Global $N_\infty$-operads

Abstract: We define $N_\infty$-operads in the globally equivariant setting and completely classify them. These global $N_\infty$-operads model intermediate levels of equivariant commutativity in the global world, i. e. in the setting where objects have compatible actions by all compact Lie groups. We classify global $N_\infty$-operads by giving an equivalence between the homotopy category of global $N_\infty$-operads and the partially ordered set of global transfer systems, which are much simpler, algebraic objects. We also explore the relation between global $N_\infty$-operads and $N_\infty$-operads for a single group, recently introduced by Blumberg and Hill. One interesting consequence of our results is the fact that not all equivariant $N_\infty$-operads can appear as restrictions of global $N_\infty$-operads.