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It is a classical fact in Poisson geometry that the cotangent bundle of aPoisson manifold has the structure of a Lie algebroid. Manifestations of thisstructure are the Lichnerowicz differential on multivector fields (calculatingPoisson cohomology) and the Koszul bracket of differential forms. "Raisingindices" by the Poisson tensor maps the de Rham differential to theLichnerowicz differential and the Koszul bracket to the Schouten bracket. Inthis paper, we present a homotopy analog of the above results. When an ordinaryPoisson structure is replaced by a homotopy one, instead of a single Koszulbracket there arises an infinite sequence of "higher Koszul brackets" definingan $L_{\infty}$-algebra structure on forms (Khudaverdian--VoronovarXiv:0808.3406). We show how to construct a non-linear transformation, whichis an $L_{\infty}$-morphism, from this $L_{\infty}$-algebra to the Liesuperalgebra of multivector fields with the canonical Schouten bracket. This isdone by using the new notion of "thick morphisms" of supermanifolds recentlyintroduced (see arXiv:1409.6475 and arXiv:1411.6720).

Thick morphisms, higher Koszul brackets, and $L_{\infty}$-algebroids