Projections of Hopf braces

TL;DR

This paper explores the structure of Hopf braces within a monoidal category, introducing bosonization techniques and establishing categorical equivalences related to projections of Hopf braces.

Contribution

It introduces the concept of bosonizable Hopf braces and proves a categorical equivalence with certain strong projection categories in a monoidal setting.

Findings

Defined braided monoidal category of Yetter-Drinfeld modules.

Introduced bosonizable Hopf braces and their bosonization.

Established categorical equivalence between bosonizable Hopf braces and v4-strong projections.

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 ${\mathbb H}$.