Pretriangulated dg-categories over a commutative ring are, roughly speaking, dg-categories whose homotopy category is canonically triangulated. These form a model for stable k-linear (∞,1)-categories, in a sense which is made precise below (at least in characteristic zero). In other words pretriangulated dg-categories can be viewed as enhanced triangulated categories. For this reason some authors call them stable dg-categories.
The notion of pretriangulated dg-category goes back to (Bondal-Kapranov 1990).
Goncalo Tabuada demonstrated the existence of a model structure on the category of small dg-categories, the quasi-equiconic model structure on dg-categories, where the fibrant objects are the pretriangulated dg-categories. See (Tabuada 07, Theorem 2.2 and Proposition 2.10). This model structure can be Bousfield localized to the Morita model structure on dg-categories, where the fibrant objects are the idempotent complete pretriangulated dg-categories. In (Cohn 13) it is shown that the associated (infinity,1)-category is equivalent to the (infinity,1)-category of stable k-linear (∞,1)-categories.
Let be a dg-category and the dg-category of dg-presheaves or right dg-modules over . The Yoneda embedding induces a fully faithful functor on the homotopy categories. The category has a canonical triangulated structure (which can be written down directly).
Let be a dg-category.
The -translation of an object is an object representing the functor
The cone of a closed morphism of degree zero is an object representing the functor
The dg-category is called strongly pretriangulated if it admits a zero object, all translations of all objects, and all cones of all morphisms.
Let be a dg-category. A strongly pretriangulated envelope of is the data of a strongly pretriangulated dg-category and a fully faithful functor such that any functor to a strongly pretriangulated dg-category factors uniquely through a functor .
A strongly pretriangulated envelope always exists, and may be constructed by taking to be the full triangulated subcategory of generated by the representable presheaves, and to be the functor induced by the Yoneda embedding. Here denotes the dg-category of dg-presheaves on . There is also another construction using twisted complexes, see Bondal-Kapranov.
Now we have the following characterization of pretriangulated dg-categories.
Let be a dg-category and be a strongly pretriangulated envelope of . is pretriangulated if and only if the induced fully faithful functor is essentially surjective (and hence an equivalence of categories).
As an immediate corollary, note that for a pretriangulated dg-category , its homotopy category inherits a canonical triangulated structure.
In the below paper it is shown that the triangulated categories of quasicoherent sheaves on quasiprojective varieties and some of their cousins (like categories of perfect complexes on quasiprojective varieties) have unique dg-enhancements. Fernando Muro has developed an obstruction theory for the existance and measuring non-uniqueness of dg-enhancements in more general settings (unpublished).
Similarly, pretriangulated dg-categories have proven to give a good model for derived noncommutative algebraic geometry in the sense of Maxim Kontsevich. See there for relevant references. In this connection see also the work of Goncalo Tabuada who has developed a theory of noncommutative motives in this framework.
The model structure presenting pretriangulated dg-categories is discussed in
See also paragraph 2.3 of
For a summary of the various model structures on dg-categories, see Section 2 of the paper
The relation to stable (infinity,1)-categories is discussed in