nLab
pretriangulated dg-category

Idea

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 $H^0(C)\to D$ where $C$ is a pretriangulated dg-category and $D$ a triangulated category is called an enhanced triangulated categories.

Definition

For $E$ a dg-category let $\mathrm{PreTr}(E)$ be its dg-category of twisted complexes.

$E$ is pretriangulated if for every twisted complex $K\in \mathrm{PreTr}(E)$ the corresponding dg-functor

$$\mathrm{PreTr}(-,K):E^{\mathrm{op}}\to C(\mathrm{Ab})$$PreTr(-,K) : E^{op} \to C(Ab)

is representable.

In other words, twisted complexes in $\mathrm{PreTr}(E)$ have representatives in $E$.

Proposition

For $E$ a pretriangulated dg-category, the homotopy category $H^0(E)$ is naturally a triangulated category.

The morphism

$$H^0(\mathrm{PreTr}(E))\to H^0(E)$$H^0(PreTr(E)) \to H^0(E)

is an equivalence of triangulated categories.

References

• 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.