$n$-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.

notation in axiom F2 clarified, 16 pages; v3: new section 5.5, minor corrections, to appear in Crelle's Journal; v4: published version

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Cited by in corpus (45)

• The representation type of Jacobian algebras
• The axioms for n-angulated categories
• Higher Nakayama algebras I: Construction
• Transport of structure in higher homological algebra
• An introduction to higher Auslander-Reiten theory
• Simplicial structures in higher Auslander-Reiten theory
• $n$-exangulated categories
• 2-CY-tilted algebras that are not Jacobian
• Auslander-Reiten $(d+2)$-angles in subcategories and a $(d+2)$-angulated generalisation of a theorem by Brüning
• Higher Auslander correspondence for dualizing $R$-varieties
• $d$-abelian quotients of $(d+2)$-angulated categories
• Tubular Jacobian Algebras
• Maximal ${\tau }_d$-rigid pairs
• Derived equivalences from cohomological approximations, and mutations of $\Phi$-Yoneda algebras
• On the existence of Auslander-Reiten (d+2)-angles in (d+2)-angulated categories
• Tensor products of n-complete algebras
• Tropical friezes and the index in higher homological algebra
• The Donovan--Wemyss Conjecture via the Derived Auslander--Iyama Correspondence
• Grothendieck groups of $d$-exangulated categories and a modified Caldero-Chapoton map
• $n$-Exact categories arising from $(n+2)$-angulated categories
• A resolution theorem for extriangulated categories with applications to the index
• Right $n$-angulated categories arising from covariantly finite subcategories
• A functorial approach to $n$-abelian categories
• $n$-exact categories arising from $n$-exangulated categories
• Derived equivalences in n-angulated categories
• n-angulated quotient categories induced by mutation pairs
• A characterisation of higher torsion classes
• Higher Auslander-Reiten sequences via morphisms determined by objects
• N-extension closed subcategories of (n+2)-angulated categories
• Higher Auslander's defect and classifying substructures of n-exangulated categories
• The long $n$-exact sequence theorem in $n$-abelian categories
• Frobenius $n$-exangulated categories
• Torsion classes and t-structures in higher homological algebra
• Symmetric approximation sequences, Beilinson-Green algebras and derived equivalences
• Grothendieck groups and Auslander-Reiten (d+2)-angles
• From $n$-exangulated categories to $n$-abelian categories
• Two new classes of n-exangulated categories
• The category of extensions and idempotent completion
• Auslander-Reiten-Serre duality for n-exangulated categories
• $n$-angulated categories from self-injective algebras
• Ideal approximation in $n$-angulated categories
• From right (n+2)-angulated categories to n-exangulated categories
• One-sided n-suspended categories
• A new characterization of n-exangulated categories with (n+2)-angulated structure
• $n$-abelian quotient categories