nLab
pullback bundle

Idea

Given a bundle $p \colon E \to X$, and a morphism $f \colon Y \to X$, then the pullback bundle $f^\ast E \to Y$ is (if it exists) simply the pullback of $p$ along $f$, regarded as a bundle over $Y$.

Universal bundles

Over a paracompact topological space every vector bundle is isomorphic to a pullback bundle of a universal vector bundle over a classifying space. See there for more.

Related concepts

Notions of pullback:

• pullback, fiber product (limit over a cospan)

• wide pullback

• lax pullback, comma object (lax limit over a cospan)

• (∞,1)-pullback, homotopy pullback, ((∞,1)-limit over a cospan)

• base change, context extension

• pullback bundle

• contravariant functor

  • pullback in cohomology, d-invariant

  • pullback of differential forms

  • pullback of a distribution