  In this work, we use Ising chain and Kitaev chain to check the validity of an
earlier proposal in arXiv:2011.02859 that enriched fusion (higher) categories
provide a unified categorical description of all gapped/gapless quantum liquid
phases, including symmetry-breaking phases, topological orders, SPT/SET orders
and certain gapless quantum phases. In particular, we show explicitly that, in
each gapped phase realized by these two models, the spacetime observables form
a fusion category enriched in a braided fusion category. We also study the
categorical descriptions of the boundaries of these models. In the end, we
provide a classification of and the categorical descriptions of all
1-dimensional (the spatial dimension) gapped quantum phases with a finite
onsite symmetry.
