Foliated groupoids and their infinitesimal data

In this work, we study Lie groupoids equipped with multiplicative foliations and the corresponding infinitesimal data. We determine the infinitesimal counterpart of a multiplicative foliation in terms of its core and sides together with a partial connection satisfying special properties, giving rise to the concept of IM-foliation on a Lie algebroid. The main result of this paper shows that if $G$ is a source simply connected Lie groupoid with Lie algebroid $A$, then there exists a one-to-one correspondence between multiplicative foliations on $G$ and IM-foliations on the Lie algebroid $A$.