Biased permutative equivariant categories

TL;DR

This paper introduces a new finitely generated G-operad, $Q_G$, that models biased permutative equivariant categories, extending the categorical G-Barratt-Eccles operad with novel algebraic structures.

Contribution

It constructs and analyzes the finitely generated operad $Q_G$, proving its properties and providing presentations for specific groups, advancing the understanding of equivariant operads.

Findings

$Q_G$ is finitely generated and a genuine $E_ abla$ G-operad.

$P_G$ is not finitely generated.

Presentations for $Q_G$ when G is cyclic of order 2 or 3.

Abstract

For a finite group G, we introduce the complete suboperad $Q_G$ of the categorical G-Barratt-Eccles operad $P_G$. We prove that $P_G$ is not finitely generated, but $Q_G$ is finitely generated and is a genuine $E_\infty$ G-operad (i.e., it is $N_\infty$ and includes all norms). For G cyclic of order 2 or 3, we determine presentations of the object operad of $Q_G$ and conclude with a discussion of algebras over $Q_G$, which we call biased permutative equivariant categories.