  The quantum nature of electron spin is crucial for establishing topological
invariants in real materials. Since the spin does not in general commute with
the Hamiltonian, some of the topological features of the material can be
extracted from its study. In insulating materials, the spin operator induces a
projected operator on valence states called the spin valence operator. Its
spectrum contains information with regard to the different phases of the spin
Chern class. If the spin valence spectrum is gapped, the spin Chern numbers are
constant along parallel planes thus defining spin Chern insulating materials.
If the spin valence spectrum is not gapped, the changes in the spin Chern
numbers occur whenever this spectrum is zero. Materials whose spin valence
spectrum is gapless will be denoted spin Weyl topological insulators and their
definition together with some of their properties will be presented in this
work. The classification of materials from the properties of the spin valence
operator provides a characterization that complements the existing list of
topological invariants.
