Some Researches On K-Categories

In this article, we disscuss the construction and representations of k-categories. This paper mainly consists of three sections. In the first section, we consider k-categories with special structure (e.g. Hopf module category and G-graded categories). We show that the structures which mentioned before are perserved under the process of trivial extensions and karoubianness. After that we prove some dual theorem about idepotent completion and idepotent completion of these categories.In the second section, we propose the definition of the samsh product category and the skew category and these constructions agree with the usual ones for algebras. We can see these algebras in [25],for example. Firstly we describe functorial relations between the representation theories of a category and of a smash product of it. After that we recover in a categorical generalized setting the Duality Theorems of S.X. Liu in [9]. Finally we discuss transitivity of the smash product.In the third section, we consider a class of k-category with special structure of morphism set, that is to say, G-garded categories. We prove that the category of idepotent completion and trivial extensions of a G-graded category is also G-graded respectively. And then we know that there is a duality theorem for G-categories with idepotent completion and trivial extensions.