三角范畴定义中八面体公理的等价命题及其应用

The Equivalent Propositions of the Octahedral Axiom of Triangulated Categories and Applications

三角范畴是一个带有自同构的加法范畴,并且满足4条公理,其中的1条重要公理是八面体公理.由Grothen-dick-Verdier在上个世纪60年代提出的八面体公理相对于其它3条公理形式比较复杂,应用起来比较不方便.因此研究八面体公理的其它等价命题引起了人们的兴趣.本文在王济荣工作的基础上给出八面体公理的第1个等价命题,再利用对偶的思想导出八面体公理的第2个等价命题。最后利用homotopy cartesian得到八面体公理的第3个等价命题,并利用第3个等价命题简化Peng和Tan的证明.

A triangulated category is an addition category with an automorphism. It satisfies four axioms. One of which is the octahedral axiom. It plays an important role. The notion of the octahedral axiom was first introduced by Grothendieck-Verdier,inearly sixties of the last century. Its form is quite complex. And it is not convenient to use. Then studying the other equivalent propositions arouses people＇s interest. In this paper,the homotopy Cartesian is equivalent with the octahedral axiom if theother three axioms are satisfied are proved. Moreover,two general equivalent propositions of the octahedral axiom are given. As an application,the proof of Peng and Tan is simplified by the third equivalent proposition.