The leaf space of a multiplicative foliation

TL;DR

This paper demonstrates that under certain conditions, the leaf space of a multiplicative foliation on a groupoid inherits a groupoid structure, with applications to Dirac and Poisson groupoids.

Contribution

It establishes conditions under which the leaf space of a multiplicative foliation forms a groupoid and applies this to relate Dirac and Poisson groupoids.

Findings

Leaf space inherits a groupoid structure under involutivity and completeness.

Special Dirac groupoids project to Poisson groupoids via forward Dirac maps.

Provides a new geometric framework for understanding Dirac and Poisson structures.

Abstract

We show that if a smooth multiplicative subbundle $S\subseteq TG$ on a groupoid $G\rr P$ is involutive and satisfies completeness conditions, then its leaf space $G/S$ inherits a groupoid structure over the space of leaves of $TP\cap S$ in $P$. As an application, a special class of Dirac groupoids is shown to project by forward Dirac maps to Poisson groupoids.