# What natural monoidal structure and braiding exists on the category of modules of the convolution algebra of an action groupoid?

Let $S$ be a set with an action $\triangleright$ of a finite group $G$. The action groupoid $S // G$ has as objects the set $S$, and the morphisms from $s_1$ to $s_2$ are just the $g \in G$ that satisfy $g \triangleright s_1 = s_2$.

It is easy to see that representation category of finite connected groupoids is equivalent to that of the stabiliser group of any object (see this question). But the representation theory of the convolution algebra seems to have more interesting extra structure, apparently. Let $\mathbb{C}[S//G]$ be the algebra spanned by the morphisms from $S//G$, and $g_1 \cdot g_2$ is the composite of the morphisms if they're composable, and 0 otherwise.

Now, already when taking the groupoid with two objects $A, B$ and two nontrivial morphisms $f: A \to B$ and $f^{-1}$, something interesting is happening: The groupoid is itself equivalent to the trivial category, but its convolution algebra is 2$\times$2 complex matrices, $M_\mathbb{C}(2)$. The representation category of $M_\mathbb{C}(2)$ is equivalent to $\mathrm{Vect}$, but not monoidally equivalent: The simple object tensor squares to two copies of itself. The reason behind this seems to be the coproduct on the algebra: It is not $\Delta f = f \otimes f$ like in the group case, but $\Delta f = 1_A \otimes f + f \otimes 1_B$. I'm assuming that in general for $f : A \to B$, a formula like this must hold: $$\Delta f = \sum_{\substack{C \in \operatorname{ob} S//G \\ g: A \to C \\ h: C \to B \\ hg = f}} g \otimes h$$ So in general, we should have a Hopf algebra, with an antipode coming from the inverse. The coproduct can even be noncommutative.

One example is particularly striking: Let $G //_{\text{ad}} G$ be the action groupoid of $G$ acting on itself with the adjoint action. Then the convolution algebra is the Drinfel'd double $D(G)$! Its representations are the Drinfel'd centre of $\operatorname{Rep}G$, which has the modular braiding.

Has the braided monoidal structure on the representations of a groupoid been studied? Is there a natural $R$-matrix on the convolution algebra? When is the braiding modular?

Edit: It occurs to me that in many cases (like $G//_\text{ad} H$ for a subgroup $H$), the algebra is not Hopf, but only weak Hopf, i.e. the unit $\eta = \sum_{s \in \operatorname{ob} S // G} 1_s$ is not always preserved by the coproduct. Does that mean that the category of modules is multifusion?