  For a positive integer r, an r-spin topological quantum field theory is a
2-dimensional TQFT with tangential structure given by the r-fold cover of SO_2
. In particular, such a TQFT assigns a scalar invariant to every closed r-spin
surface Sigma. Given a sequence of scalars indexed by the set of diffeomorphism
classes of all such Sigma, we construct a symmetric monoidal category C and a
C-valued r-spin TQFT which reproduces the given sequence. We also determine
when such a sequence arises from a TQFT valued in an abelian category with
finite-dimensional Hom spaces. In particular, we construct TQFTs with values in
super vector spaces that can distinguish all diffeomorphism classes of r-spin
surfaces, and we show that the associated algebras are necessarily
non-semisimple.
