The group of strong Galois objects associated to a cocommutative Hopf quasigroup

Let H be a cocommutative faithfully flat Hopf quasigroup in a strict symmetric monoidal category with equalizers. In this paper we introduce the notion of (strong) Galois H-object and we prove that the set of isomorphism classes of (strong) Galois H-objects is a (group) monoid which coincides, in the Hopf algebra setting, with the Galois group of H-Galois objects introduced by Chase and Sweedler

Smash (co)products and skew pairings

Let [tau] be an invertible skew pairing on (B,H), where B and H are Hopf algebras in a symmetric monoidal category C with (co)equalizers. Assume that H is quasitriangular. Then we obtain a new algebra structure such that B is a Hopf algebra in the braided category HHYD and there exists a Hopf algebra isomorphism w: B[infinity]H --> B [tau]H in C, where B[infinity]H is a Hopf algebra with (co)algebra structure the smash (co)product and B [tau] H is the Hopf algebra defined by Doi and Takeuchi