Volume 18, issue 5 (2018)

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Javier J Gutiérrez and David White

Algebraic & Geometric Topology 18 (2018) 2919–2962
Abstract

We prove a conjecture of Blumberg and Hill regarding the existence of ${N}_{\infty }$–operads associated to given sequences $\mathsc{ℱ}={\left({\mathsc{ℱ}}_{n}\right)}_{n\in ℕ}$ of families of subgroups of $G×{\Sigma }_{n}$. For every such sequence, we construct a model structure on the category of $G$–operads, and we use these model structures to define ${E}_{\infty }^{\mathsc{ℱ}}$–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 ${E}_{\infty }^{\mathsc{ℱ}}$–operads, obtaining some new results as well for ${N}_{\infty }$–operads.

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