N-stable Exact Sequence

The n-exact and n-abelian categories were introduced by Jasso [12].They are analogs of abelian categories and exact categories from the point of view of higher homological algebra. In this thesis, we are mainly focused on n-exact categories,n-abelian categories and the new definitional n-stable exact categories. This master’s degree thesis consists of four chapters.In first chapter, we recall some backgrounds needed in the sequel and list the main results of this thesis.In second chapter, we recall some basic definitions and properties,and we present notions of n-pullback diagrams and n-pushout diagrams,then we prove the close link between n-pullback diagrams, n-kernel and n-cokernel.In third chapter,firstly, we introduce the definition of n-exact categories, and we present properties of n-exact categories. Then we give a link between admissible n-exact sequence and n-pullback diagrams, n-pushout diagrams. After that, we introduce the projection and injective objects under the n-exact structure,and get their basic properties. Finally, we introduce n-abelian categories, and we prove the relationship between weak cokernel, weak kernel and split epimorphism under idempotent complete additive categorj’.In forth chapter, we present the definitions of n-semi stable kernel,n-semi stable cokernel and n-stable exact sequence. Firstly, we demonstrate n-pullback diagrams of n-pullback diagrams also form n-pullback diagrams. Then we prove that n-cokerel and n-cokerel form a commutative diagram is n-pushout diagrams. On the basis, we give a further characterization of the properties of n-stable sequence.