The Leaf Space Of A Multiplicative Foliation

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