Crossed Burnside rings for groupoids

In this paper, we extend the classical theory of crossed $G$-sets and the crossed Burnside ring from a finite group $G$ to a finite groupoid $\mathcal{G}$. We introduce a natural monoidal structure on the category of crossed $\mathcal{G}$-sets over a $\mathcal{G}$-monoid and construct the corresponding crossed Burnside ring of a $\mathcal{G}$-monoid. Finally, we prove a decomposition theorem that expresses the crossed Burnside ring of a groupoid as a product of crossed Burnside rings of groups.