Stable Homotopy theory
Pretriangulated dg-categories are models for stable (∞,1)-categories in terms of dg-categories, much like simplicial categories are models for (∞,1)-categories.
The zeroth cohomology category of a pretriangulated dg-category is an ordinary triangulated category, hence a morphism from where is a pretriangulated dg-category and a triangulated category is called an enhanced triangulated categories.
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
is called strongly pretriangulated if it admits a zero object, all translations of all objects, and all cones of all morphisms.
For every DG category , there exists a strongly pretriangulated DG category and a fully faithful DG functor such that for any DG functor to a strongly pretriangulated DG category , there exists a unique lift .
See below for a construction of .
The DG category is called pretriangulated if the induced functor is an equivalence.
For a pretriangulated dg-category, the homotopy category is naturally a triangulated category.
If is a DG functor between two pretriangulated DG categories, then
- commutes with translation and preserves cones;
- the induced functor is a triangulated functor on the homotopy categories;
- is a quasi-equivalence? if and only if is a triangulated equivalence.
Definition using twisted complexes
For a dg-category let be its dg-category of twisted complexes.
is pretriangulated if for every twisted complex the corresponding dg-functor
In other words, twisted complexes in have representatives in .
- 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)
See enhanced triangulated category for more links to references.