# Resolutions of proper Riemannian Lie groupoids

## TL;DR

This paper demonstrates that proper Riemannian Lie groupoids can be desingularized into regular ones, preserving geometric properties and invariance under Morita equivalence, with implications for the study of differentiable stacks.

## Contribution

It introduces a method to desingularize proper Riemannian Lie groupoids into regular ones, maintaining geometric and Morita invariance properties.

## Key findings

- Every proper Lie groupoid admits a desingularization to a regular proper Lie groupoid.
- The desingularization can be made arbitrarily close in Gromov-Hausdorff distance when equipped with a Riemannian metric.
- The construction is invariant under Morita equivalence, ensuring it applies to the underlying differentiable stack.

## Abstract

In this paper we prove that every proper Lie groupoid admits a desingularization to a regular proper Lie groupoid. When equipped with a Riemannian metric, we show that it admits a desingularization to a regular Riemannian proper Lie groupoid, arbitrarily close to the original one in the Gromov-Hausdorff distance between the quotient spaces. We construct the desingularization via a successive blow-up construction on a proper Lie groupoid. We also prove that our construction of the desingularization is invariant under Morita equivalence of groupoids, showing that it is a desingularization of the underlying differentiable stack.