  We consider principal bundles as generalized morphisms between topological
groupoids. In the category of these generalized morphisms two topological
groupoids are isomorphic if and only if they are Morita equivalent. We show
that the fibers of a generalized morphism from H to G induce a singular
foliation of the topological groupoid H, and we prove a Reeb-Thurston stability
theorem for such foliations. Next, we use generalized morphisms to study some
Morita invariants of topological groupoids, in particular the homotopy groups
of a topological groupoid and the Connes convolution algebra of an etale
groupoid.
