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 $G\times {\Sigma }_n$. 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.

arXiv commentThis version has been accepted to Algebraic & Geometric Topology

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Cited by in corpus (27)

• Genuine equivariant operads
• Combinatorial $N_{\infty }$ operads
• Infinity Operads and Monoidal Categories with Group Equivariance
• Left Bousfield localization without left properness
• Multiplicativity of the idempotent splittings of the Burnside ring and the G-sphere spectrum
• Equivariant stable categories for incomplete systems of transfers
• Equivariant dendroidal Segal spaces and $G$-$\infty$-operads
• An algebraic model for rational naive-commutative equivariant ring spectra
• The genuine operadic nerve
• 1-smooth pro-p groups and Bloch-Kato pro-p groups
• Equivariant dendroidal sets and simplicial operads
• Categorifying the algebra of indexing systems
• Real topological Hochschild homology of schemes
• Characterizations of equivariant Steiner and linear isometries operads
• Idempotent characters and equivariantly multiplicative splittings of K-theory
• Lifting $N_{\infty }$ operads from conjugacy data
• Global transfer systems of abelian compact Lie groups
• Global $N_{\infty }$-operads
• Homotopy theory of equivariant operads with fixed colors
• Equivariant homotopy commutativity for $G=C_{pqr}$
• Goerss--Hopkins obstruction theory for $\infty$-categories
• On the homotopy theory of equivariant colored operads
• ${\mathbb{E}}_{\infty }$ automorphisms of motivic Morava $E$-theories
• Symmetric monoidal G-categories and their strictification
• Realization of saturated transfer systems on cyclic groups of order $p^n{q}^m$ by linear isometries $N_{\infty }$-operads
• Transfer systems for rank two elementary Abelian groups: characteristic functions and matchstick games
• Genuine-commutative ring structure on rational equivariant $K$-theory for finite abelian groups