  We construct topological quantum field theories (TQFTs) and commuting
projector Hamiltonians for any 1+1d gapped phases with non-anomalous fusion
category symmetries, i.e. finite symmetries that admit SPT phases. The
construction is based on two-dimensional state sum TQFT whose input datum is an
$H$-simple left $H$-comodule algebra, where $H$ is a finite dimensional
semisimple Hopf algebra. We show that the actions of fusion category symmetries
$\mathcal{C}$ on the boundary conditions of these state sum TQFTs are
represented by module categories over $\mathcal{C}$. This agrees with the
classification of gapped phases with symmetry $\mathcal{C}$. We also find that
the commuting projector Hamiltonians for these state sum TQFTs have fusion
category symmetries at the level of the lattice models and hence provide
lattice realizations of gapped phases with fusion category symmetries. As an
application, we discuss the edge modes of SPT phases based on these commuting
projector Hamiltonians. Finally, we mention that we can extend the construction
of topological field theories to the case of anomalous fusion category
symmetries by replacing a semisimple Hopf algebra with a semisimple
pseudo-unitary connected weak Hopf algebra.
