Some Researches On Two Kinds Of Pre-additive Categories

Category theory has promoted the development of many branches of mathe-matics since S. Eilenberg and S.Maclane introduced the concept of category in1945. Study of the structures of the categories and the preserving problems for categories under the extending actions are hot topics for mathematicians. In this thesis, we investigate two kinds of pre-additive categories:one is pre-additive with two object-s, the other is idempotent completion category (specially Krull-Schmidt category). Discuss the preserving problems for these two kinds of pre-additive categories un-der taking their functor categories, push-out categories, product categories, complex categories and stable categories.The first chapter presents the background as well as the structure of this dis-sertation.The second chapter uses Morita context to characterize the pre-additive cate-gory with two objects.In chapter three, based on the conception and the properties of idempotent completion categories, we presents an interesting result:the functor category of an idempotent completion category is an idempotent completion category. Moreover, we discuss the exchanging relation for an additive category taking its functor cate-gory and its idempotent completion, and give a positive answer that the idempotent completion category of an additive category taking its functor category is equivalent to the functor category of this additive category taking its idempotent completion.The fourth chapter researches the push-out categories of idempotent comple-tion categories, and obtains the result goes that a category with zero object is an idempotent completion category if and only if its push-out category is also an idem-potent completion category. With this idea, we present a category with zero object is additive(or abelian) if and only if its push-out category is also additive(or abelian).The fifth chapter studies the product categories、complex categories and the stable categories of idempotent completion categories. And we put forward that all of the product categories、complex categories and the stable categories of idempo-tent completion categories are idempotent completion. As applications, we get the following result:if A is a Krull-schmidt category,(C(A)) denotes the idempotent completion of the stable category of the complex category of l, and C(A) denotes the stable category of the complex category of the idempotent completion category for l, then C(A) and (C(A)) are equivalent as triangulated categories.