  Given a bundle of chain complexes, the algebra of functions on its shifted
cotangent bundle has a natural structure of a shifted Poisson algebra. We show
that if two such bundles are homotopy equivalent, the corresponding Poisson
algebras are homotopy equivalent. We apply this result to $L_\infty$-algebroids
to show that two homotopy equivalent bundles have the same $L_\infty$-algebroid
structures and explore some consequence on the theory of shifted Poisson
structures.
