Global $N_\infty$-operads

TL;DR

This paper introduces and classifies global $N_$-operads in the setting where objects have compatible actions by all compact Lie groups, establishing a correspondence with algebraic transfer systems and exploring their relation to single-group $N_$-operads.

Contribution

It defines global $N_$-operads, classifies them via transfer systems, and analyzes their relation to single-group $N_$-operads, revealing not all can be restrictions of global ones.

Findings

Global $N_$-operads are classified by global transfer systems.

Not all equivariant $N_$-operads are restrictions of global $N_$-operads.

The homotopy category of global $N_$-operads is equivalent to a poset of algebraic objects.

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.