STRONG HOPF MODULES FOR WEAK HOPF QUASIGROUPS

This paper is a further step in the study of the theory of modules associated to a weak Hopf quasigroup H. We introduce the category of strong H-Hopf modules, and we prove that there exists an adjoint equivalence between this category and the category of right modules over the image of the target morphism of H. In the Hopf quasigroup setting every Hopf module is strong, and we recover the results of Brzezinski. Also, in the weak Hopf case, every Hopf module is strong, and we generalize the theorem proved by Bohm, Nill and Szlachanyi that contains as a particular instance the categorical equivalence associated to the category of Hopf modules for a Hopf algebra H