(see also Chern-Weil theory, parameterized homotopy theory)
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 $Y$.
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.
Notions of pullback:
pullback, fiber product (limit over a cospan)
lax pullback, comma object (lax limit over a cospan)
(∞,1)-pullback, homotopy pullback, ((∞,1)-limit over a cospan)
Last revised on December 2, 2020 at 12:30:39. See the history of this page for a list of all contributions to it.