nLab enhanced triangulated category

Idea

Any stable (∞,1)-category $C$ gives rise to a canonical triangulated category $ho(C)$, its homotopy category. Much of the information contained in $C$ is still present in $ho(C)$, so that in many situations it can be enough to work at the level of triangulated categories. However, in general various problems with triangulated categories require one to work with some sort of higher structure lying above $ho(C)$. Such a structure is often called an enhancement of the triangulated category $ho(C)$.

Examples of enhanced triangulated categories are, in order of increasing information:

• stable derivators,
• pretriangulated dg-categories or pretriangulated A-∞-categories,
• and stable (∞, 1)-categories themselves.

History

After Jean-Louis Verdier started the development of the theory of triangulated categories in his 1963 thesis, it was quickly recognized that some sort of enhancements were necessary to get the full picture. For this purpose, Alexandre Grothendieck suggested the notion of derivator in the early 1980’s, while Alexei Bondal and Mikhail Kapranov developed the notion of pretriangulated dg-category in 1990. In 2006 Jacob Lurie developed the notion of stable (∞,1)-category.

Problems with triangulated categories

• non-functoriality of the mapping cone,
• more generally, non-existence of homotopy colimits and homotopy limits,
• …

(to be expanded on)

Comparison results

In (Cohn 13) it is shown that the (∞,1)-category of idempotent complete pretriangulated dg-categories over a commutative ring $k$ is equivalent to the (∞,1)-category of stable k-linear (∞,1)-categories. See pretriangulated dg-category for details.

In (Renaudin 2006) it is shown that the 2-category of locally presentable derivators is equivalent to the localization of the 2-category of combinatorial model categories at the Quillen equivalences. Combinatorial model categories are equivalent to locally presentable (∞,1)-categories in a sense made precise there. However, a comparison between the stability conditions in the derivator and (∞,1)-category settings seems still to be open.

Related concepts

• triangulated category

• stable derivator

• pretriangulated dg-category, pretriangulated A-infinity category

• stable (∞,1)-category

References

• Alexei Bondal, Mikhail Kapranov, Enhanced triangulated categories, Mat. Sb. (Russian) 181:5 (1990), 669–683 link; Engl. transl. in Sbornik Math. 70:1 (1991) 93–107 doi:10.1070/SM1991v070n01ABEH001253 MR1055981

• Valery A. Lunts, Dmitri O. Orlov, Uniqueness of enhancement for triangulated categories, J. Amer. Math. Soc. 23 (2010), no. 3, 853–908

• Benjamin Antieau, On the uniqueness of infinity-categorical enhancements of triangulated categories, arxiv/1812.01526

Brief overview and list of further references:

• Stefan Schwede, Markus Hausmann, Enhancements of triangulated categories, Graduate Seminar Topology, Bonn 2019-2020 (pdf)