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

The authors introduce the concept of finitely coloured equivalence for unital ^*-homomorphisms between \mathrm C^*-algebras, for which unitary equivalence is the 1-coloured case. They use this notion to classify ^*-homomorphisms from separable, unital, nuclear \mathrm C^*-algebras into ultrapowers of simple, unital, nuclear, \mathcal Z-stable \mathrm 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 the authors calculate the nuclear dimension of non-AF, simple, separable, unital, nuclear, \mathcal Z-stable \mathrm 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, the authors derive a "homotopy equivalence implies isomorphism" result for large classes of \mathrm C^*-algebras with finite nuclear dimension.