  Courant algebroids correspond to degree-2 symplectic differential graded
manifolds or NQ-manifolds for short. We review how a similar construction shows
that locally the gauge structure of Double Field Theory corresponds to degree-2
symplectic pre-NQ manifolds. To illustrate first steps towards a global
understanding of the pre-NQ case, we apply the local constructions to
3-dimensional nilmanifolds carrying an abelian gerbe. These are among prime
examples where T-duality is well-understood and allow us to investigate classic
results in the graded language.
