Covering Dimension of C*-Algebras and 2-Coloured Classification

We introduce the concept of nitely 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 classication 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 nite topological covering dimension, this conrms 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 nite nuclear dimension.