On topologically zero-dimensional morphisms

We investigate $^*$-homomorphisms with nuclear dimension equal to zero. In the framework of classification of $^*$-homo-morphisms, we characterise such maps as those that can be approximately factorised through an AF-algebra. Along the way, we obtain various obstructions for the total invariant of zero-dimensional morphisms and show that in the presence of real rank zero, nuclear dimension zero can be completely determined at the level of the total invariant. We end by characterising when unital embeddings of $\mathcal{Z}$ have nuclear dimension equal to zero.