Cluster categories have led to new developments in the theory of the canonical basis and particularly its dual. They are providing insight into cluster algebras and their related combinatorics, and they have also been used to define a new kind of tilting theory, known as cluster-tilting theory. It is an important result of cluster theory, due to B. Keller and I. Reiten [Adv. Math. 211, No. 1, 123–151 (2007; Zbl 1128.18007)] and S. Koenig and B. Zhu [Math. Z. 258, No. 1, 143–160 (2008; Zbl 1133.18005)], that certain quotients of triangulated categories are abelian. Indeed, Let \(\mathcal{T}\) be a triangulated category. If \(T\) is a cluster tilting object and \(I = [\text{add } T]\) is the ideal of morphisms factoring through an object of \(\text{add } T\), then the quotient category \(\mathcal{T}/I\) is abelian. This result was generalized by Jacobson and Jørgensen to \((n+2)\)-angulated category and \(n\)-abelian category. They showed that if \(\mathcal{T}\) is a suitable \((n+2)\)-angulated category for an integer \(n\geq 1\), \(T\) is a cluster tilting object in the sense of Oppermann-Thomas and \(I = [\text{add } T]\) is the ideal of morphisms factoring through an object of \(\text{add } T\), then \(\mathcal{T}/I\) is \(n\)-abelian.
The main result of this article is a generalisation of the above result. Indeed, they show that if \(\mathcal{C}\) is an \((n+2)\)-angulated category with an \(n\)-suspension functor \(\Sigma^n\) and \(X\) is a cluster-tilting subcategory of \(\mathcal{C}\) then the quotient category \(\mathcal{C} /X\) is an \(n\)-abelian category, and \(\mathcal{C} /X\) is equivalent to an \(n\)-cluster tilting subcategory of an abelian category \(\text{mod}(\Sigma^n X )\).