Anomalous Actions of Groups on Tensor Categories

TL;DR

This paper introduces a method to construct anomalous actions of certain 3-groups derived from a group and a 4-cocycle on tensor categories, expanding the understanding of group actions in higher category theory.

Contribution

It provides a novel construction method for anomalous actions of 3-groups on tensor categories based on a given group and 4-cocycle.

Findings

Constructed anomalous actions using 3-group frameworks

Extended the theory of group actions to higher categorical structures

Provided explicit methods for implementation in tensor categories

Abstract

For a group $G$ and a 4-cocycle $\pi\in Z^{4}(G,\mathbb{k}^{\times})$, a $\pi$-anomalous action of $G$ on a linear monoidal category $C$ is a linear monoidal 2-functor between 3-groups $3\text{-Gr}(G,\pi)\rightarrow \text{2-Aut}_\otimes(C)$ where the latter denotes the 3-group of autoequivalences of $C$. Given $G$ and $\pi$, we provide a method of constructing anomalous actions of $3\text{-Gr}(G,\pi)$ on a tensor categories.