Some Studies On Extriangulated Categories

Extriangulated categories were introduced by Nakaoka and Palu in 2016.Exact categories and triangulated categories are examples of extriangulated cat-egories.Of course,there exist examples of extriangulated categories which are not exact nor triangulated.At presentit has become popular in the category theory.The main purposes of this thesis is to study the properties of extriangu-lated categories by using the methods of homological algebra.At the same time,we study the construction of triangulated quotient categories,the properties of right triangulated categories and right triangulated quotient categories.This thesis consists of six chapters.In chapter one,we recall some notions and classic results,which are need in the sequel.In particular,we introduce some correlation theory about extri-angylated categories.In chapter two,we prove that the extension subcategory of two covariantly finite subcategories in an extriangulated category is also covariantly finite.As an application,we give a simultaneous generalization of a result of Chen and of a result of Gentle-Todorov.In chapter three,we prove that certain quotient categories of extriangu-lated categories are abelian.As an application,we obtain that cluster-tilting subcategories of extriangulated categories are abelian.This unifies a result by Koenig-Zhu for triangulated categories and a result by Demonet-Liu for exact categories.In chapter four,from mutation pairs in triangulated categories,we give three equivalent characterization on triangulated quotient categories.As an application,this generalize a result of J(?)rgensen.In chapter five,we give a equivalent characterization on the octahedral axiom for right triangulated categories,and show that the morphism axiom for right angulated categories is redundant by this equivalent characterization.In chapter six,we get that certain quotient categories of right triangulated categories are right triangulated.This result generalizes many results.We also obtain that right triangulated version of Wakamatsu's lemma.