Morita Invariance of Saturated and Split Subgroupoids for Orbifold Embeddings

Abstract

Borzellino and Brunsden introduced a general notion of suborbifold and proved that a suborbifold is the image of a complete orbifold embedding if and only if it is both saturated and split. Their definitions and proofs are formulated in the classical language of orbifold charts. In this paper we translate these notions into the language of proper étale Lie groupoids.We introduce the groupoid analogues of saturation and splitting, which we call \emph{orbit saturation} and \emph{stabilizer splitting}, and prove that they coincide with the chart-based conditions. Our main theorem states that a subgroupoid is the image of a complete orbifold embedding if and only if it is both orbit saturated and stabilizer split. We then prove that these notions are Morita invariant: if two orbifold groupoids are Morita equivalent, then their orbit saturated and stabilizer split subgroupoids correspond naturally to each other. This solves an open problem raised by Mestre and Weilandt and explains why the embedding definition of Cho--Hong--Shin fails to be Morita invariant. Along the way we provide detailed examples, including the diagonal embedding, the non-split example of Borzellino and Brunsden, and explicit Morita equivalences.