Hopf bracoids
José Manuel Fernández Vilaboa, Ramón González Rodríguez, Brais Ramos Pérez
TL;DR
Hopf bracoids are introduced as the quantum analogue of skew bracoids within a braided monoidal category, generalizing Hopf braces to the Hopf bracoid framework. The paper develops the category ${\sf HBrcd}$ of Hopf bracoids, establishes its monoidal (and, in symmetric settings, symmetric) structure, and connects Hopf bracoids to Hopf braces via natural functors, including a trivial functor and adjoint relationships with skew bracoids. A central achievement is the construction of a 1-cocycle formalism: functors $F:{\sf 1C}\to {\sf HBrcd}$ and $G:{\sf HBrcd}\to {\sf 1C}$ yield a categorical isomorphism between appropriate subcategories, yielding equivalences that generalize the brace–1-cocycle correspondence to the braided setting. The results unify and extend known correspondences between braces and cocycles, and link Hopf brace theory with generalized skew bracoid structures, providing a framework with potential applications to quantum symmetries and solutions of the Yang–Baxter equation in braided contexts.
Abstract
The present article is devoted to introduce the notion of Hopf bracoid in the braided monoidal framework as the quantum version of skew bracoids, which have been presented by Martin-Lyons and Paul J. Truman. Taking into account that Hopf braces are particular examples of Hopf bracoids, in this paper we generalize many properties of Hopf braces to the Hopf bracoid setting and we obtain a categorical isomorphism between certain full subcategories of the Hopf bracoids category and the 1-cocycles category.