  In a recent work arXiv:2108.08835, Kong, Wen and Zheng use Ising chain to
show that the macroscopic observables in a 1d (space dimension) gapped quantum
phase form an enriched fusion category. In this work, we generalize their
results to the phases with finite abelian onsite symmetries. By carefully
analyzing the observables on both the symmetry preserving and symmetry breaking
cases, we explicitly show that the spacetime observables of these 1d phases
form fusion categories enriched in their monoidal centers. These are actually
concrete examples of the holographic duality based on topological wick
rotation. Our categorical descriptions also reproduce an existing
classification result and shed light on a unified definition of all quantum
phases.
