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# Homology, Homotopy and Applications

## Volume 23 (2021)

### Number 1

### Biased permutative equivariant categories

Pages: 77 – 100

DOI: https://dx.doi.org/10.4310/HHA.2021.v23.n1.a6

#### Authors

#### Abstract

For a finite group $G$, we introduce the complete suboperad $\mathcal{Q}_G$ of the categorical $G$-Barratt–Eccles operad $\mathcal{P}_G$. We prove that $\mathcal{P}_G$ is not finitely generated, but $\mathcal{Q}_G$ 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 $\mathcal{Q}_G$ and conclude with a discussion of algebras over $\mathcal{Q}_G$, which we call biased permutative equivariant categories.

#### Keywords

equivariant symmetric monoidal category, operad

#### 2010 Mathematics Subject Classification

18D10, 18D50, 55P48, 55P91

Copyright © 2020, Kayleigh Bangs, Skye Binegar, Young Kim, Kyle Ormsby, Angélica M. Osorno, David Tamas-Parris and Livia Xu. Permission to copy for private use granted.

This article was revised on June 29, 2022 to correct the names used for internal cross-references.

Received 19 August 2019

Received revised 23 January 2020

Accepted 18 February 2020

Published 19 August 2020