  Let A be a DGA over a field and X a module over H_*(A). Fix an
$A_\infty$-structure on H_*(A) making it quasi-isomorphic to A. We construct an
equivalence of categories between A_{n+1}-module structures on X and length n
Postnikov systems in the derived category of A-modules based on the bar
resolution of X. This implies that quasi-isomorphism classes of A_n-structures
on X are in bijective correspondence with weak equivalence classes of
rigidifications of the first n terms of the bar resolution of X to a complex of
A-modules. The above equivalences of categories are compatible for different
values of n. This implies that two obstruction theories for realizing X as the
homology of an A-module coincide.
