The authors introduce the concept of finitely coloured equivalence for unital    -homomorphisms between    -algebras, for which unitary equivalence is the  -coloured case. They use this notion to classify    -homomorphisms from separable, unital, nuclear    -algebras into ultrapowers of simple, unital, nuclear,  -stable    -algebras with compact extremal trace space up to  -coloured equivalence by their behaviour on traces; this is based on a  -coloured classification theorem for certain order zero maps, also in terms of tracial data.

As an application the authors calculate the nuclear dimension of non-AF, simple, separable, unital, nuclear,  -stable    -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, the authors derive a “homotopy equivalence implies isomorphism” result for large classes of    -algebras with finite nuclear dimension.