A fast convergence theorem for nearly multiplicative connections on proper Lie groupoids

Motivated by the study of global geometric properties of differentiable stacks presented by proper Lie groupoids, we investigate the existence of multiplicative connections on such groupoids. We show that one can always deform a given connection which is only approximately multiplicative into a genuinely multiplicative connection. The proof of this fact presented here relies on a recursive averaging technique. We regard our results as a preliminary step towards the construction of an obstruction theory for multiplicative connections on proper Lie groupoids.