A singular Serre-Swan theorem via tepui fibrations

The classical Serre-Swan theorem asserts that any finitely generated projective module over the algebra $C^{\mathrm{\infty }}(M)$ of smooth functions of a manifold $M$ can be realized as the sections of a vector bundle over $M$. In this article, we extend this theorem beyond the projective case by introducing a notion of singular vector bundle whose sections can realize all finitely generated $C^{\mathrm{\infty }}(M)$-modules, up to invisible elements. We introduce tepui fibrations as the underlying geometric objects of these singular vector bundles, and show how these tepui fibrations can model singular foliations, their holonomy groupoids, and singular subalgebroids.