Symmetries of a rigid braided category

TL;DR

This paper explores natural symmetries in rigid higher braided categories, constructing functorial actions of continuous groups that reveal deep structural invariances and relate to known symmetries in specific cases.

Contribution

It introduces a functorial action of the group al 4444 on rigid higher braided categories and relates these symmetries to known categorical identities.

Findings

Constructed a functorial al 4444 action on _{n-1}-monoidal categories.

Established a canonical al 4444 action on the moduli space of objects.

Compared continuous symmetries to known symmetries in small parameter cases.

Abstract

We identify natural symmetries of each rigid higher braided category. Specifically, we construct a functorial action by the continuous group $\Omega \mathsf{O}(n)$ on each $\mathcal{E}_{n-1}$-monoidal $(g,d)$-category $\mathcal{R}$ in which each object is dualizable (for $n\geq 2$, $d \geq 0$, $d \leq g \leq \infty$). This action determines a canonical action by the continuous group $\Omega \mathbb{R}\mathbb{P}^{n-1}$ on the moduli space of objects of each such $\mathcal{R}$. In cases where the parameters $n$, $d$, and $g$ are small, we compare these continuous symmetries to known symmetries, which manifest as categorical identities.