Generalizing the Weil–Cartan equivalence to Lie groupoids and Lie n-groupoids

Determine whether the equivalence between the Weil and Cartan models of equivariant cohomology for a compact connected Lie group—implemented by the automorphism Φ = exp(ξ^α \widehat{\iota}_α) on T[1]g[1] × T[1]M that identifies the basic complex with (S(g^*) ⊗ Ω^•(M))^G and its Cartan differential—extends to the setting of Lie groupoids and Lie n-groupoids; specifically, develop appropriate Weil and Cartan models for equivariant cohomology in these higher contexts and construct an analogue of Φ that intertwines them.

Background

In the Lie group case, the paper reviews the Weil algebra W(g) and shows that an explicit automorphism Φ on T[1]g[1] × T[1]M transforms the Weil model to the Cartan model, yielding a concrete isomorphism of complexes computing equivariant cohomology.

The authors note that while this equivalence is classical for groups, a generalization to Lie groupoids or Lie n-groupoids—requiring suitable notions of Weil and Cartan models and a corresponding intertwining automorphism—has not been established.