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

• fiber product

• base change