# nLab enhanced triangulated category

### Context

#### Homological algebra

homological algebra

and

nonabelian homological algebra

diagram chasing

### Theorems

#### Stable homotopy theory

stable homotopy theory

# Contents

## Idea

The notion of an enhanced triangulated category is a refinement of that of triangulated category which remembers more of the full structure of a stable (∞,1)-category (see Cohn 13).

Triangulated categories are a somewhat rough sort of a localization, and lack some naturally expected properties, for example the functoriality of the mapping 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 $A_\infty$-categories, 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 $dg$-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.

## References

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

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 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, J. Amer. Math. Soc. 23 (2010), 853-908, journal, 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).

The relation to stable (infinity,1)-categories is discussed in

Revised on April 4, 2014 23:26:31 by Tim Porter (2.26.36.99)