Multiplicative connections and their Lie theory

We define and study multiplicative connections in the tangent bundle of a Lie groupoid. Multiplicative are linear satisfying an appropriate compatibility with groupoid structure. Our definition is natural sense that connection on if only its torsion tensor Bursztyn–Drummond [Lie theory tensors, Mat. Ann. 375 (2019) 1489–1554, arXiv:1705.08579] geodesic spray vector field. identify obstruction to existence connection. also discuss infinitesimal version bundle, we call infinitesimally (IM) prove integration theorem for IM connections. Finally, present few toy examples.