Some Researches On The Localization Of Cocomma Categories

Localization is one of the most important means for algebraic topology research. The localization theory introduced by Grothedieck and Adams is extensively applied to homological algebra, representation theory of algebra. Localizations of certain special categories, such as additive categories, abelian categories, triangulated cate-gories, have been well developed. The introduction summarizes the background and recent development of which we investigate. This dissertation includes five chapters, and concentrates on the localization of cocomma categories.The first chapter applies Verdier localization theory to construct the left lo-calization of a cocomma category by a localizing class of the original category. Particularly, the localization of a morphism category determined by an object is available.In the second chapter, several morphism categories determined by objects, in-cluding the morphism categories determined by finite objects and a special class of morphism categories on an exact category, are studied. Furthermore, their proper-ties are given.The third chapter considers a pair of cocomma categories induced by an adjoint pair. It focus on the construction of the left localization of one cocomma category by the left localizing class of the other, under certain conditions. Afterwards, the relationships between their left localizations are studied.As the consequences of the third chapter, the fourth chapter investigates a series of cocomma categories and comma categories induced by a sequence of adjoint functors, and then constructs their localizations.The fifth chapter sums up the main results of this dissetation.