Serre fibrations in the Morita category of topological groupoids

TL;DR

This paper extends the concept of Serre fibrations to the Morita category of topological groupoids, enabling the computation of homotopy groups for complex structures like foliation groupoids.

Contribution

It introduces a generalized notion of Serre fibration in the Morita category and derives a long exact sequence of homotopy groups for these structures.

Findings

Derived long exact sequence of homotopy groups for topological groupoids.

Calculated homotopy groups of foliation groupoids.

Extended classical fibrations to a broader categorical context.

Abstract

In this paper, we generalize the notion of Serre fibration to the Morita category of topological groupoids and derive the associated long exact sequence of homotopy groups. We use this results for calculation of homotopy groups of various groupoids, such as the foliation groupoid of a Riemannian foliation.