nLab bundle






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

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

One often refers to such a bundle simply as EE, even though BB is really part of the data.

For xBx \in B a generalized element of BB, the fiber E xE_x of the bundle over xx is the pullback x *Ex^* E.

Given two bundles E 1E_1 and E 2E_2 over BB, then a morphism of bundles over BB is a morphism E 1E 2E_1 \to E_2 which makes this diagram commute:

E 1 E 2 p 1 p 2 B. \array{ E_1 && \longrightarrow && E_2 \\ & {}_{\mathllap{p_1}}\searrow && \swarrow_{\mathrlap{p_2}} \\ && B } \,.

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

Bundles with extra structure and properties

One generally considers bundles with extra properties or structure:

The collection of all bundles

The category of bundles over a given object BB is the over category 𝒞/B\mathcal{C}/B. The collection of all bundles in a given category 𝒞\mathcal{C} therefore arranges itself into the codomain fibration cod:[I,𝒞]𝒞cod : [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


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

Last revised on September 3, 2022 at 03:47:12. See the history of this page for a list of all contributions to it.