nLab sub-bundle

Definition

A sub-bundle is a subobject in a category of bundles.

If $\left[ E \overset{fb}{\to} X\right]$ is a bundle, then a sub-bundle is a monomorphism

$$S \hookrightarrow E$$

in the given slice category over $X$, hence

$$\array{ S && \hookrightarrow && E \\ & {}_{\mathllap{\exists!}}\searrow && \swarrow_{\mathrlap{fb}} \\ && X }$$

Examples

• Given a real vector bundle with orthogonal structure, then the sub-bundle whose fibers are the unit spheres is called the unit sphere bundle of the vector bundle.