Multiplicative Ehresmann connections

We develop the theory of multiplicative Ehresmann connections for Lie groupoid submersions covering identity, as well their infinitesimal counterparts. construct obstructions to existence such connections, and we prove several interesting classes groupoids algebroids, including all proper groupoids. show that many notions from principal bundle have analogues in this general setup, 1-forms, curvature 2-forms, Bianchi etc. In [19] provide a non-trivial application results obtained here local models Poisson geometry obtain linearization around submanifolds.