Fixed-point constructions of (1+1)d fusion category symmetry enriched topological phases

With the broadened understanding of symmetries, we initiate the investigation of (1+1)d topological phases enriched by fusion category symmetries. With a focus on symmetry protected topological phases (SPT), we start with Hopf algebra as the on-site version of the anomaly-free fusion category symmetry to realize the anomaly-free fusion category symmetry on the lattice. Under such a definition of ``onsite" symmetry, we define the trivial phase for SPT protected by non-invertible symmetries. We introduce the concept of local degeneracy and a SPT phase is viewed as the one, the local degeneracy of which is one in the bulk. Such an understanding serves as an alternative interpretation of non-invertible SPTs apart from their relation with fiber functors. We construct specific examples of SPTs protected by $\Rep(D_8)$ and $\Rep(H_8)$ symmetries. In particular, we propose one method to permute three SPT phases protected by $\Rep(D_8)$ symmetry.