nLab bundle

Definition

In the most general sense, a bundle over an object $B$ in a category $\mathcal{C}$ is simply an object $E$ of $\mathcal{C}$ equipped with a morphism in $\mathcal{C}$ from $E$ to $B$:

$$\array { E \\ \downarrow^{\mathrlap{p}} \\ B }$$

One often refers to such a bundle simply as $E$, even though $B$ is really part of the data.

For $x \in B$ a generalized element of $B$, the fiber $E_x$ of the bundle over $x$ is the pullback $x^* E$.

Given two bundles $E_1$ and $E_2$ over $B$, then a morphism of bundles over $B$ is a morphism $E_1 \to E_2$ which makes this diagram commute:

$$\array{ E_1 && \longrightarrow && E_2 \\ & {}_{\mathllap{p_1}}\searrow && \swarrow_{\mathrlap{p_2}} \\ && B } \,.$$

This way bundles over $B$ form a category, also called the slice category $\mathcal{C}_{/B}$ of $\mathcal{C}$ over $B$.

Bundles with extra structure and properties

One generally considers bundles with extra properties or structure:

• fiber bundle,

• microbundle

• principal bundle

• vector bundle,

• bundle with connection

  • Simons-Sullivan structured bundle

• etc.

The collection of all bundles

The category of bundles over a given object $B$ is the over category $\mathcal{C}/B$. The collection of all bundles in a given category $\mathcal{C}$ therefore arranges itself into the codomain fibration $cod : [I,\mathcal{C}] \to \mathcal{C}$. As such, the descent for bundles may be expressed as monadic descent with respect to the codomain bifibration. This does in general not work inside one of the more restrictive subcategories of bundles with extra structure and property, as the push-forward operation typically does not respect these extra conditions. For more on this see monadic descent of bundles.

Relation to other concepts

• Bundles tend to have sheaves of local sections.

• In physics, gauge fields may be described in terms of bundles with connection.

Related concepts

• bundle, display map

• sub-bundle

• ex-space

• fiber bundle / fiber ∞-bundle

  • associated bundle

  • vector bundle

    • line bundle

    • stable vector bundle

  • sphere bundle

  • natural bundle

  • equivariant bundle

  • bundle of spectra

  • fiber bundles in physics

• principal bundle / torsor / associated bundle

• principal 2-bundle / gerbe / bundle gerbe

• principal 3-bundle / bundle 2-gerbe

• principal ∞-bundle / associated ∞-bundle

• (∞,1)-vector bundle / (∞,n)-vector bundle

  • spherical fibration

References

A useful collection of introductory notes to fiber bundles, vector bundles and fiber bundles with connection is at

• Steven Bradlow, Vector bundles and introduction to gauge theory