(also nonabelian homological algebra)
Pretriangulated dg-categories over a commutative ring $k$ 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 $A$ be a dg-category and $P(A)$ the dg-category of dg-presheaves or right dg-modules over $A$. The Yoneda embedding induces a fully faithful functor $h : ho(A) \hookrightarrow ho(P(A))$ on the homotopy categories. The category $ho(P(A))$ has a canonical triangulated structure (which can be written down directly).
The dg-category $A$ is called pretriangulated if the functor $h$ is stable under the suspension functor (and its inverse), and under taking mapping cones in $ho(P(A))$.
A dg-category $A$ is pretriangulated if and only if it is a fibrant object in the quasi-equiconic model structure on dg-categories.
See (Tabuada 07, Proposition 2.10).
Let $A$ be a dg-category.
The $n$-translation of an object $X \in A$ is an object $X[n] \in A$ representing the functor
The cone of a closed morphism $f : X \to Y$ of degree zero is an object $\Cone(f) \in A$ representing the functor
which is a mapping cone in chain complexes.
The dg-category $A$ is called strongly pretriangulated if it admits a zero object, all translations of all objects, and all cones of all morphisms.
Let $A$ be a dg-category. A strongly pretriangulated envelope of $A$ is the data of a strongly pretriangulated dg-category $tri(A)$ and a fully faithful functor $A \hookrightarrow tri(A)$ such that any functor $u: A \to B$ to a strongly pretriangulated dg-category $B$ factors uniquely through a functor $tri(A) \to B$.
A strongly pretriangulated envelope $A \hookrightarrow tri(A)$ always exists, and may be constructed by taking $tri(A)$ to be the full dg-subcategory of $P(A)$ spanned by the objects of the full triangulated subcategory of $ho(P(A))$ generated by the representable presheaves, and $A \hookrightarrow tri(A)$ to be the functor induced by the Yoneda embedding. There is also another construction using twisted complexes, see Bondal-Kapranov.
Now we have the following characterization of pretriangulated dg-categories.
Let $A$ be a dg-category and $A \hookrightarrow tri(A)$ be a strongly pretriangulated envelope of $A$. $A$ is pretriangulated if and only if the induced fully faithful functor $ho(A) \hookrightarrow ho(tri(A))$ is essentially surjective (and hence an equivalence of categories).
As an immediate corollary, note that for a pretriangulated dg-category $A$, its homotopy category $ho(A)$ inherits a canonical triangulated structure.
Let $u : A \to B$ be a functor between two dg-categories. If $A$ and $B$ are pretriangulated then the induced functor $ho(u): ho(A) \to ho(B)$ is triangulated. Further, $u$ is a quasi-equivalence if and only if $ho(u)$ is a triangulated equivalence.
The use of pretriangulated dg-categories as enhanced triangulated categories has been especially successful in the study of the various triangulated categories of sheaves on algebraic varieties.
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).
V. A. Lunts, D. O. Orlov, Uniqueness of enhancement for triangulated categories, J. Amer. Math. Soc. 23 (2010), 853-908, journal, arXiv:0908.4187.
Valery Lunts, Olaf M. Schnuerer, New enhancements of derived categories of coherent sheaves and applications, 2014, arXiv.
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
Last revised on October 25, 2017 at 13:45:53. See the history of this page for a list of all contributions to it.