The Zilber-Pink conjecture is a very general statement that implies many well-known results in diophantine geometry, e.g., Manin-Mumford, Mordell-Lang, André-Oort and Falting's Theorem. It explains the behaviour of intersections between an algebraic variety and a countable family of "special varieties” in an ambient space that usually is an abelian variety, a Shimura variety or even a family of abelian varieties. After a general introduction to the problem, I will report on joint work with Gabriel Dill in which we proved that the Zilber-Pink conjecture for a complex abelian variety A can be deduced from the same statement for its trace, i.e., the largest abelian subvariety of A that can be defined over the algebraic numbers. This gives some unconditional results, e.g., the conjecture for curves in complex abelian varieties (over the algebraic numbers this is due to Habegger and Pila) and the conjecture for arbitrary subvarieties of powers of elliptic curves that have transcendental j-invariant. While working on this project we realised that many definitions, statements and proofs were formal in nature and we came up with a categorical setting that contains most known examples and in which (weakly) special subvarieties can be defined and Zilber-Pink and Mordell-Lang statements can be formulated. We obtained some conditional as well as some unconditional results .-- This seminar is part of the activities of the Dipartimento di Eccellenza CUP B83C23001390001 and it is funded by the European Union – Next Generation EU.