Monodromy and faithful representability of Lie groupoids

For any topological groupoid G and any homomorphism ρ from a locally compact Hausdorff topological group K to G, we construct an associated monodromy group Mon(G;ρ). We prove that Morita equivalent topological groupoids have the same monodromy groups. We show how the monodromy groups can be used to test if a Lie groupoid lacks faithful representations.