  We build two-dimensional quantum field theories on spin surfaces starting
from theories on oriented surfaces with networks of topological defect lines
and junctions. The construction uses a combinatorial description of the spin
structure in terms of a triangulation equipped with extra data. The amplitude
for the spin surfaces is defined to be the amplitude for the underlying
oriented surface together with a defect network dual to the triangulation.
Independence of the triangulation and of the other choices follows if the line
defect and junctions are obtained from a Delta-separable Frobenius algebra with
involutive Nakayama automorphism in the monoidal category of topological
defects. For rational conformal field theory we can give a more explicit
description of the defect category, and we work out two examples related to
free fermions in detail: the Ising model and the so(n) WZW model at level 1.
