\(n\)-angulated categories. (English) Zbl 1271.18013

Triangulated categories were invented at the end of the 1950s by Grothendieck-Verdier [J.-L. Verdier, Lect. Notes Math. 569, 262–311 (1977; Zbl 0407.18008)] and, independently, D. Puppe [Colloq. algebr. Topology, Aarhus 1962, 65–71 (1962; Zbl 0139.41106)]. Their aim was to axiomatize the properties of derived categories respectively of stable homotopy categories. In the case of the derived category, the triangles are the ‘shadows’ of the \(3\)-term exact sequences of complexes. Longer exact sequences of complexes are splicings of \(3\)-term exact sequences and thus they also have their shadows in the derived category. Cluster tilting theory [A. B. Buan, et al., Adv. Math. 204, No. 2, 572–618 (2006; Zbl 1127.16011)] and in particular Iyama’s higher Auslander-Reiten-theory [O. Iyama, Adv. Math. 226, No. 1, 1–61 (2011; Zbl 1233.16014)] have lead to the surprising discovery that there is a large class of categories which are naturally inhabited by shadows of \(n\)-term exact sequences without being home to shadows of \(3\)-term exact sequences.
In this paper, the authors axiomatize this remarkable class of categories by introducing the new notion of \(n\)-angulated category and construct large classes of examples using cluster tilting theory. They define \(n\)-angulated categories by modifying the axioms of triangulated categories in a natural way. Then they show that Heller’s parametrization of pre-triangulations [A. Heller, Bull. Am. Math. Soc. 74, 28–63 (1968; Zbl 0177.25605)] extends to pre-\(n\)-angulations. The authors obtain a large class of examples of \(n\)-angulated categories by considering \((n-2)\)-cluster tilting subcategories of triangulated categories which are stable under the \((n-2)\)nd power of the suspension functor. As an application, they show how \(n\)-angulated Calabi-Yau categories yield triangulated Calabi-Yau categories of higher Calabi-Yau dimension. Finally, a link to algebraic geometry and string theory is sketched.


18E30 Derived categories, triangulated categories (MSC2010)
Full Text: DOI arXiv