Encoding equivariant commutativity via operads

Abstract

We prove a conjecture of Blumberg and Hill regarding the existence of N 1 -operads associated to given sequences F D .F n / n2N of families of subgroups of G n . For every such sequence, we construct a model structure on the category of Goperads, and we use these model structures to define E F 1 -operads, generalizing the notion of an N 1 -operad, and to prove the Blumberg-Hill conjecture. We then explore questions of admissibility, rectification, and preservation under left Bousfield localization for these E F 1 -operads, obtaining some new results as well for N 1operads.