Some Researches On Pull-back Exact Categories

Exact category is not only the foundation of Abelian category but also its nat-ural generalization. Since last 50s or 60s, experts and authors have studied exact categories, especially in the ones based.on additive categories commonly attributed to Quillen, which have played an important role in algebraic K-theory, the represen-tation theory, category theory, etc. The class of exact categories, whose admissible monomorphisms are closed under pull-backs, is investigated in the thesis. These exact categories can be commonly found in torsion theory and they are exactly dif-ferent from abelian categories. In this thesis, we characterize the lattice structures of pull-back exact categorries and also give a study on equivalence and universal propertities of the localizations of modular pull-back exact categories.At the beginning of the thesis, we make a review on the researches, explore the trends related to this thesis, sum up the groundwork and give some notations presented in the thesis. The thesis is divided into four chapters.Chapter one:We first introduce the notion of pull-back exact category and give some descriptions of its basic properties. Then we give some examples to illustrate the existence of these exact categories. We also show the existence of non-abelian pull-back exact categories. At last, we study the recollements of pull-back exact categories.Chapter two:The lattice structures on a pull-back exact category are investi-gated. Using admissible monomorphisms and admissible epimorphisms. we define two classes of modular lattices ((?)) and ((?)), and show the existence of a lattice isomorphism between these two classes. We also use distributive elements to characterize lattice category on a pull-back exact category and then we obtain exact structures of a lattice category induced by those of a pull-back exact category.Chapter three:We use Serre classes and direct limits to define two classes of localization categories of modular pull-back exact categories, respectively, and show these two classes are equivalent.Chapter four:We give a further study on the localization of modular pull-back exact categories under universal properties.