Braided bi-Galois theory II: The cocommutative case

We study groups of bi-Galois objects over a Hopf algebra H in a braided monoidal category B. We assume H to be cocommutative in a certain sense; this does not mean that H is a cocommutative coalgebra with respect to the braiding given in B, but it is cocommutative with respect to a different braiding subject to specific axioms. The type of cocommutative Hopf algebras under consideration (investigated in previous papers) occurs naturally, for example in Majid's transmutation construction. We show that for cocommutative H the suitably defined cocommutative bi-Galois objects form a subgroup in the group of H-H-bi-Galois objects. We also show that all cocycles on H are lazy, and that second (lazy) cohomology describes the subgroup of cleft bi-Galois extensions in the group of cocommutative ones.