Projections of Hopf braces

This paper is devoted to the study of Hopf braces projections in a monoidal setting. Given a cocommutative Hopf brace H in a strict symmetric monoidal category C, we define the braided monoidal category of left Yetter-Drinfeld modules over H. For a Hopf brace A in this category, we introduce the concept of bosonizable Hopf brace and we prove that its bosonization A-0.15cm H is a new Hopf brace in C that gives rise to a projection of Hopf braces satisfying certain properties. Finally, taking these properties into account, we introduce the notions of vᵢ-strong projection over 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 H and the category of v₄-strong projections over H.