  We generalize the AKSZ construction of topological field theories to allow
the target manifolds to have possibly-degenerate (homotopy) Poisson structures.
Classical AKSZ theories, which exist for all oriented spacetimes, are described
in terms of dioperads. The quantization problem is posed in terms of extending
from dioperads to properads. We conclude by relating the quantization problem
for AKSZ theories on R^d to the formality of the E_d operad, and conjecture a
properadic description of the space of E_d formality quasiisomorphisms.
