Higher equivariant excision

We develop a theory of Goodwillie calculus for functors between G-equivariant homotopy theories, where G is a finite group. We construct J-excisive approximations for any finite G-set J. These combine into a poset, the Goodwillie tree, that extends the classical Goodwillie tower. We prove convergence results for the tree of a functor on pointed G-spaces that commutes with fixed-points, and we reinterpret the Tom Dieck-splitting as an instance of a more general splitting phenomenon that occurs for the fixed-points of the equivariant derivatives of these functors. As our main example we describe the layers of the tree of the identity functor in terms of the equivariant Spanier-Whitehead duals of the partition complexes.