nLab
enhanced triangulated category

Idea

Triangulated categories are a bit rough sort of a localization, and lack some naturally expected properties, for example the functoriality of the cones. Therefore, it has been a wish since the 1960s to replace or repair them with a more coherent structure.

Good candidates at present are $A_{\mathrm{\infty }}$-categories, $\mathrm{dg}$-categories and stable (∞,1)-categories. In practice, triangulated categories are still used to an extent, so one likes to have both the full structure and the underlying truncation which exists as a triangulated category in all three cases.

It is known that all 3 approaches give the same result over a field of characteristic zero. The $\mathrm{dg}$-enhancement is most documented and studied in genuine applications so far and is the first to be historically understood.

Triangulated categories may arise as homotopy categories of stable (∞,1)-categories.

An enhancement of a triangulated category to a (pretriangulated) differential graded category may be, in characteristic zero, considered as a way to retain the information in the stable (∞,1)-category. Therefore in that case, the enhanced triangulated categories are models for stable (∞,1)-categories in terms of dg-categories, much like simplicial categories are models for (∞,1)-categories.

Details

It is well-known that the concept of a triangulated category is suffering many deficiencies for the purposes of homological algebra, geometry and topology; and even categorical properties like the non-functoriality of the cones. For that reason, Bondal and Kapranov in

• A. I. Bondal, M. M. Kapranov, Оснащенные триангулированные категории, Матем. Сборник, Том 181 (1990), No.5, 669–683 pdf (Russian); Engl. transl. Enhanced triangulated categories, USSR Math. USSR Sbornik, vol. 70 (1991), No. 1, pp. 93–107, (MR91g:18010) (Bondal-Kapranov Enhanced triangulated categories pdf)

introduce a notion of dg-enhancement of a triangulated category.

A dg-enhancement of a triangulated category $T$ is a (pretriangulated) differential graded category $A$ together with an equivalence $H^0(A)\to T$. The triangulated category $T$ with that structure is called enhanced. According to the new results presented in

• Valery Lunts, Uniqueness of enhancement for a triangulated category, talk at Workshop on triangulated categories, Swansea, Wales, Dec 10-12, 2008, (report on a joint work with D. Orlov)

• V. A. Lunts, D. O. Orlov, Uniqueness of enhancement for triangulated categories, arXiv:0908.4187.

the triangulated categories of quasicoherent sheaves on quasiprojective varieties and some of their cousins (like categories of perfect complexes on quasiprojective varieties) have essentially unique dg-enhancements. F. Muro has developed an obstruction theory for the existance and measuring non-uniqueness of dg-enhancements in more general setting (unpublished).