Projections of Hopf braces
José Manuel Fernández Vilaboa, Ramón González Rodríguez, Brais Ramos Pérez, Ana Belén Rodríguez Raposo
TL;DR
The paper develops a categorical framework for projections of Hopf braces in a strict braided monoidal setting, introducing a dedicated Yetter-Drinfeld theory for Hopf braces and the bosonization process. It defines a hierarchy of projection notions (strong, $\mathrm{v}_1$–$\mathrm{v}_4$) and proves that bosonization yields new Hopf braces; crucially, there is an equivalence between the category of bosonizable Hopf braces in $^{\mathbb{H}}_{\mathbb{H}}\mathrm{YD}$ and the category of $\mathrm{v}_4$-strong projections over a fixed cocommutative $\mathbb{H}$. This yields a structural bridge between brace projections and bosonized braces, extending Radford–Majid type correspondences to the brace setting. The results provide a robust toolkit to construct and classify Hopf braces from projections, with implications for Yang–Baxter solutions in braided categories. Overall, the work broadens the scope of Hopf brace theory by situating it firmly within a monoidal-categorical projection framework and establishing categorical equivalences that generalize classical Radford–Majid correspondences.
Abstract
This paper is devoted to the study of Hopf braces projections in a monoidal setting. Given a cocommutative Hopf brace ${\mathbb H}$ in a strict symmetric monoidal category ${\sf C}$, we define the braided monoidal category of left Yetter-Drinfeld modules over ${\mathbb H}$. For a Hopf brace ${\mathbb A}$ in this category, we introduce the concept of bosonizable Hopf brace and we prove that its bosonization ${\mathbb A}\blacktriangleright\hspace{-0.15cm}\blacktriangleleft {\mathbb H}$ is a new Hopf brace in ${\sf C}$ that gives rise to a projection of Hopf braces satisfying certain properties. Finally, taking these properties into account, we introduce the notions of v$_{i}$-strong projection over ${\mathbb H}$, $i=1,2,3,4$, and we prove that there is a categorical equivalence between the categories of bosonizable Hopf braces in the category of left Yetter-Drinfeld modules over ${\mathbb H}$ and the category of v$_{4}$-strong projections over ${\mathbb H}$.