$n$-angulated categories are a generalization to integers $n\geq 3$ of a triangulated category, which is obtained for $n = 3$. They are introduced in
Its abstract reads:
We define $n$-angulated categories by modifying the axioms of triangulated categories in a natural way. We show that Heller’s parametrization of pre-triangulations extends to pre-$n$-angulations. We obtain a large class of examples of $n$-angulated categories by considering $(n-2)$-cluster tilting subcategories of triangulated categories which are stable under the $(n-2)$nd power of the suspension functor. As an application, we show how $n$-angulated Calabi-Yau categories yield triangulated Calabi-Yau categories of higher Calabi-Yau dimension. Finally, we sketch a link to algebraic geometry and string theory.
Other works are
P.A. Bergh, M. Thaule, The axioms for $n$-angulated categories, arXiv:1112.2533; The Grothendieck group of an $n$-angulated category, arxiv/1205.5697
Gustavo Jasso, $n$-abelian and $n$-exact categories, arxiv/1405.7805