Multiplicative Ehresmann connections

We develop the theory of multiplicative Ehresmann connec-tions for Lie groupoid submersions covering the identity, as well as their infinitesimal counterparts. We construct obstruc-tions to the existence of such connections, and we prove exis-tence for several interesting classes of Lie groupoids and Lie algebroids, including all proper Lie groupoids. We show that many notions from the theory of principal bundle connections have analogues in this general setup, including connections 1-forms, curvature 2-forms, Bianchi identity, etc. In [19] we provide a non-trivial application of the results obtained here to construct local models in Poisson geometry and to obtain linearization results around Poisson submanifolds.& COPY; 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons .org /licenses /by /4 .0/).