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 Brzeziński. Also, in the weak Hopf case, every Hopf module is strong, and we generalize the theorem proved by Böhm, Nill and Szlachányi that contains as a particular instance the categorical equivalence associated to the category of Hopf modules for a Hopf algebra $H$.