Editing Pullback (category theory) (section)

===Fiber bundles===
Another example of a pullback comes from the theory of [[fiber bundle]]s: given a bundle map {{math|''π'' : ''E'' → ''B''}} and a [[continuous map]] {{math|''f''&nbsp;:&nbsp;''X''&nbsp;→&nbsp;''B''}}, the pullback (formed in the [[category of topological spaces]] with [[Continuous function (topology)|continuous maps]]) {{math|''X''&nbsp;×<sub>''B''</sub>&nbsp;''E''}} is a fiber bundle over {{mvar|X}} called the [[pullback bundle]]. The associated commutative diagram is a morphism of fiber bundles. A special case is the pullback of two fiber bundles {{math|''E''<sub>''1''</sub>, ''E''<sub>2</sub> → ''B''}}. In this case {{math|''E''<sub>1</sub> × ''E''<sub>2</sub>}} is a fiber bundle over {{math|''B × B''}}, and pulling back along the diagonal map {{math|''B'' → ''B × B''}} gives a space homeomorphic (diffeomorphic) to {{math|''E''<sub>1</sub> ×<sub>''B''</sub> ''E''<sub>2</sub>}}, which is a fiber bundle over {{math|''B''}}. All statements here hold true for differentiable [[Differentiable manifold|manifolds]] as well. Differentiable maps {{math|''f''&nbsp;:&nbsp;''M''&nbsp;→&nbsp;''N''}} and {{math|''g''&nbsp;:&nbsp;''P''&nbsp;→&nbsp;''N''}} are [[Transversality (mathematics)|transverse]] if and only if their product{{math| ''M × P'' → ''N × N''}} is transverse to the diagonal of {{math|''N''}}.<ref>{{Citation |last=Lee |first=John M. |title=Smooth Manifolds |date=2003 |work=Graduate Texts in Mathematics |pages=1–29 |url=https://doi.org/10.1007/978-0-387-21752-9_1 |access-date=2025-02-28 |place=New York, NY |publisher=Springer New York |isbn=978-0-387-95448-6}}</ref> Thus, the pullback of two transverse differentiable maps into the same [[differentiable manifold]] is also a differentiable manifold, and the [[tangent space]] of the pullback is the pullback of the tangent spaces along the differential maps.