On quiver skew braces, their ideals and products

Abstract

Quiver skew braces or skew bracoids are equivalent to braided groupoids, that is, groupoids with a constraint of abelianity. They are the quiver-theoretic version of skew braces, an increasingly studied structure lying in the intersection of group and ring theory. In this paper, we define ideals and quotients for quiver skew braces, with respect to two notions of morphisms. Following the track of a previous work of ours (2025), we define a classical semidirect product \`a la Brown, and a categorical semidirect product \`a la Bourn and Janelidze, for the category of quiver skew braces. It is known that connected groupoids can be expressed as the datum of a group of loops and a set of vertices. We demonstrate how no such decomposition holds for quiver skew braces, which makes their theory richer than the theory of groupoids.