The axioms for<i>n</i>–angulated categories

Abstract

We discuss the axioms for an n-angulated category, recently introduced by Geiss, Keller and Oppermann in In particular, we introduce a higher "octahedral axiom", and show that it is equivalent to the mapping cone axiom for an n-angulated category. For a triangulated category, the mapping cone axiom, our octahedral axiom and the classical octahedral axiom are all equivalent. 18E30 1 Introduction Triangulated categories were introduced independently in algebraic geometry by Verdier [7; These constructions have since played a crucial role in representation theory, algebraic geometry, commutative algebra, algebraic topology and other areas of mathematics (and even theoretical physics). Recently, Geiss, Keller and Oppermann introduced in [1] a new type of categories, called n-angulated categories, which generalize triangulated categories: the classical triangulated categories are the special case n D 3. These categories appear for instance when considering certain .n 2/-cluster tilting subcategories of triangulated categories. Conversely, certain n-angulated Calabi-Yau categories yield triangulated Calabi-Yau categories of higher Calabi-Yau dimension.