In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward.

Precomposition

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 In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications ...

and their

cohomology classes In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...

; see *

Pullback (differential geometry) Suppose that is a smooth map between smooth manifolds ''M'' and ''N''. Then there is an associated linear map from the space of 1-forms on ''N'' (the linear space of sections of the cotangent bundle) to the space of 1-forms on ''M''. This lin ...

* Pullback (cohomology)

Fiber-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 mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\ti ...

. In that example, the base space of a

fiber bundle In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a ...

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 In category theory, a branch of mathematics, a pullback (also called a fiber product, fibre product, fibered product or Cartesian square) is the limit of a diagram consisting of two morphisms and with a common codomain. The pullback is ofte ...

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 bundles in

algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classif ...

and differential geometry. See also: *

Pullback (category theory) In category theory, a branch of mathematics, a pullback (also called a fiber product, fibre product, fibered product or Cartesian square) is the limit of a diagram consisting of two morphisms and with a common codomain. The pullback is often ...

* Fibred category * Inverse image sheaf

Functional analysis

When the pullback is studied as an operator acting on

function space In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a ve ...

s, it becomes a

linear operator In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...

, and is known as the

transpose In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations). The tr ...

or composition operator. Its adjoint is the push-forward, or, in the context of

functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined ...

, the transfer operator.

Relationship

The relation between the two notions of pullback can perhaps best be illustrated by

sections Section, Sectioning or Sectioned may refer to: Arts, entertainment and media * Section (music), a complete, but not independent, musical idea * Section (typography), a subdivision, especially of a chapter, in books and documents ** Section sig ...

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.

References

{{reflist Mathematical analysis