Chapter 8 extends the 10-fold way of gapped phases from one to any dimension of space. This is done by presenting the homotopy groups of the classifying spaces of normalized Dirac masses. There follows two applications. First, there is the interplay between Anderson localization and the topology of classifying spaces for disordered quantum wires. Second, it is possible to derive the breakdown of the 10-fold way due to short-range interactions in any dimension. The chapter closes with the relationship between invertible topological phases and invertible topological field theories.