Some Researches On Subobjects In Pull-back Exact Categories

The pull-back exact categories are such categories that they are between A-belian categories and exact categories. These categories have all properties of exact categories, but different from Abelian categories, they are the real promotions. Such kind of categories have shown their important roles in different fields such as algebra-ic K-theory and the representation theory of algebras etc., see[20,21,23,27-29]. The objects are important elements in a category,and the study of the objects is also important, so the researches of the intersection and the sum of the objects have been put forward naturally, see[10,11,17,32,34]for further details.We consider the objects in pull-back exact categories in the thesis. At the beginning of this thesis, we summarize the background and the developments of which we investigate, elaborate the main results. Than we outline the main findings of this article and describe some notations. This paper is divided into four sections.In the first chapter, we give the concept of pull-back exact categories, intro-duce the existing important properties. We promote the related properties in A-belian categories to pull-back exact categories, then obtain some similar conclu-sions to which in Abelian categories, that is Kerf(?) Kergf(?) Kerg×B Coimf and Kerg(?)Kerg цD Coimf(?) Coimgf are both ε-short exact sequences, where D=Kerg×B Coimf.In the second chapter, we show the notions and some basic properties of single subobjects, maximal subobjects,small subobjects and essential subobjects, intro-duce the definition of the projective cover in pull-back exact categories.In the third chapter,we show some related results of single extension. Under the condition of the left-exact functors, we demonstrate that the single extension of a pull-back exact categories is also a pull-back exact categories. We also consider the preservation problem of single subobjects, small subobjects and essential subobjects with single extension.The fourth chapter sums up the main results of this thesis.