Encoding equivariant commutativity via operads

Abstract

We prove a conjecture of Blumberg and Hill regarding the existence of $N_{\infty }$–operads associated to given sequences $\mathcal{ℱ}={\left({\mathcal{ℱ}}_n\right)}_{n\in \mathbb{N}}$ of families of subgroups of $G\times {\mathrm{\Sigma }}_n$. 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.