  We construct a family of one-dimensional (1D) quantum lattice models based on
$G$-graded unitary fusion category $\mathcal{C}_G$. This family realize an
interpolation between the anyon-chain models and edge models of 2D
symmetry-protected topological states, and can be thought of as edge models of
2D symmetry-enriched topological states. The models display a set of
unconventional global symmetries that are characterized by the input category
$\mathcal{C}_G$. While spontaneous symmetry breaking is also possible, our
numerical evidence shows that the category symmetry constrains the the models
to the extent that the low-energy physics has a large likelihood to be gapless.
