  We study lattice constructions of gapped fermionic phases of matter. We show
that the construction of fermionic Symmetry Protected Topological orders by Gu
and Wen has a hidden dependence on a discrete spin structure on the Euclidean
space-time. The spin structure is needed to resolve ambiguities which are
otherwise present. An identical ambiguity is shown to arise in the fermionic
analog of the string-net construction of 2D topological orders. We argue that
the need for a spin structure is a general feature of lattice models with local
fermionic degrees of freedom and is a lattice analog of the spin-statistics
relation.
