Groupoids in categories with partial covers

Meyer and Zhu survey the general theory of groupoids, groupoid actions, groupoid principal bundles, and various kinds of morphisms between groupoids in the framework of categories with pretopology. We modify the categorical framework there to allow for "partial" notion. A category with partial covers is equipped with a notion of "partial cover", which allows to define partial sheaves, partial bibundle actors, partial Hilsum-Skandalis morphisms, partial Morita equivalence, partial groupoid fibration, partial groupoid covering, and so on. The categories of topological spaces, finite or infinite dimensional manifolds are examples of such categories. We study extra assumptions on stronger pretopologies that are needed for this theory. We check these extra assumptions in several categories with partial covers. We will see that the generalized groupoid action of H an G may be transformed along a Morita equivalence G∼K to an action of H on K. We proved that in a groupoid fibration G→L→H, if the groupoids G and H are basic then so is L.