Did anybody here ever get a good handle on what 2-equivariance means? There is a kind of appealing sketch in Lurie's Elliptic Cohomology paper but I don't see how to state what a 2-equivariant cohomology theory is abstractly, or any desiderata restricting what it assigns to K(Z,3) etc.
For ordinary equivariant cohomology theories the main thing that one might not guess is the existence of Thom isomorphisms for equivariant vector bundles. Is there an analogous geometric operation that becomes available in the 2-equivariant world? I think it can't be as simple as inverting BS^1-representatio…