Some Researches On The Trivial Extensions Of Categories And Category Representations

In this thesis, we focus on the trivial extensions of categories and representation categories, systematically study the preservability of special categories under trivial extension, structures of the representation categories, and also some properties of it, finally obtain several interesting results.The first chapter introduces the background and the latest trends of our subject, gives an outline of main results of this dissertation.In the second chapter, we use extremal surjection、pushout、pullback to discuss the relationship between an abelian category and its trivial extension category, trying to present the corresponding projective cover、injectivc envelope. Moreover, we show a category is perfect or semiperfect if and only if its trivial extension category is perfect or semiperfect respectively, and apply the results to ring theoryThe third chapter is aimed at describing the structure of representation cate-gories. First, we discuss the relationship between projective reprsentations、cyclical representations and indecomposable representations. Second, we describe the simple categories in two ways. On one hand, we give a characterization of the simple cat-egories by using the connective property of themselves. On the other hand, we put forward an alternative characterization by studying their representation categories.There is an interesting problem which reads:when is a category equivalent to a representation category? In order to solve this problem, we recall two impor-tant kinds of comma categories, using colimits and dense generators, then show a necessary and sufficient condition for the problem in the fourth chapter.