Models for Gapped Boundaries and Domain Walls

We define a class of lattice models for two-dimensional topological phases with boundary such that both the bulk and the boundary excitations are gapped. The bulk part is constructed using a unitary tensor category C as in the Levin-Wen model, whereas the boundary is associated with amodule category over C.We also consider domainwalls (or defect lines) between different bulk phases.Adomainwall is transparent to bulk excitations if the corresponding unitary tensor categories are Morita equivalent. Defects of higher codimension will also be studied. In summary, we give a dictionary between physical ingredients of lattice models and tensor-categorical notions