Bundles of metric structures as left ultrafunctors

TL;DR

This paper establishes an equivalence between categories of left ultrafunctors and bundles of models for continuous theories, generalizing classical structures like Banach bundles and continuous fields of C*-algebras.

Contribution

It introduces the concept of bundles of $ ext{T}$-models in the context of ultracategories of complete metric structures, extending the framework of continuous model theory.

Findings

Equivalence between left ultrafunctors and bundles of $ ext{T}$-models for continuous theories.

The new notion of bundle recovers classical structures like Banach bundles and C*-algebra fields.

Provides a categorical framework linking ultracategories and classical metric structures.

Abstract

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.