A concrete instance of the last example goes as follows. Let M be a smooth manifold and { U α α } an open cover of M . Its Čech groupoid G 1 ⇉ ⇉ G 0 is defined by the disjoint unions G 0 := ⨆ ⨆ α α U α α and G 1 := ⨆ ⨆ α α , β β U α α β β , where U α α β β = U α α ∩ ∩ U β β ⊂ ⊂ M . The source and target map are defined as the embeddings s : U α α β β → → U α α and t : U α α β β → → U β β , and the multiplication is the obvious one if we read the U α α β β as subsets of M (compatible points in U α α β β and U β β γ γ actually are the same in M and also lie in U α α γ γ ). The Čech groupoid is in fact the pullback groupoid, under the obvious submersion p : G 0 → → M , of the unit groupoid M ⇉ ⇉ M . As such, Čech groupoids associated to different open covers of M are Morita equivalent.