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We introduce the concept of fermionic matrix product operators, and show thatthey provide a natural representation of fermionic fusion tensor categories.This allows for the classification of two dimensional fermionic topologicalphases in terms of matrix product operator algebras. Using this approach wegive a classification of fermionic symmetry protected topological phases withrespect to a group $G$ in terms of three cohomology groups:$H^1(G,\mathbb{Z}_2)$, describing which matrix product operators are ofMajorana type, $H^2(G,\mathbb{Z}_2)$, describing the fermionic nature of thefusion tensors that arise when two matrix product operators are multiplied, andthe supercohomolgy group $\bar{H}^3(G,U(1))$ which corresponds to theassociator that changes the order of fusion. We also generalize the tensornetwork description of the string-net ground states to the fermionic setting,yielding simple representations of a class that includes the fermionic toriccode.

Fermionic Matrix Product Operators and Topological Phases of Matter