Idempotent Completion Preserving And M-v Construction Preserving On Categories

Motivated by the facts:any additive category can be idempotent completed and any two additive categories can induce a new additive category by M-V construction. We consider idempotent completion preservings and M-V construction preservings on some categories in this thesis.In chapter one,let((?),Ω,△)be a left triangulated category.If(s((?)),Ω,△)de-notes the idempotent completion of the stabilization of((?),Ω,△)and(S((?)),Ω,△) denotes the stabilization of the idempotent completion of((?)Ω,△),then we get a triangle-equivalence(S((?)),Ω,△)≌(S((?)),Ω,△). Duality,in chapter two,if (R((?)),Ω,△)denotes the costabilization of((?),Ω,△),then they have the other triangle-equivalence(R((?)),Ω,△)≌(R((?)),Ω,△).In chapter three,let((?),ε)be an exact category. The quotient category of the idempotent completed category((?),(?))modulo (?) is proved to be equivalent to the idempotent completed category of the quotient category((?)/M,ε/M),that is ((?)/M,ε/M)≌((?)/M,ε/M),under the assumption that M is an ideal of the additive category (?) and the conditions①②③,M={α∈Mor(?)|α∈M}.In the last chapter,we study the M-V construction preserving original prop-erties on idempotent completed categories,exact categories and quasi-abelian cat-egories. By the way,we find an equivalence on exact categories and quasi-abelian categories.