  We construct (2+1)-dimensional lattice systems, which we call fusion surface
models. These models have finite non-invertible symmetries described by general
fusion 2-categories. Our method can be applied to build microscopic models
with, for example, anomalous or non-anomalous one-form symmetries, 2-group
symmetries, or non-invertible one-form symmetries that capture non-abelian
anyon statistics. The construction of these models generalizes the construction
of the 1+1d anyon chains formalized by Aasen, Fendley, and Mong. Along with the
fusion surface models, we also obtain the corresponding three-dimensional
classical statistical models, which are 3d analogues of the 2d
Aasen-Fendley-Mong height models. In the construction, the "symmetry TFTs" for
fusion 2-category symmetries play an important role.
