On The Homotopy Type of Higher Orbifolds and Haefliger Classifying Spaces

Abstract

We describe various equivalent ways of associating to an orbifold, or more generally a higher \'etale 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 X arising from a Lie groupoid G, the weak homotopy type of X agrees with that of BG. 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 of an almost free action of a Lie group G on a smooth manifold M as the classifying space of a category whose objects consists of smooth maps R^n to M which are transverse to all the G-orbits, where n=dim M - dim G. We also prove a generalization of Segal's theorem, which presents the weak homotopy type of Haefliger's groupoid $\Gamma^q$ as the classifying space of the monoid of self-embeddings of R^q, and our generalization gives analogous presentations for the weak homotopy type of the Lie groupoids $\Gamma^{Sp}_{2q}$ and $R\Gamma^q$ 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.