Category of vector bundles over a manifold

Definition

Fix a finite-dimensional Hausdorff second-countable smooth manifold $M$ and $\mathbb F\in\{\mathbb R,\mathbb C\}$ . The category of finite-rank smooth $\mathbb F$ -vector bundles over $M$ , denoted $\mathbf{Vect}_{\mathbb F}(M)$ , has smooth finite-rank $\mathbb F$ -vector bundles $E\to M$ as objects. A morphism $E\to F$ is a smooth vector-bundle morphism whose base map is exactly $\operatorname{id}_M$ .

The house convention is that $M$ has no boundary. If $M$ is disconnected, the rank may be any finite locally constant function on $M$ ; it need not be bounded across all connected components.

Composition is composition of total-space maps. Fiberwise addition and scalar multiplication make each morphism set an $\mathbb F$ -vector space, and direct sum gives a biproduct. Bundle isomorphisms over $M$ are precisely the isomorphisms in this category.

Why the base is fixed

A general vector-bundle morphism may cover a smooth map $f:M\to N$ . Such morphisms belong to a larger varying-base category. They cannot all be placed in $\mathbf{Vect}_{\mathbb F}(M)$ , because their source and target have different bases.

The fixed-base convention is essential for the Serre–Swan equivalence. A map $\Phi:E\to F$ over $\operatorname{id}_M$ sends a section $s$ to $\Phi\circ s$ and thereby gives a $C^\infty(M,\mathbb F)$ -linear map. If $\Phi$ covers a nonidentity map, postcomposition does not have this source and target and is not a homomorphism between section modules over one fixed ring.

Sheaf formulation

Taking local sections defines a covariant functor from $\mathbf{Vect}_{\mathbb F}(M)$ to finite-rank locally free $C^\infty_M(\mathbb F)$ -module sheaves. This sheaf-level construction is local and includes arbitrary finite locally constant rank on a disconnected base.

The global-section Serre–Swan equivalence uses a smaller full subcategory when $M$ is disconnected. Write

\mathbf{Vect}^{\mathrm{bd}}_{\mathbb F}(M) \subseteq \mathbf{Vect}_{\mathbb F}(M)

for bundles whose ranks are globally bounded across the connected components. These, and only these, correspond to finitely generated projective $C^\infty(M,\mathbb F)$ -modules. For connected $M$ , the two bundle categories agree.

References

Dale Husemoller, Fibre Bundles, 3rd ed., Springer, 1994. DOI record. Relevant: vector bundles, bundle maps, and direct sums.
Jet Nestruev, Smooth Manifolds and Observables, Springer, 2003. DOI record. Relevant: Chapter 11, vector bundles and projective modules.