On the homotopy type of higher orbifolds and Haefliger classifying spaces (Q266144)

This paper explains how to functorially associate to an orbifold, or more generally to a higher étale differentiable stack, a canonical weak homotopy type. The authors are able to express the weak homotopy type of an \(n\)-dimensional higher étale differentiable stack as the homotopy colimit of a diagram of spaces indexed by the monoid \(Emb({\mathbb R}^{n})\) of smooth embeddings of \({\mathbb R}^{n}\). The authors recover Segal's theorem expressing the weak homotopy type of the classifying space \(B\Gamma^{n}\) of Haefliger's groupoid \(\Gamma^{n}\) as the classifying space \(B(Emb({\mathbb R}^{n}))\). The flavor is highly categorical and the authors provide an appendix on \(\infty\)-catgories and another appendix on principal bundles for Lie groupoids. There are numerous illustrative examples that serve as motivation for the material.