Some Properties Of N-Exact Categories And Weak N-Exact Categories

n-Exact categories were introduced by Gustavo Jasso in 2016,which are natural extension of exact categories.This thesis mainly studies homology properties of n-exact categories,introduces the concept of weak n-exact categories and studies some properties.The first Chapter introduces some relevant background of the content of this paper.In the second Chapter,we introduce some basic concepts such as additive categories and exact categories and some preliminary knowledge of related properties.In the third Chapter,we first introduced the concept of n-exact categories and some related properties.On this basis,we study some homology properties of n-exact categories,introduce the concepts of projective object and injective object in n-exact categories,and give the concept and some properties of projective n-resolution.For example,the uniqueness of projective n-resolution is proved.In the fourth Chapter,we first introduce n-cluster-tilting subcategories in exact categories,and the relationship between n-cluster-tilting subcategories and n-cluster-tilting subcategories of the exact categories.Referring to Kvamme’s conclusion,we give some necessary conditions that n-exact categories are equivalent to n-cluster-tilting subcategories of some exact categories.Next,we introduce the concept of n-extension of n-exact categories,and prove some related properties,so as to generalize the related results of n-extension in n-abelian categories.In the fifth Chapter,we generalize the concept of weakly exact categories,introduce the concept of weakly n-exact categories,and give some related properties.We also introduce the concept of left(right)weak n-exact categories and prove that the left weak n-exact structure and the right weak n-exact structure can form a weak n-exact structure.