We study equivariant vector bundles over configuration spaces with diagonals included, viewed as orbifold quotients M^n/\mathfrakS_n by symmetric groups. Working in the equivalent language of equivariant vector bundles, we construct an induced-equivariance functor and prove its adjunction with restriction. We then define Hadamard and Cauchy tensor products and show that they form a symmetric 2-monoidal structure. We construct the corresponding tensor and symmetric algebra bundles and prove that, for a local vector bundle V →M, the bundle \mathbfS^⊠(\mathbfS^⊗(V)) is the free commutative 2-algebra generated by V. Finally, we show that any skew-symmetric bundle map k : V ⊠V →\mathbfI_⊗ induces a compatible Poisson bracket on this 2-algebra bundle.
@unpublished{nguyen2026equivariant,title={Equivariant 2-Poisson Algebra Bundles over Configuration Spaces},author={Nguyên, Hai Châu},year={2026},note={Preprint},}