## 1. Algebra from action groupoids

Let $G$ be a finite group acting on a finite set $X$ from the right (denoted in element as $x^{g}$). We have an algebra (of the action groupoid) over $\mathbb{C}$: the vector space

$$ H = <\delta_{x} | x \in X> \otimes <g | g \in G>$$

with the algebra structure given as $g \cdot g' = gg'$, $\delta_x \cdot \delta_{x'} = \chi_{x = x'} \delta_x$, $g \cdot \delta_{x} = \delta_{x^{(g^{-1})}} \cdot g$, and $1 = \sum_{x \in X} \delta_{x} \otimes e$, where $e$ denotes the group identity element of $G$.

## 2. Bi-algebra

To upgrade this algebra to a bi-algebra, it would suffice to require $X$ to be a finite group $H$ such that the action satisfies Leibniz rule:

$$ (h h')^{g} = (h^{g}) (h'^{g}).$$

Indeed, one can define the co-algebra structure by $\Delta(g) = g \otimes g$ and $\Delta(\delta_{h}) = \sum_{h' \in H} \delta_{h'} \otimes \delta_{h'^{-1}h}$. Note that this turns the representation category into a (fusion) tensor category.

## 3. Braided Bi-algebra

I'm interested in all ways to extend this to a braided (fusion) category. In general, this is a hard question (e.g. it was a big deal of Drinfeld's construction of interesting braided bi-algebras, leading to quantum groups.) However, there is a known extra condition that makes it possible: one requires that there exists a group homomorphism $\partial: H \to G$ such that

- $\partial (h^{g}) = g^{-1} (\partial h) g$ and
- $h^{\partial h'} = h'^{-1} h h'$

Indeed, we now have a braided structure, namely for any representation $V_{1}$ and $V_{2}$, define [1, page 4]

$$R: V_{1} \otimes V_{2} \to V_{2} \otimes V_{1}$$ $$v_{1} \otimes v_{2} \mapsto \sum_{h \in H} (\partial h)v_{2} \otimes \delta_{h} v_{1}$$

This is a miracle to me given that finding braided $R$-matrices
is a hard task in general! **What's even more mysterious to me is
that the condition turns the pair $(G, H)$ into a crossed-module
(or a strict $2$-group [2])!** I wonder if there's a deeper reason
why a braided structure shows up. However, having been thinking
of this question for months, I have no clue.

### Question

How to interpret the existence of the braided structure from the fact that the action ($G$ on $H$) and the boundary morphism $\partial$ forms a $2$-group?

In general, is there any hope to fit this into some $2$-representation theory of $2$-groups?

### Reference

- [1] Characters of Crossed Modules and Premodular Categories. P. Bantay, arXiv:math/0512542.
- [2] Higher-Dimensional Algebra V: 2-Groups