  Recently, it has been shown that two-dimensional bosonic symmetry-protected
topological(SPT) phases with on-site unitary symmetry $G$ can be completely
classified by the group cohomology class $H^3(G, \mathrm{U}(1))$. Later, group
super-cohomology class was proposed as a partial classification for SPT phases
of interacting fermions. In this work, we revisit this problem based on the
mathematical framework of $G$-extension of unitary braided tensor
category(UBTC) theory. We first reproduce the partial classifications given by
group super-cohomology, then we show that with an additional $H^1(G,
\mathbb{Z}_2)$ structure, a complete classification of SPT phases for
two-dimensional interacting fermion systems for a total symmetry group
$G\times\mathbb{Z}_2^f$ can be achieved. We also discuss the classification of
interacting fermionic SPT phases protected by time-reversal symmetry.

Strongly Correlated Electrons