Projections of Hopf braces

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 $\mathsf{C}$, 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 $\mathbb{A}▶◀\mathbb{H}$ is a new Hopf brace in $\mathsf{C}$ 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 $\mathbb{H}$.