  The string-net approach by Levin and Wen, and the local unitary
transformation approach by Chen, Gu, and Wen, provide ways to classify
topological orders with gappable edge in 2D bosonic systems. The two approaches
reveal that the mathematical framework for 2+1D bosonic topological order with
gappable edge is closely related to unitary fusion category theory. In this
paper, we generalize these systematic descriptions of topological orders to 2D
fermion systems. We find a classification of 2+1D fermionic topological orders
with gappable edge in terms of the following set of data $(N^{ij}_k, F^{ij}_k,
F^{ijm,\alpha\beta}_{jkn,\chi\delta},d_i)$, that satisfy a set of non-linear
algebraic equations. The exactly soluble Hamiltonians can be constructed from
the above data on any lattices to realize the corresponding topological orders.
When $F^{ij}_k=0$, our result recovers the previous classification of 2+1D
bosonic topological orders with gappable edge.
