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Right \(n\)-angulated categories arising from covariantly finite subcategories. (English) Zbl 1371.18010
In [A. Beligiannis and N. Marmaridis, Commun. Algebra 22, No. 12, 5021–5036 (1994; Zbl 0811.18005)], the notion of right (resp., left) triangulated category was introduced as a one-sided version of a triangulated category. Using this notion, some classical results about triangulated categories can be extended, e.g., just like the stable category of a Frobenius category has a natural triangulated structure, given an exact category with enough injectives (resp., projectives), the quotient category over the ideal of maps that factor through some injective (projective) is naturally right (left) triangulated.
In this paper, the notions of right and left \(n\)-angulated category are introduced for any \(n\geq 1\), where left/right \(3\)-angulated categories are just the left/right triangulated categories. Furthermore, an \(n\)-angulated category in the sense of [C. Geiss et al., J. Reine Angew. Math. 675, 101–120 (2013; Zbl 1271.18013)] is exactly a left and right \(n\)-angulated category such that the “shift to the right” and “shift to the left” functors are one the quasi-inverse of the other.
The main result of the paper gives a sufficient condition for the quotient of an additive category over a contravariantly finite subcategory to have a natural right \(n\)-angulated structure. As a corollary, one gets that the quotient category of an \(n\)-exact category (see [G. Jasso, Math. Z. 283, No. 3–4, 703–759 (2016; Zbl 1356.18005)]) with enough injectives (resp., projectives) over the ideal of maps that factor through some injective (projective) is naturally right (left) \(n+2\)-angulated. In particular, the stable category of an \(n\)-exact Frobenius category is \(n+2\)-angulated.

MSC:
18E30 Derived categories, triangulated categories (MSC2010)
18E10 Abelian categories, Grothendieck categories
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[1] DOI: 10.1080/00927879408825119 · Zbl 0811.18005
[2] DOI: 10.2140/agt.2013.13.2405 · Zbl 1272.18008
[3] DOI: 10.1112/jlms/jdv064 · Zbl 1371.18009
[4] DOI: 10.1016/j.jpaa.2013.06.007 · Zbl 1291.18015
[5] Geiss C., J. Reine Angew. Math. 675 pp 101– (2013)
[6] DOI: 10.1017/CBO9780511629228
[7] DOI: 10.1007/s00209-016-1619-8 · Zbl 1356.18005
[8] DOI: 10.1007/s10587-015-0220-3 · Zbl 1363.18009
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