arxiv/1201.6118