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| {{see also|Pull back (disambiguation)}}
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| In mathematics, a '''pullback''' is either of two different, but related processes: precomposition and fiber-product. Its "dual" is [[pushforward]].
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| ==Precomposition==
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| 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''(''x'').
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| : <math>f(y(x)) \equiv g(x) \, </math>
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| It is such a fundamental process, that it is often passed over without mention, for instance in elementary calculus: this is sometimes called ''omitting pullbacks'', and pervades areas as diverse as [[fluid mechanics]] and [[differential geometry]].
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| 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]]
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| and their [[de Rham cohomology|cohomology classes]].
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| See:
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| *[[Pullback (differential geometry)]]
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| *[[Pullback (cohomology)]]
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| ==Fibre-product==
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| The notion of pullback as a fibre-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 and differential geometry.
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| See:
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| *[[Pullback (category theory)]]
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| *[[Inverse image sheaf]]
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| *[[Pullback bundle]]
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| *[[Fibred category]]
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| ==Functional analysis==
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| When the pullback is studied as an operator acting on [[function space]]s, it becomes a [[linear operator]], and is known as the [[composition operator]]. Its adjoint is the push-forward, or, in the context of [[functional analysis]], the [[transfer operator]].
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| ==Relationship==
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| The relation between the two notions of pullback can perhaps best be illustrated by [[section (fiber bundle)|sections]] of [[fibre bundle]]s: if ''s'' is a section of a fibre bundle ''E'' over ''N'', and ''f'' is a map from ''M'' to ''N'', then the pullback (precomposition) <math> f^* s=s\circ f</math> of ''s'' with ''f'' is a section of the pullback (fibre-product) bundle ''f''*''E'' over ''M''.
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| ==See also==
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| * [[Inverse image functor]]
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| [[Category:Mathematical analysis]]
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