The Equivalent Propositions Of The Octahedral Axiom Of Triangulated Categories And Applications

The notion of a triangulated category was first introduced and studied by Grothendieck-Verdier in early sixties of the last century, which marks a new starting point of the development of modern algebras. A triangulated category is an addition category with an automorphism, and satisfies four axioms. One of the axioms is called the octahedral axiom and plays an important role. Our main results are the equivalent propositions of the octahedral axiom and applications. This academic dissertation is divided into three chapters altogether.In chapter one, we introduce the research direction and the trends of the development related to the thesis, and sum up the groundwork of this text.In chapter two, we recall a pre-triangulated category, a triangulated category and the octahedral axiom of triangulated categories. We list some propositions of triangulated categories and eight proved equivalent propositions of the octahedral axiom. The eight equivalent propositions are (TR4-1)—(TR4-8).In chapter three, we deal with the other equivalent propositions of the octahedral axiom and applications. In section one, we obtian a equivalent proposition (TR4-9) by (TR4-1). In section two, we contiue to study homotopy cartesian and the octahedral axiom, then we obtain seven equivalent propositions (TR4-10)—(TR4-16). In section three, we recall the mapping cone and show that the mapping cone is equivalent with the octahedral axiom if the other three axioms are satisfied. In section four, we point out that the mapping cylinder is not equivalent with the octahedral axiom. In section five, we study the applications of the octahedral axiom.