Some Researches On Semi-Pullback-Pushout Exact Categories

Since 1950’s, people have studied the exact categories, with the definition mainly by two ways. In this dissertation, we take the definition in the way of Quillen’s exact category based on the additive categories. Exact category is more extensive than Abel category. Abel category is exact, however, exact category may not be abel. Therefore, it is significant to study exact category and con-duct some appropriate generalizations. In this paper, we focus on exact categories with some conditions, that is, semi-pullback(pushout) exact categories and modular semi-pullback(pushout) exact categories, which can be found as examples in torsion theory. They are different from Abel categories. In the paper, it will prove this example in the fourth chapter, We make the localization to determined modular semi-pullback of exact categories and use the lattice theory to investigate the lattice structures of semi-pullback-pushout exact categories.The dissertation is divided into four chapters.In the first chapter, we carry on the introduction to the research directions and the trends of the development related to dissertation, and sum up the groundwork of this text.In the second chapter, we pay attention to the localization of modular semi-pullback exact categories. Firstly, we introduce the definition of semi-pullback(pushout) exact categories and modular semi-pullback(pushout) exact categories, generalizing the Noether isomorphism theorem in pre-abelian exact category. Then we show a class of A-weak separable admissible morphism, which is recorded as SA, and prove SA is a localization of modular semi-pullback exact categories.In the third chapter, we present the categories of equivalence class which ad-missible monomorphisms end with E and the categories of equivalence class which admissible epimorphisms begin with E, denoted as (?) and (?) respectively. Then we construct the lattice structures on semi-pullback-pushout exact category, and prove ((?), ∨, ∧) and ((?),∨,∧) are lattices. Finally, under certain conditions, we study how can ((?), ∨,∧) (or ((?), ∨,∧)) be a modular lattices.In the fourth chapter, we consider semi-pullback-pushout exact category, and illustrate it is not abel.