N-abelian Categories And Exactness

In 2016,n-abelian and n-exact categories were introduced by Gustavo Jasso,which are analogs of abelian and exact categories from the point of view of higher homological algebra.In this thesis,we are mainly focused on two special classes with n-exact structure on n-abelian category.Two articles from Sieg.D have been inspired,so we defined l-strict n-exact sequence and n-semi stable exact sequence.Then the class of all of l-strict n-exact se-quence and the class of all of n-semi stable exact sequence both are discussed in n-abelian category.This master's degree thesis consists of three chapters.In first chapter,we are mainly recall some basic concepts and related properties of this thesis,as well as the signed symbols.Finally,we present the historical background and list the main result of this thesis.In second chapter,we list the definition of n-abelian category and n-exact category respectively.and some lemmas related to this article.Also,several theorems are proved.In third chapter,Sieg.D has defined l-strict in pre-additive category.In the first section,we have redefined l-strict in semisimple abelian category,which satisfy the two axioms of(A0)and(A1)in n-abelian category.So that,defines l-strict n-exact sequence.Finally,we prove that all the classes of l-strict n-exact sequence in a category,which is not only a semisimple abelian category but n-abelian one,is an n-exact structure on n-exact category.In the second section,firstly,we promote semisimple stable kernel and semisimple stable cokernel in reference[25]to n-semi stable kernel and n-semi stable cokernel respectively.Finally,we prove that the class of n-semi stable exact sequence in n-abelian category is an n-exact structure on n-exact category.