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arXiv · 2610.15347

A categorical approach to the set-theoretic Yang-Baxter equation

A. L. Agore·A. Chirvasitu·G. Militaru
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Abstract

We investigate set-theoretic solutions of the Yang--Baxter equation from a categorical perspective. The paper has two main objectives. On the one hand, several existing constructions and classification results in the literature are recast in categorical terms, thereby showing that certain hypotheses previously imposed on such solutions are in fact superfluous. On the other hand, the categorical approach leads naturally to new classes of Yang-Baxter solutions. Our approach consists in identifying suitable subcategories of the category $\mathcal{YB}$ of Yang-Baxter solutions and providing categorical characterisations of the solutions they encode. As an application, we prove that the category of Płonka bi-magmas is isomorphic to the subcategory of $\mathcal{YB}$ consisting of solutions of the Baaj-Long-Skandalis equation. Along the way, we introduce and study a new class of Lyubashenko type solutions and examine the extent to which non-degenerate bijective solutions of the braid equation arise from skew left braces.

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A. L. Agore, A. Chirvasitu, G. Militaru. 2026-10-11. A categorical approach to the set-theoretic Yang-Baxter equation. https://arxiv.org/abs/2610.15347

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