Some Studies On N-angulated Categories

Following the development of triangulated categories and the introduction of higher dimensional algebra,the notion of n-angulated category is presented naturally.It appeared when considering some kind of n-cluster tilting subcategories of triangulated category,and aroused the heated concern of many mathematics workers.An n-angulated category is a generalization of triangulated category,and the classic triangulated category is the special case of n-angulated categories in n(28)3.This paper studies the equivalent statements of higher mapping cone axiom of n-angulated categories,the localization theory of n-angulated categories and the stabilization of left n-angulated categories.This paper is organized as follows.In chapter two,we introduce the homotopy cartesian diagrams in n-angulated categories.We prove that the higher homotopy cartesian axiom is equivent to the higher mapping cone axiom,then give several other somewhat new equivalent statements of the higher octahedral axiom.In chapter three,we introduce the compatible localizing class of n-angulated categories and prove that the quotient of an n-angulated category with respect to its compatible localizing class admits an n-angulated structure.In chapter four,we first prove that the stabilization category of a left n-angulated category is an n-angulated category,then prove that the Grothendieck group of a left n-angulated category and the Grothendieck group of its stabilization category are isomorphic.