nLab
enhanced triangulated category

Context

Homological algebra

homological algebra

and

nonabelian homological algebra

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

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 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 dgdg-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 TT is a (pretriangulated) differential graded category AA together with an equivalence H 0(A)TH^0(A)\to T. The triangulated category TT 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

References

  • Valery Lunts, Olaf M. Schnuerer, New enhancements of derived categories of coherent sheaves and applications, arXiv.

Revised on July 6, 2014 08:00:23 by Adeel Khan (77.9.210.144)