  We classify the different ways in which matrix product states (MPSs) can stay
invariant under the action of matrix product operator (MPO) symmetries. This is
achieved through a local characterization of how the MPSs, that generate a
ground space, remain invariant under a global MPO symmetry. This
characterization yields a set of quantities satisfying the coupled pentagon
equations, associated with a module category over the fusion category that
describes the MPO symmetry. Equivalence classes of these quantities provide
complete invariants for an MPO symmetry protected phase: they are robust under
continuous deformations of the MPS tensor, and two phases with the same
equivalence class can be connected by a symmetric gapped path. Our techniques
match and extend the known renormalization fixed point classifications and
facilitate the numerical study of these systems. For MPO symmetries described
by a group, we recover the symmetry protected topological order classification
for unique and degenerate ground states. Moreover, we study the interplay
between time reversal symmetry and an MPO symmetry and we also provide examples
of our classification, together with explicit constructions based on groups.
Finally, we elaborate on the connection between our setup and gapped boundaries
of two-dimensional topological systems, where MPO symmetries also play a key
role.
