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

Kayleigh Bangs (Department of Mathematics, Reed College, Portland, Oregon, U.S.A.)

Skye Binegar (School of Mathematics, Georgia Institute of Technology, Atlanta, Ga., U.S.A.)

Young Kim (Department of Mathematics, University of California at Santa Cruz)

Kyle Ormsby (Department of Mathematics, Reed College, Portland, Oregon, U.S.A.)

Angélica M. Osorno (Department of Mathematics, Reed College, Portland, Oregon, U.S.A.)

David Tamas-Parris (Department of Mathematics, Reed College, Portland, Oregon, U.S.A.)

Livia Xu (Department of Mathematics, University of Chicago, Illinois, U.S.A.)

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

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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.

Received 19 August 2019

Received revised 23 January 2020

Accepted 18 February 2020

Published 19 August 2020