Covering dimension of C∗-algebras and 2-coloured classification

We introduce the concept of finitely coloured equivalence for unital ∗-homomorphisms between C∗-algebras, for which unitary equivalence is the 1- coloured case. We use this notion to classify ∗-homomorphisms from separable, unital, nuclear C∗-algebras into ultrapowers of simple, unital, nuclear, Z-stable C∗- algebras with compact extremal trace space up to 2-coloured equivalence by their behaviour on traces; this is based on a 1-coloured classification theorem for certain order zero maps, also in terms of tracial data. As an application we calculate the nuclear dimension of non-AF, simple, separable, unital, nuclear, Z-stable C∗-algebras with compact extremal trace space: it is 1. In the case that the extremal trace space also has finite topological covering dimension, this confirms the remaining open implication of the Toms-Winter conjecture. Inspired by homotopy-rigidity theorems in geometry and topology, we derive a "homotopy equivalence implies isomorphism" result for large classes of C∗-algebras with finite nuclear dimension.