Some Researches On The Properties Conservation And Extension Of Functor Categories

Functor category is an important for category-theoretic, and according to the Yoneda embedding theorem, gives an arbitrary category can be embedded in a functor category, so the functor category can be regarded as a category extension. In this thesis, we focus on a functor category, systematically study the preservability of special category under functor category, structures of the representation, 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.The second chapter studies preservability of the nature of the relationships between the extension and the original category and got the equivalent characteri-zations of several special properties category, that is, if and only if Dc is also and the null object of Dc and coproduct of the constant functor of C â†' D are constant functors (i.e. theorem2.1.7). Moreover, functor category and common category such as loop category, Artin category, weakly exact category, etc. in category theory have similar results. Thus, the functor category Dc has maintained good structures and properties of the original category D. Thus can execute many many new construc-tion which the original category doesn’t have on the functor category.The main consideration of the third chapter is the commutation relation be-tween a functor category and the η-extension category. Then get a conclusion that the preadditive category (D(η))c is isomorphic to the preadditive category Dc(η). This conclusion extended the results of Robert M. Fossum, Phillip A. Griffith, Idun Reiten in [24] of (R-Mod)(?)(M(?)R-)≌(R(?)M)-Mod to representation category of the small preadditive category...