  For symmorphic crystalline interacting gapped systems we derive a
classification under adiabatic evolution. This classification is complete for
non-degenerate ground states. For the degenerate case we discuss some
invariants given by equivariant characteristic classes. We do not assume an
emergent relativistic field theory nor that phases form a topological spectrum.
We also do not restrict to systems with short-range entanglement, stability
against stacking with trivial systems nor assume the existence of
quasi-particles as is done in SPT and SET classifications respectively. Using a
slightly generalized Bloch decomposition and Grassmanians made out of ground
state spaces, we show that the $P$-equivariant cohomology of a $d$-dimensional
torus gives rise to different interacting phases, where $P$ denotes the point
group of the crystalline structure. We compare our results to bosonic
symmorphic crystallographic SPT phases and to non-interacting fermionic
crystallographic phases in class A. Finally we discuss the relation of our
assumptions to those made for crystallographic SPT and SET phases.
