# The Serre-Swan theorem for normed modules

## TL;DR

This paper investigates the structure of $L^0$-normed $L^0$-modules over metric measure spaces, establishing conditions for their representation as sections of measurable Banach bundles and exploring categorical equivalences.

## Contribution

It provides a detailed analysis of the conditions under which $L^0$-modules correspond to measurable Banach bundles, extending the Serre-Swan theorem to this setting.

## Key findings

- Identifies conditions for $L^0$-modules to be sections of Banach bundles
- Establishes an equivalence of categories between modules and bundles
- Advances the differential calculus on metric measure spaces

## Abstract

The aim of this note is to analyse the structure of the $L^0$-normed $L^0$-modules over a metric measure space. These are a tool that has been introduced by N. Gigli to develop a differential calculus on spaces verifying the Riemannian Curvature Dimension condition. More precisely, we discuss under which conditions an $L^0$-normed $L^0$-module can be viewed as the space of sections of a suitable measurable Banach bundle and in which sense such correspondence can be actually made into an equivalence of categories.