n-angulated categories

Abstract

We define n-angulated categories by modifying the axioms of triangulated categories in a natural way. We show that Heller's parametrization of pre-triangulations extends to pre-n-angulations. We 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, we show how n-angulated Calabi-Yau categories yield triangulated Calabi-Yau categories of higher Calabi-Yau dimension. Finally, we sketch a link to algebraic geometry and string theory.

Highern-angulations from local rings

“…We recall the definition of an n-angulated category from [6]. Let C be an additive category with an automorphism Σ : C → C , and n an integer greater than or equal to 3.…”

“…Moreover, two such n-angulations coincide if and only if the defining units u and v satisfy up = vp in the ring R. These various n-angulations arise from global automorphisms on the underlying category, introduced by Balmer [2]. One can obtain the result by applying [6,Proposition 3.4].…”

“…Geiss, Keller and Oppermann recently introduced in [6] 'higher-dimensional' analogues of triangulated categories, called n-angulated categories, and showed that certain cluster tilting subcategories of triangulated categories give rise to n-angulated categories. For n = 3, an n-angulated category is the same as a classical triangulated category.…”

An introduction to higher Auslander-Reiten theory

“…We recall that a (right) ‐module is ‐ cluster‐tilting if Note that a 1‐cluster‐tilting ‐module is precisely a representation generator of . The intrinsic properties of ‐cluster‐tilting subcategories have been investigated in . Theorem There is a one‐to‐one correspondence between the equivalence classes of ‐cluster‐tilting modules for Artin ‐algebras and Morita‐equivalence classes of ‐Auslander ‐algebras, that is Artin ‐algebras satisfying The correspondence is given by , where is a ‐cluster‐tilting ‐module.…”

“…Note that a 1-cluster-tilting Λ-module is precisely a representation generator of mod Λ. The intrinsic properties of d-cluster-tilting subcategories have been investigated in [14,25,26].…”