We 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. The categories of topological spaces and finite or infinite dimensional manifolds are examples of such categories. We study extra assumptions on pretopologies that are needed for this theory. We check these extra assumptions in several categories with pretopologies.
Functors between groupoids may be localised at equivalences in two ways. One uses spans of functors, the other bibundles (commuting actions) of groupoids. We show that both approaches give equivalent bicategories. Another type of groupoid morphism, called an actor, is closely related to functors between the categories of groupoid actions. We also generalise actors using bibundles, and show that this gives another bicategory of groupoids.
Keywords: Grothendieck topology; cover; groupoid; groupoid action; groupoid sheaf; principal bundle; Hilsum--Skandalis morphism; anafunctor; bicategory; comorphism; infinite dimensional groupoid
2010 MSC: 51H25
Theory and Applications of Categories, Vol. 30, 2015, No. 55, pp 1906-1998.