  It is a classical fact in Poisson geometry that the cotangent bundle of a
Poisson manifold has the structure of a Lie algebroid. Manifestations of this
structure are the Lichnerowicz differential on multivector fields (calculating
Poisson cohomology) and the Koszul bracket of differential forms. "Raising
indices" by the Poisson tensor maps the de Rham differential to the
Lichnerowicz differential and the Koszul bracket to the Schouten bracket. In
this paper, we present a homotopy analog of the above results. When an ordinary
Poisson structure is replaced by a homotopy one, instead of a single Koszul
bracket there arises an infinite sequence of "higher Koszul brackets" defining
an $L_{\infty}$-algebra structure on forms (Khudaverdian--Voronov
arXiv:0808.3406). We show how to construct a non-linear transformation, which
is an $L_{\infty}$-morphism, from this $L_{\infty}$-algebra to the Lie
superalgebra of multivector fields with the canonical Schouten bracket. This is
done by using the new notion of "thick morphisms" of supermanifolds recently
introduced (see arXiv:1409.6475 and arXiv:1411.6720).
