Encoding Equivariant Commutativity via Operads

Abstract

In this paper, we prove a conjecture of Blumberg and Hill regarding the existence of $N_{\infty }$-operads associated to given sequences $\mathcal{F}=({\mathcal{F}}_n{)}_{n\in \mathbb{N}}$ of families of subgroups of $G\times \hat{I}{\pounds }_n$. For every such sequence, we construct a model structure on the category of $G$-operads, and we use these model structures to define $E_{\infty }^{\mathcal{F}}$-operads, generalizing the notion of an $N_{\infty }$-operad, and to prove the Blumberg-Hill conjecture. We then explore questions of admissibility, rectification, and preservation under left Bousfield localization for these $E_{\infty }^{\mathcal{F}}$-operads, obtaining some new results as well for $N_{\infty }$-operads.

This version has been accepted to Algebraic & Geometric Topology