{{Short description|Process in mathematics}} {{about|the uses of the term "pullback" in mathematics|other uses|Pull back (disambiguation)}}{{refimprove|date=November 2025}}In [[mathematics]], a '''pullback''' is either of two related processes: precomposition and fiber-product; precomposition is a special case of the general fiber-product.<ref name="7sketches">{{cite book|last1=Fong|first1=Brendan|last2=Spivak|first2=David|authorlink2=David Spivak|title=An Invitation to Applied Category Theory: Seven Sketches in Compositionality|url=https://dspivak.net/7sketches.pdf|archive-url=https://web.archive.org/web/20220315151140/http://www.dspivak.net/7sketches.pdf |archive-date={{date|2022-03-15}} |url-status=dead|arxiv=1803.05316|isbn=978-1108711821|pages=112–113|publisher=[[Cambridge University Press]]|publication-date={{date|2019-07-18}}}}</ref> Its dual is a [[Pushforward (disambiguation)|pushforward]]<!--intentional link to DAB page-->.
==Precomposition== Precomposition with a [[Function (mathematics)|function]] probably provides the most elementary notion of pullback: in simple terms, a function <math>f</math> of a variable <math>y,</math> where <math>y</math> itself is a function of another variable <math>x,</math> may be written as a function of <math>x.</math> This is the pullback of <math>f</math> by the function <math>y.</math> <math display=block>f(y(x)) \equiv g(x)</math>It is such a fundamental process that it is often passed over without mention.
However, it is not just functions that can be "pulled back" in this sense. Pullbacks can be applied to many other objects such as [[differential forms]] and their [[de Rham cohomology|cohomology classes]]; see
* [[Pullback (differential geometry)]] * [[Pullback (cohomology)]]
==Fiber-product== {{Main|Pullback bundle}}
The pullback bundle is an example that bridges the notion of a pullback as precomposition, and the notion of a pullback as a [[Pullback_(category_theory)| Cartesian square]]. In that example, the base space of a [[fiber bundle]] is pulled back, in the sense of precomposition, above. The fibers then travel along with the points in the base space at which they are anchored: the resulting new pullback bundle looks locally like a Cartesian product of the new base space, and the (unchanged) fiber. The pullback bundle then has two projections: one to the base space, the other to the fiber; the product of the two becomes coherent when treated as a [[fiber product]].
===Generalizations and category theory===
The notion of pullback as a fiber-product ultimately leads to the very general idea of a [[Category theory|categorical]] pullback, but it has important special cases: inverse image (and pullback) sheaves in [[algebraic geometry]], and [[pullback bundle]]s in [[algebraic topology]] and differential geometry.
==Functional analysis== {{See also|Transpose of a linear map}} When the pullback is studied as an operator acting on [[function space]]s, it becomes a [[linear operator]], and is known as the [[Transpose of a linear map|transpose]] or [[composition operator]]. Its adjoint is the push-forward, or, in the context of [[functional analysis]], the [[transfer operator]].
==Relationship== The relation between the two notions of pullback can perhaps best be illustrated by [[Section (fiber bundle)|sections]] of fiber bundles: if <math>s</math> is a section of a fiber bundle <math>E</math> over <math>N,</math> and <math>f : M \to N,</math> then the pullback (precomposition) <math>f^* s = s\circ f</math> of ''s'' with <math>f</math> is a section of the pullback (fiber-product) bundle <math>f^*E</math> over <math>M.</math>
==See also==
* {{annotated link|Inverse image functor}} * [[Pullback (category theory)]] * [[Fibred category]] * [[Inverse image sheaf]]
==References== {{reflist}}
[[Category:Mathematical analysis]]