and
nonabelian homological algebra
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.
For a dg-category let be its dg-category of twisted complexes.
is pretriangulated if for every twisted complex the corresponding dg-functor
is representable.
In other words, twisted complexes in have representatives in .
Proposition
For a pretriangulated dg-category, the homotopy category is naturally a triangulated category.
The morphism
is an equivalence of triangulated categories.
pretriangulated dg-category, enhanced triangulated category
See enhanced triangulated category for more links to references.