Crossed Burnside rings for groupoids

TL;DR

This paper generalizes the concept of crossed Burnside rings from finite groups to finite groupoids, introducing a new monoidal structure and a decomposition theorem for these rings.

Contribution

It extends classical theory to groupoids, constructs the crossed Burnside ring for groupoids, and proves a decomposition theorem relating it to group-based rings.

Findings

Defined a monoidal structure on crossed groupoid-sets.

Constructed the crossed Burnside ring for a groupoid.

Proved a decomposition theorem for the crossed Burnside ring.

Abstract

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.