and
nonabelian homological algebra
A differential graded category is a category enriched over complexes of modules for some commutative ring $k$. Given a differential graded category $D$, one can form a $k$-linear category $H^{0}(D)$ with the same objects as $D$ but with morphisms between two objects $X$ and $Y$ defined to be the $0$th cohomology group of the complex of morphisms between $X$ and $Y$ in $D$.
The derived category of an abelian category $\mathcal{A}$ having enough injectives (or projectives) can be constructed as $H^{0}$ of the differential graded category consisting of complexes of injective (or projective) objects in $\mathcal{A}$ with morphisms between two complexes $X$ and $Y$ being the Hom-complex from $X$ to $Y$. One can check that $H^{0}$ of the Hom-complex between $X$ and $Y$ is precisely homotopy classes of chain maps from $X$ to $Y$.
A dg-category or differential graded category is a category enriched over a symmetric monoidal category of chain complexes, usually taken to be that of chain complexes of $k$-vector spaces for some field $k$: $dgCat := Ch(Mod_k)\Cat$.
Notice that a dg-category $\mathbf{B}A$ with a single object is a differential graded algebra (see dg-algebra), $A$.
Tim : Would it be better to say cochain complexes as both Keller and Toen use the cochain convention? This can be confusing. (In other words I am confused!) Things related to this seem central to several questions elsewhere. Perhaps a word here would be a good idea.
To get from a simplicially enriched category to a chain enriched one is easy (it is linearisation plus Dold-Kan, looked at by Tabuada in arXiv:0711.3845) but to get to a dg-category in the sense of Keller or Toen (i.e. cochain enriched) is more interesting and complicated.
Therefore, following the terminology for horizontal categorification a dg-category might more descriptively be addressed as a differential graded algebroid. (A similar comment applies for instance to C*-category, which is a $C^*$-algebroid.)
A left dg-module over a dg-category $C$ is a dg-functor $L : C\to \mathrm{Com}_k$ where $\mathrm{Com}_k$ is the dg-category of complexes of $k$-vector spaces (that is Ch(Mod_k) with inner hom); similarly a right dg-module is a contravariant right dg-module. Morphisms between left dg-modules $L$ and $L'$ are elements of $Z^0\mathbf{Hom}(L,L')$ where the inner hom $\mathbf{Hom}$ is the complex of graded morphisms. Left dg-modules and their morphisms make a category $dgMod C$ which has a natural structure of Quillen exact category, which is in fact Frobenius. There is a Yoneda functor $Z^0(C)\to dgMod C$ given by $X\mapsto C(-,X)$.
Zoran Škoda: If one talks about left modules then we have category dgMod-C, and if talk about right modules then C-dgMod. When it is all the time clear which one we talk about than one can use either notation, but as long as we talk about left and right, then we should use A-Mod for left and Mod-A for right as it is usual in noncommutative algebra.
Toby: Good system, but there shouldn't be a minus sign in either version! (You can get a hyphen, if you really need one, either by moving out of the dollar signs (the quick and dirty way) or (more properly) by using the Unicode hyphen: $dgMod$-$C$ or $dgMod‐C$.)
Intuitively, a dg-category is pre-triangulated if its homotopy category is a triangulated category. More precisely, it is pre-triangulated if the image of the Yoneda functor is closed under translations (in both directions) and extensions.
Pre-triangulated dg-categories linear over field $k$ of characteristic 0 are equivalent to $k$-linear A-infinity-categories and both are models for stable (infinity,1)-categories.
Hanno Becker: Hello! Maybe one could add that in a pretriangulated dg-category $C$, the Frobenius-structure on dg-mod(C) pulls back to a Frobenius-structure on $Z^0(C)$ via the Yoneda-functor, and that dividing out its injective-projective objects turns out to be equivalent to passing to the homotopy category. Thus, the stable category of $Z^0(C)$ equals the homotopy category, which is then triangulated.
Do you assume the existence of a zero-object in a pretriangulated dg-category (there are interesting examples of dg-categories not possessing a zero object, e.g. the dg-categories of matrix-factorizations)?
Zoran: I took the conventions from Keller’s article, as far as I recall. You are welcome to enter and explain the equivalent variants of the definition, and other variations if you are currently thinking on this subject. If you can cite the references for the proofs or reasoning supporting it, even better. I am concentrating on another subject at the very moment.
There is a model structure on dg-categories.
The homotopy theory of $k$-linear dg-categories (for instance under the above model category) structure is that of $k$-linear stable (infinity,1)-categories (Cohn 13).
A. I. Bondal, Mikhail Kapranov, Enhanced triangulated categories, Матем. Сборник, Том 181 (1990), No.5, 669–683 (Russian); transl. in USSR Math. USSR Sbornik, vol. 70 (1991), No. 1, pp. 93–107, (MR91g:18010) (Bondal-Kapranov Enhanced triangulated categories pdf)
Bernhard Keller, On differential graded categories International Congress of Mathematicians. Vol. II, 151–190, Eur. Math. Soc., Zürich, 2006. (arXiv)
Bertrand Toën, Lectures on dg-categories (pdf) (these lectures also inspired Dmitry Kaledin’s Seoul lectures)
Bernhard Keller, Deriving DG categories, Ann. Sci. École Norm. Sup. (4) 27 (1994), no. 1, 63–102 (numdam)
B. Keller, A remark on tilting theory and DG algebras, Manuscripta Math. 79 (1993), no. 3-4, 247–252.
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.
M. Batanin, What do dg-categories form (after Tamarkin), talks at Paris 7 and Australian category seminar (abstract), math.CT/0606553
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.; 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.
G. Tabuada, Homotopy theory of DG categories, Thesis, Paris, 2007, pdf (some chapters in English and some in French)
G. Tabuada, Homotopy theory of dg categories via localizing pairs and Drinfeld’s dg quotient, Homology, homotopy and applications 12 (2010), No. 1, pp.187-219, files.
A. I. Bondal, M. Larsen, V. A. Lunts, Grothendieck ring of pretriangulated categories (arXiv)
Oren Ben-Bassat, Jonathan Block, Cohesive DG categories I: Milnor descent, arxiv/1201.6118
The relation to stable (infinity,1)-categories is discussed in
See also motives and dg-categories.