Homotopy equivalence of shifted cotangent bundles

TL;DR

This paper proves that homotopy equivalent bundles of chain complexes have homotopy equivalent shifted cotangent bundles' Poisson algebras, with applications to $L_$-algebroids and shifted Poisson structures.

Contribution

It establishes a homotopy invariance result for shifted cotangent bundles and their Poisson algebras, extending to $L_$-algebroids and shifted Poisson geometry.

Findings

Homotopy equivalent bundles have homotopy equivalent shifted cotangent bundle Poisson algebras.

Homotopy equivalence preserves $L_$-algebroid structures.

Implications for shifted Poisson structures in derived geometry.

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.