We define mutation pair in an n-angulated category and prove that given such a mutation pair, the corresponding quotient category carries a natural n-angulated structure. This result generalizes a theorem of Iyama-Yoshino in classical triangulated category. As an application, we obtain that the quotient category of a Frobenius n-angulated category is also an n-angulated category.2010 Mathematics Subject Classification. 18E30.

Abstract. We define the notion of right n-angulated category, which generalizes the notion of right triangulated category. Let C be an additive category or n-angulated category and X a covariantly finite subcategory, we show that under certain conditions the quotient C/X is a right n-angulated category. This result generalizes some previous work.

Let C be an additive category equipped with an automorphism Σ. We show how to obtain n-angulations of (C, Σ) using some particular periodic injective resolutions. We give necessary and sufficient conditions on (C, Σ) admitting an n-angulation. Then we apply these characterizations to explain the standard construction of n-angulated categories and the n-angulated categories arising from some local rings. Moreover, we obtain a class of new examples of n-angulated categories from quasi-periodic selfinjective algebras.

We mainly investigate abelian quotients of the categories of short exact sequences. The natural framework to consider the question is via identifying quotients of morphism categories as modules categories. These ideas not only can be used to recover the abelian quotients produced by clustertilting subcategories of both exact categories and triangulated categories, but also can be used to reach our goal. Let (C, E) be an exact category. We denote by E(C) the category of bounded complexes whose objects are given by short exact sequences in E and by SE(C) the full subcategory formed by split short exact sequences. In general, E(C) is just an exact category, but the quotient E(C)/[SE(C)] turns out to be abelian. In particular, if (C, E) is Frobenius, we present three equivalent abelian quotients of E(C) and point out that the equivalences are actually given by left and right rotations. The abelian quotient E(C)/[SE(C)] admits some nice properties. We explicitly describe the abelian structure, projective objects, injective objects and simple objects, which provide a new viewpoint to understanding Hilton-Rees Theorem and Auslander-Reiten theory. Furthermore, we present some analogous results both for n-exact versions and for triangulated versions.

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