Some Researches On Comma Categories

In this thesis,we study the the projective covers,(semi-)perfect properties and AR sequences of comma categories,as well as the recollement of generalized comma categories.At the beginning of this thesis,we summarize the background of this paper,which includes the history and the trend of categories,recollement,extension and perfect properties of the categories.On this basis,this academic dissertation is devoted to the following three aspects of the research.In the first chapter,we introduce basic concepts and some lemmas of the con-struction of MacPherson and Vilonen,and point out that comma category is a kind of special categories A(ζ)in the the construction of MacPherson and Vilonen.Based on this,we obtain the sufficient and necessary conditions of projective cover in com-ma categories and prove the(semi-)perfect properties of comma categories.Then,we apply it to the ring and obtain the corresponding conclusions.In the second chapter,we use the relationships between the comma categories and the trivial extension categories to obtain the sufficient and necessary conditions of exact sequence in comma categories.Furthermore,the paper shows preservation of AR sequence in comma categories and examples.In the third chapter,we induce the functors of generalized comma categories A(ζ)from abelian categories,which use the methods of inducing the functor of com-ma categories form abelian categories.On this basis,we construct the recollement of generalized comma categories from ablian categories.