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 \(\mathcal {E}\) -projective objects, where \(\mathcal {E}\) is the class of cleft extensions. One then proves that, for any cocommutative Hopf algebra, there exists a weak \(\mathcal {E}\) -universal normal (=central) extension. This fact allows one to apply the methods of categorical Galois theory to classify normal \(\mathcal {E}\) -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 homology sequence that can be seen as a Hopf-theoretic analogue of the classical Stallings-Stammbach exact sequence in group theory.
