Pullback And Pushout In Abelian Categories And In The Categories Of Sheaves Of Abelian Groups

This dissertation contains two parts: Pullback and Zassenhaus Lemma in Abelian categories, Pullback and pushout in the categories of sheaves of Abelian groups.In the second chapter, using the properties of pullback in the Abelian categories, we prove the Zassenhaus Lemma in Abelian categories , and then we prove the Jordan-H(?)lder Theorem in Abelian categories.In the third chapter, we descripts the characterization of pullback in the categories of sheaves of Abelian groups on a topological space X, and we also descripts the characterization of pushout by using an adjoint pairs in the categories of the sheaves of Abelian groups. Meanwhile we show that if (F;α,β) is the pullback of (φ,ψ;H), then (f_*F; f_*α, f_*β) is the pullback of (f_*φ, f_*ψ; f_*H); and if (φ,ψ;H) is the pushout of (F;α,β). then (f^-1φ, f^-1ψ: f^-1H) is the pushout of (f^-1F;f^-1α,f^-1β).