Poisson bivectors and Poisson brackets on affine derived stacks

Let A be a commutative dg algebra concentrated in degrees (−∞,m], and let SpecA be the associated derived stack. We give two proofs of the existence of a canonical map from the moduli space of shifted Poisson structures (in the sense of [16]) on SpecA to the moduli space of homotopy (shifted) Poisson algebra structures on A. The first makes use of a more general description of the Poisson operad and of its cofibrant models, while the second is more computational and involves an explicit resolution of the Poisson operad.