Hopf formulae for cocommutative Hopf algebras

The adjunction between coalgebras and Hopf algebras, first described by Takeuchi, allows one to prove that the semi-abelian category of cocommutative Hopf algebras has enough E-projective objects with respect to the class E of cleft extensions. One then proves that, for any cocommutative Hopf algebra, there exists a weak E-universal normal (=central) extension. This fact allows one to apply the methods of categorical Galois theory to classify normal E-extensions and to provide an explicit description of the fundamental group of a cocommutative Hopf algebra in terms of a generalized Hopf formula. Moreover, with any cleft extension, we associate a 5-term exact sequence in homology that can be seen as a Hopf-theoretic analogue of the classical Stallings-Stammbach exact sequence in group theory.