Homotopy equivalence of shifted cotangent bundles

Abstract

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.

Several improvements and corrected a mistake in the proof of the global version of the main theorem