Some Researches On The Quotient Category Of The Quasi-right Exact Category And Its Localization Category

Research on category theories,including Abelian category,exact category and so on,has begun as early as the fifties last century.Just because Abelian category is the exact category,it is more significant to study the exact category.There are two kinds of different,research methods about the exact category,what this paper studies is one of them,namely,it is based on the Quillen additive category.The key problem to the exact category is that the so-called short exact sequence.If the kernel morphism of the short exact sequence is taken,some one-sided exact categories are formed under certain condition.In this paper,we study the quasi-right exact category which is made of kernel morphism having some certain conditions which have certain connection with weakly right exact category introduced by A.Jafari and other right exact categories.Meanwhile,we also study the quotient category of the quasi-right exact category and its localization category.The paper is divided into four chapters.Chapter one:The introduction section.We mainly introduce the research background、direction and some necessary preparatory knowledge.Chapter two:We give the concepts of several kinds of common one-sided exact categories,which include BC right exact category、BC strongly right exact catego-ry、Rump right exact category、Z right exact category、weakly right exact category and quasi-right exact category,and we study the relationships between them.Weak-ly right exact category is closely related with the quasi-right exact category.Finally,we popularize and prove the snake lemma of the weakly right exact category.Chapter three:We explore the quotient category conditions which are based on the quasi-right exact category,namely,if((A,H)is quasi-right exact category,A/R will be quotient,category.We also examine the conditions of A/R being quasi-right exact category,which are called R conditions to meet the need.Chapter four:We explore the localization problems and discuss the conditions of the localization category of the quasi-right exact category also being the quasi-right exact category.