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Inmathematics, apullback is either of two different, but related processes: precomposition and fiber-product. Its dual is apushforward.

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

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 asdifferential forms and theircohomology classes; see

• Pullback (differential geometry)
• Pullback (cohomology)

The pullback bundle is an example that bridges the notion of a pullback as precomposition, and the notion of a pullback as aCartesian square. In that example, the base space of afiber 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 afiber product.

The notion of pullback as a fiber-product ultimately leads to the very general idea of acategorical pullback, but it has important special cases: inverse image (and pullback) sheaves inalgebraic geometry, andpullback bundles inalgebraic topology and differential geometry.

See also:

• Pullback (category theory)
• Fibred category
• Inverse image sheaf

When the pullback is studied as an operator acting onfunction spaces, it becomes alinear operator, and is known as thetranspose orcomposition operator. Its adjoint is the push-forward, or, in the context offunctional analysis, thetransfer operator.

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

• Inverse image functor – functor between categories of Abelian-group-valued sheaves induced by a continuous map between topological spaces; sheafification of the presheaf associating to an open set U the inductive limit of the groups associated to open supersets of U’s imagePages displaying wikidata descriptions as a fallback

• Mathematical analysis

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