Symmetric monoidal categories of conveniently-constructible Banach bundles

TL;DR

This paper proves that certain finitely-generated Banach bundles over compact spaces are locally trivial and explores their categorical properties, including a symmetric monoidal structure and the characterization of algebraically finitely-generated bundles.

Contribution

It establishes local triviality for finitely-generated Banach bundles and characterizes algebraically finitely-generated bundles as rigid objects in a symmetric monoidal category.

Findings

Finitely-generated Banach bundles over compact spaces are locally trivial.

The category of topologically finitely-generated Banach bundles is symmetric monoidal.

Algebraically finitely-generated bundles are precisely the rigid objects in this category.

Abstract

We show that a continuously-normed Banach bundle $\mathcal{E}$ over a compact Hausdorff space $X$ whose space of sections is algebraically finitely-generated (f.g.) over $C(X)$ is locally trivial (and hence the section space is projective f.g over $C(X)$); this answers a question of I. Gogi\'c. As a preliminary we also provide sufficient conditions for a quotient bundle to be continuous phrased in terms of the Vietoris continuity of the unit-ball maps attached to the bundles. Related results include (a) the fact that the category of topologically f.g. continuous Banach bundles over $X$ is symmetric monoidal under the (fiber-wise-maximal) tensor product, (b) the full faithfulness of the global-section functor from topologically f.g. continuous bundles to $C(X)$-modules and (c) the consequent identification of the algebraically f.g. bundles as precisely the rigid objects in the aforementioned symmetric monoidal category.