The study of higher homological algebra has started by Iyama in the late 2000’s. This new theory quickly attracted a lot of attention, with many authors generalising classical notions to the setting of higher homological algebra. Examples of such generalisations are the introduction of n-abelian categories by Jasso and n-torsion classes by Jørgensen. Recently, it was shown by Kvamme and, independently, by Ebrahimi and Nasr-Isfahani that every small n-abelian category is the n-cluster-tilting subcategory of an abelian category.
In the first part of this talk we will explain the relation between the n-torsion classes in an n-abelian category and the torsion classes of its ambient abelian category and we will discuss some consequence of this relation. In the second part of the talk we will restrict ourselves to functorially finite n-torsion classes and we will state some results connecting them to higher Auslander-Reiten theory.
The results presented in the fist half of the talk are joint work with J. Asadollahi, P. Jørgensen and S. Schroll. The rest are part of an ongoing project in collaboration with J. August, J. Haugland, K. Jacobsen, S. Kvamme and Y. Palu.