(1)Īn object T ∈ C is called rigid if Hom C ( T, Σ T ) = 0. Īssume that C is a triangulated category with a shift functor Σ. We now recall that the notion of cluster tilting objects and related objects from, ,. The n-versions of triangulated or abelian categories appear as an attempt to get a better understanding of cluster tilting subcategories: some n-cluster tilting subcategories in triangulated categories are ( n + 2 )-angulated categories and n-cluster tilting subcategores in abelian categories are n-abelian categories.Ĭluster tilting objects (or subcategories) and cluster categories provide insight into cluster algebras and their related combinatorics. It has gained more and more attentions since the introduction of ( n + 2 )-angulated categories by Geiss-Keller-Oppermann, and the introduction of n-abelian categories by Jasso. Higher homological algebra emerged from the higher Auslander-Reiten theory by Iyama in.
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