Nuclear dimension and $\mathcal{Z}$-stability of pure C?-algebras

In this article I study a number of topological and algebraic dimension type properties of simple C^∗-algebras and their interplay. In particular, a simple C^∗-algebra is defined to be (tracially) (m,`(m)])(m,\bar{m})-pure, if it has (strong tracial) m-comparison and is (tracially) `(m)]\bar{m}-almost divisible. These notions are related to each other, and to nuclear dimension.