On dynamical skew braces and skew bracoids

TL;DR

This paper explores the relationship between dynamical skew braces and braided groupoids, establishing a correspondence and analyzing their combinatorial properties based on integer invariants.

Contribution

It proves that solutions from dynamical skew braces are braided groupoids and shows that connected braided groupoids can be represented as dynamical skew braces, linking these structures.

Findings

Solutions from dynamical skew braces are braided groupoids

Connected braided groupoids can be parallelised to dynamical skew braces

The combinatorics depend on specific integer invariants

Abstract

Dynamical skew braces are known to produce solutions to the quiver-theoretic Yang--Baxter equation. Under a technical hypothesis, we prove that these solutions are braided groupoids (and hence skew bracoids in the sense of Sheng, Tang and Zhu). Conversely, every connected braided groupoid can be parallelised, making it isomorphic to a dynamical skew brace. We study the combinatorics of these objects, depending on some strings of integer invariants.