Bundles of metric structures as left ultrafunctors

We pursue the study of Ultracategories initiated by Makkai and more recently Lurie by looking at properties of ultracategories of complete metric structures, i.e. coming from continuous model theory, instead of ultracategories of models of first order theories.

Our main result is that for any continuous theory $\mathbb{T}$, there is an equivalence between the category of left ultrafunctors from a compact Hausdorff space $X$ to the category of $\mathbb{T}$-models and a notion of bundle of $\mathbb{T}$-models over $X$.

The notion of bundle of $\mathbb{T}$-models is new but recovers many classical notions like Bundle of Banach spaces, or (semi-)continuous field of $\mathrm{C}^*$-algebras or Hilbert spaces.