  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.
