MONOIDAL FUNCTORS AND EXACT SEQUENCES OF GROUPS FOR HOPF QUASIGROUPS

In this paper we introduce the notion of strong Galois H-progenerator object for a finite cocommutative Hopf quasigroup H in a symmetric monoidal category C. We prove that the set of isomorphism classes of strong Galois H-progenerator objects is a subgroup of the group of strong Galois H-objects introduced in [3]. Moreover, we show that strong Galois H-progenerator objects are preserved by strong symmetric monoidal functors and, as a consequence, we obtain an exact sequence involving the associated Galois groups. Finally, to the previous functors, if H is finite, we find exact sequences of Picard groups related with invertible left H-(quasi)modules and an isomorphism Pic((H)Mod) congruent to Pic(C)circle plus G(H*) where Pic((H)Mod) is the Picard group of the category of left H-modules, Pic(C) the Picard group of C, and G(H*) the group of group-like morphisms of the dual of H.