Title: The Extension Dimension of Abelian Categories

Let $\mathcal{A}$ be an abelian category having enough projective objects and enough injective objects. We prove that if $\mathcal{A}$ admits an additive generating object, then the extension dimension and the weak resolution dimension of $\mathcal{A}$ are identical, and they are at most the representation dimension of $\mathcal{A}$ minus two. By using it, for a right Morita ring $\Lambda$, we establish the relation between the extension dimension of the category ${\rm mod}\;\Lambda$ of finitely generated right $\Lambda$-modules and the representation dimension as well as the global dimension of $\Lambda$. In particular, we give an upper bound for the extension dimension of ${\rm mod}\;\Lambda$ in terms of the projective dimension of certain class of simple right $\Lambda$-modules and the radical layer length of $\Lambda$. In addition, we investigate the behavior of the extension dimension under some ring extensions.