Some Researches And Applications Of The Direct Limits And The Limit Categories

Categories and extensions of the categories are important branches in mathe-matics study, develop crosswise with many diciplines, and yield a series of profound and the challenging research results. This thesis takes direct limits and limit cat-egories as research objects, and gives three equivalent characterizations of direct limit. As for further studies, it researches on the recollements, Kt groups of lim-it categories, the communicativity between the idempotent completion (respectively loop categories) and limit categories.The first chapter introduces the background and the recent development, and gives an outline of main results of this dissertation.Started from the known concepts of generalized pushout, origin object and representable functor, the second chapter shows three equivalent characterizations for direct limits of general categories by constructing new categories. Finally, as an application, it also gives a new proof for the existence of direct limits in module category.The third chapter gives a rccollement of a cocomplctc abelian category and its limit category by constructing six functors, and then obtains that the ki groups of cocomplctc abelian categories are summands of their limit categories. Finally, it can applied to the modules category as well.Based on the concepts and properties of idempotent completion category, the fourth chapter studies the relationship between the idempotent completion category and its limit category, and obtains an interesting result:the limit category of an idempotent completion category is also idempotent completed. Then it obtains three equivalent propositions for a category (with zero object) being idempotent completed. Moreover, for a comcomplete category, it also shows that the idempotent completion of its limit category is equivalent to the limit category of its idempotent completion category.For a category, the fifth chapter discusses the relationship between the loop category of its limit category and the limit category of its loop category. What’s more, the aforementioned two categories are equivlent for the cocomplete abelian categoires.