On the homotopy type of higher orbifolds and Haefliger classifying spaces

We describe various equivalent ways of associating to an orbifold, or more generally a higher étale differentiable stack, a weak homotopy type. Some of these ways extend to arbitrary higher stacks on the site of smooth manifolds, and we show that for a differentiable stack XX arising from a Lie groupoid GG, the weak homotopy type of XX agrees with that of BGBG. Using this machinery, we are able to find new presentations for the weak homotopy type of certain classifying spaces. In particular, we give a new presentation for the Borel construction M×GEGM×GEG of an almost free action of a Lie group G on a smooth manifold M   as the classifying space of a category whose objects consist of smooth maps Rn→MRn→M which are transverse to all the G  -orbits, where n=dim⁡M−dim⁡Gn=dim⁡M−dim⁡G. We also prove a generalization of Segal's theorem, which presents the weak homotopy type of Haefliger's groupoid ΓqΓq as the classifying space of the monoid of self-embeddings of RqRq, B(Emb(Rq)), and our generalization gives analogous presentations for the weak homotopy type of the Lie groupoids Γ2qSp and RΓqRΓq which are related to the classification of foliations with transverse symplectic forms and transverse metrics respectively. We also give a short and simple proof of Segal's original theorem using our machinery.