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

We study several properties and applications of the ultrapower $M_{\mathcal{U}}$ of a metric space $M$. We prove that the Lipschitz-free space $\mathcal{F}(M_{\mathcal{U}})$ is finitely representable in $\mathcal{F}(M)$. 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 $\mathcal{F}(M)$ is finitely representable in $\mathcal{F}(X)$. 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.