  We propose an axiomatic approach towards studying unlikely intersections by
introducing the framework of distinguished categories. This includes
commutative algebraic groups and mixed Shimura varieties. It allows us to
define all basic concepts of the field and prove some fundamental facts about
them, e.g. the defect condition.
  In some categories that we call very distinguished, we are able to show some
implications between Zilber-Pink statements with respect to base change. This
yields unconditional results, i.e. the Zilber-Pink conjecture for a complex
curve in $\mathcal{A}_2$ that cannot be defined over $\bar{\mathbb{Q}}$, a
complex curve in the $g$-th fibered power of the Legendre family, and a complex
curve in the base change of a semiabelian variety over $\bar{\mathbb{Q}}$.
