On The Homotopy Type of Higher Orbifolds and Haefliger Classifying Spaces

TL;DR

This paper develops methods to associate a homotopy type to higher orbifolds and stacks, providing new presentations for classifying spaces and generalizing Segal's theorem for foliation groupoids.

Contribution

It introduces new techniques to compute the weak homotopy type of higher stacks and orbifolds, extending existing results to broader classes of groupoids and stacks.

Findings

New presentation for the homotopy type of classifying spaces of orbifolds.

Generalization of Segal's theorem to symplectic and metric foliation groupoids.

Simplified proof of Segal's original theorem.

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.