For C*-algebras arising from dynamical systems or more generally groupoids, one would like to read off regularity properties of the algebra from the underlying dynamics. One such regularity property, which rose to prominence for its role in the classification program for nuclear C*-algebras is finite nuclear dimension. Dynamical variants on covering dimension can give upper bounds for the nuclear dimension of the resulting C*-algebras, and such results go well beyond the usual unitality and simplicity assumptions from classification. But what these dynamical dimensions are usually picking up is a colored version of local subhomogeneity of the C*-algebra, a much stronger property than finite nuclear dimension.
