Multiplicative Ehresmann connections for Lie groupoid fibrations

Abstract

We introduce multiplicative Ehresmann connections on surjective submersions of Lie groupoids, extending both the classical notion of Ehresmann connections on fibre bundles and the more recent notion of multiplicative connections on Lie groupoid extensions. We investigate the existence of such connections, showing that, in general, they may fail to exist even for proper Lie groupoids. In contrast, positive results hold for Morita submersions, uniform Lie groupoid fibrations, locally trivial families of Lie groupoids, and proper families of Lie groupoids. Our main results concern completeness. For Lie groupoid fibrations, we prove that the completeness of a multiplicative connection is governed by the induced connection on the kernel bundle and, under connectivity assumptions, by the base connection. For families of source-proper Lie groupoids, we prove the equivalence between local triviality and the existence of complete multiplicative Ehresmann connections.