Homotopy types of topological stacks

We introduce the notion of classifying space   of a topological stack XX: it is a topological space XX with a morphism φ:X→Xφ:X→X that is a universal weak equivalence. We show that every topological stack XX has a classifying space XX that is well defined up to weak homotopy equivalence. Under a certain paracompactness condition on XX, we show that XX is indeed well defined up to homotopy equivalence. These results are formulated in terms of functors from the category of topological stacks to the (weak) homotopy category of topological spaces. We prove similar results for (small) diagrams of topological stacks.