Lipschitz-free spaces, ultraproducts, and finite representability of metric spaces

We study several properties and applications of the ultrapower MU of a metric space M. We prove that the Lipschitz-free space F(MU) is finitely representable in F(M). We also characterize the metric spaces that are finitely Lipschitz representable in a Banach space as those that biLipschitz embed into an ultrapower of the Banach space. Thanks to this link, we obtain that if M is finitely Lipschitz representable in a Banach space X, then F(M) is finitely representable in F(X). We apply these results to the study of cotype in Lipschitz-free spaces and the stability of Lipschitz-free spaces and spaces of Lipschitz functions under ultraproducts.