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Pullback

Process in mathematics

Wikipedia contributorsAdapted for Fathomly · View authors
2 min read
Source updated September 12, 2026

In mathematics, a pullback is either of two related processes: precomposition and fiber-product; precomposition is a special case of the general fiber-product. Its dual is a pushforward.

01Precomposition

Precomposition with a function probably provides the most elementary notion of pullback: in simple terms, a function f of a variable y, where y itself is a function of another variable x, may be written as a function of x. This is the pullback of f by the function y. f(y(x))\equiv g(x)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 cohomology classes; see

02Fiber-product

The pullback bundle is an example that bridges the notion of a pullback as precomposition, and the notion of a pullback as a 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 categorical pullback, but it has important special cases: inverse image (and pullback) sheaves in algebraic geometry, and pullback bundle in algebraic topology and differential geometry.

03Functional analysis

When the pullback is studied as an operator acting on function spaces, it becomes a linear operator, and is known as the transpose or composition operator. Its adjoint is the push-forward, or, in the context of functional analysis, the transfer operator.

04Relationship

The relation between the two notions of pullback can perhaps best be illustrated by sections of fiber bundles: if s is a section of a fiber bundle E over N, and f:M\to N, then the pullback (precomposition) f^{*}s=s\circ f of s with f is a section of the pullback (fiber-product) bundle f^{*}E over M.

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Sources and credits

This article is adapted from the Wikipedia article “Pullback”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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Related topics

Inverse image functor

In mathematics, specifically in algebraic topology and algebraic geometry, an inverse image functor is a contravariant construction of sheaves; here “contravariant” in the sense given a map f : X → Y {\displaystyle f:X\to Y} , the inverse image functor is a functor from the category of sheaves on Y to the category of sheaves on X. The direct image functor is the primary operation on sheaves, with the simplest definition. The inverse image exhibits some relatively subtle features.

Pullback (category theory)

In category theory, a branch of mathematics, a pullback is the limit of a diagram consisting of two morphisms f : X → Z and g : Y → Z with a common codomain. The pullback is written P = X ×f, Z, g Y. Usually the morphisms f and g are omitted from the notation, and then the pullback is written P = X ×Z Y. The pullback comes equipped with two natural morphisms P → X and P → Y. The pullback of two morphisms f and g need not exist, but if it does, it is essentially uniquely defined by the two morphisms.

Fibred category

Fibred categories are abstract entities in mathematics used to provide a general framework for descent theory. They formalise the various situations in geometry and algebra in which inverse images (or pull-backs) of objects such as vector bundles can be defined.