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

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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 * Pullback (differential geometry) * 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 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

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, and pullback bundles in

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and differential geometry. See also: * Pullback (category theory) * Fibred category * Inverse image sheaf

Functional analysis

When the pullback is studied as an operator acting on function spaces, it becomes a

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, and is known as the transpose or

composition operator In mathematics, the composition operator C_\phi with symbol \phi is a linear operator defined by the rule C_\phi (f) = f \circ \phi where f \circ \phi denotes function composition. The study of composition operators is covered bAMS category 47B3 ...

. 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 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.

References

{{reflist Mathematical analysis