The study of topological quantum field theories increasingly relies upon concepts from higher-dimensional algebra such as n-categories and n-vector spaces. We review progress towards a definition of n-category suited for this purpose, and outline a program in which n-dimensional TQFTs are to be described as n-category representations. First we describe a ‘suspension’ operation on ncategories, a...

We show that the stable module categories of certain selfinjective algebras of finite representation type having tree class An, Dn, E6, E7 or E8 are triangulated equivalent to ucluster categories of the corresponding Dynkin type. The proof relies on the “Morita” theorem for u-cluster categories by Keller and Reiten, along with the recent computation of Calabi-Yau dimensions of stable module cat...

We study higher cluster tilting objects in generalized higher cluster categories arising from dg algebras of higher Calabi-Yau dimension. Taking advantage of silting mutations of Aihara-Iyama, we obtain a class of m-cluster tilting objects in generalized m-cluster categories. For generalized m-cluster categories arising from strongly (m + 2)-Calabi-Yau dg algebras, by using truncations of minim...