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Let $G$ be a compact and connected Lie group. The Hamiltonian $G$-modelfunctor maps the category of symplectic representations of closed subgroups of$G$ to the category of exact Hamiltonian $G$-actions. Based on previous jointwork with Y. Karshon, the restriction of this functor to the momentum propersubcategory on either side induces a bijection between the sets of isomorphismclasses. This classifies all momentum proper exact Hamiltonian $G$-actions (ofarbitrary complexity).  As an extreme case, we obtain a version of the Eliashberg cotangent bundleconjecture for transitive smooth actions. As another extreme case, the momentumproper Hamiltonian $G$-actions on contractible manifolds are exactly thesymplectic $G$-representations, up to isomorphism.

Classification of momentum proper exact Hamiltonian group actions and the equivariant Eliashberg cotangent bundle conjecture