The Serre-Swan theorem for normed modules

The aim of this note is to analyse the structure of the L0-normed L0-modules over a metric measure space. These are a tool that has been introduced by Gigli to develop a differential calculus on spaces verifying the Riemannian Curvature Dimension condition. More precisely, we discuss under which conditions an L0-normed L0-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.