Bundle Map

Definition of Bundle Morphism

Where the diagram

commutes and linear on fibers.

Category

With this morphism, we can define the bundle category. Also, in this category, the Product bundle is a product between two bundle objects.

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Definition of Bundle Morphism over with specific base

FIx the base $B$,

the diagram commutes, $\mathcal{L}$ is a linear map between fibers.

Category

Similarly, we can define the bundle over $B$ category, and the coproduct as Whitney sum of bundles.

Moreover, it forms an abelian category of

$$0\to \eta \to \xi \to \xi /\eta \to 0.$$

¹

Definition of Bundle map

Let $\eta$ and $\xi$ be a two $n$-plane vector bundle. Then the bundle map $g:\eta \to \xi$ is defined as a commute diagram as following

where $g$ is an isomorphism on each fibers.

Proposition

If $\eta ={\bar{g}}^{\ast }\xi$ as vector bundle over $B\left(\eta \right)$, then $E(\eta )\cong {\bar{g}}^{\ast }E\left(\xi \right)$ such that

Proof

We can map $E(\eta )\to B(\eta ){\times }_{B(\xi )}E\left(\xi \right)$ where $e\mapsto \left(\pi (e),g(e)\right)$. Since, we are in finite dimensional vector space, by the rank-nullity theorem, it is a isomorphism.

Footnotes

1. Not sure. But I guess. ↩