Encoding Equivariant Commutativity via Operads

TL;DR

This paper proves a conjecture about the existence of specialized operads called $N_ abla$-operads, constructs model structures for them, and explores their properties, advancing the understanding of equivariant commutative structures.

Contribution

It constructs a model structure on $G$-operads for sequences of subgroup families and defines $E_ abla^{ ext{families}}$-operads, confirming the Blumberg-Hill conjecture.

Findings

Constructed model structures for $G$-operads.

Defined generalized $E_ abla^{ ext{families}}$-operads.

Proved the Blumberg-Hill conjecture.

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.