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Gapped quantum liquids (GQL) include both topologically ordered states (withlong range entanglement) and symmetry protected topological (SPT) states (withshort range entanglement). In this paper, we propose a classification of 2+1DGQL for both bosonic and fermionic systems: 2+1D bosonic/fermionic GQLs withfinite on-site symmetry are classified by non-degenerate unitary braided fusioncategories over a symmetric fusion category (SFC) $\cal E$, abbreviated as$\text{UMTC}_{/\cal E}$, together with their modular extensions and totalchiral central charges. The SFC $\cal E$ is $\text{Rep}(G)$ for bosonicsymmetry $G$, or $\text{sRep}(G^f)$ for fermionic symmetry $G^f$. As a specialcase of the above result, we find that the modular extensions of$\text{Rep}(G)$ classify the 2+1D bosonic SPT states of symmetry $G$, while the$c=0$ modular extensions of $\text{sRep}(G^f)$ classify the 2+1D fermionic SPTstates of symmetry $G^f$. Many fermionic SPT states are studied based on theconstructions from free-fermion models. But it is not clear if free-fermionconstructions can produce all fermionic SPT states. Our classification does nothave such a drawback. We show that, for interacting 2+1D fermionic systems,there are exactly 16 superconducting phases with no symmetry and no fractionalexcitations (up to $E_8$ bosonic quantum Hall states). Also, there are exactly8 $Z_2\times Z_2^f$-SPT phases, 2 $Z_8^f$-SPT phases, and so on. Besides, weshow that two topological orders with identical bulk excitations and centralcharge always differ by the stacking of the SPT states of the same symmetry.

Classification of 2+1D topological orders and SPT orders for bosonic and fermionic systems with on-site symmetries