Central series of cocommutative Hopf braces

By extending some classical results known for groups and skew braces, we define and investigate central series of cocommutative Hopf braces. Both left and right central series are defined using a $\star$-product that measures the difference between the two algebra operations, and naturally leads to introducing the notions of socle and of annihilator of a cocommutative Hopf brace. We characterize the central extensions relative to the subcategories of cocommutative Hopf algebras and of commutative and cocommutative Hopf algebras, respectively. Since the category of cocommutative Hopf braces is semi-abelian and it has enough projectives with respect to the class of cleft extensions, one can then establish suitable Hopf formulae for their homology. These are expressed in terms of the corresponding notions of relative commutators of cocommutative Hopf braces. In particular, the one relative to the subcategory of commutative and cocommutative Hopf algebras turns out to be the Huq commutator.