Encoding Equivariant Commutativity via Operads

In this paper, we prove a conjecture of Blumberg and Hill regarding the existence of $N_{\mathrm{\infty }}$-operads associated to given sequences $$\mathcal{F}$ = ($\mathcal{F}$n){n \in $\mathbb{N}$}$offamiliesofsubgroupsof$G\times \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 $E_{\mathrm{\infty }}^{\mathcal{F}}$-operads, generalizing the notion of an $N_{\mathrm{\infty }}$-operad, and to prove the Blumberg-Hill conjecture. We then explore questions of admissibility, rectification, and preservation under left Bousfield localization for these $E_{\mathrm{\infty }}^{\mathcal{F}}$-operads, obtaining some new results as well for $N_{\mathrm{\infty }}$-operads.