  Fusion surface models, as introduced by Inamura and Ohmori, extend the
concept of anyon chains to 2+1 dimensions, taking fusion 2-categories as their
input. In this work, we construct and analyze fusion surface models on the
honeycomb lattice built from braided fusion 1-categories. These models preserve
mutually commuting plaquette operators and anomalous 1-form symmetries. Their
Hamiltonian is chosen to mimic the structure of Kitaev's honeycomb model, which
is unitarily equivalent to the Ising fusion surface model. In the anisotropic
limit, where one coupling constant is dominant, the fusion surface models
reduce to Levin-Wen string-nets. In the isotropic limit, they are described by
weakly coupled anyon chains and are likely to realize chiral topological order.
We focus on three specific examples: (i) Kitaev's honeycomb model with a
perturbation breaking time-reversal symmetry that realizes chiral Ising
topological order, (ii) a $\mathbb{Z}_N$ generalization proposed by Barkeshli
et al., which potentially realizes chiral parafermion topological order, and
(iii) a novel Fibonacci honeycomb model featuring a non-invertible 1-form
symmetry.
