The notion of pushforward in mathematics is "dual" to the notion of
pullback
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: i ...
, and can mean a number of different but closely related things.
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Pushforward (differential)
In differential geometry, pushforward is a linear approximation of smooth maps on tangent spaces. Suppose that is a smooth map between smooth manifolds; then the differential of ''φ, d\varphi_x,'' at a point ''x'' is, in some sense, the be ...
, the differential of a smooth map between manifolds, and the "pushforward" operations it defines
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Pushforward (homology) In algebraic topology, the pushforward of a continuous function f : X \rightarrow Y between two topological spaces is a homomorphism f_:H_n\left(X\right) \rightarrow H_n\left(Y\right) between the homology groups for n \geq 0.
Homology is a functo ...
, the map induced in homology by a continuous map between topological spaces
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Pushforward measure In measure theory, a pushforward measure (also known as push forward, push-forward or image measure) is obtained by transferring ("pushing forward") a measure from one measurable space to another using a measurable function.
Definition
Given meas ...
, measure induced on the target measure space by a measurable function
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Pushout (category theory)
In category theory, a branch of mathematics, a pushout (also called a fibered coproduct or fibered sum or cocartesian square or amalgamated sum) is the colimit of a diagram consisting of two morphisms ''f'' : ''Z'' → ''X'' and ''g'' : ''Z'' &r ...
, the categorical dual of pullback
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Direct image sheaf In mathematics, the direct image functor is a construction in sheaf theory that generalizes the global sections functor to the relative case. It is of fundamental importance in topology and algebraic geometry. Given a sheaf ''F'' defined on a topo ...
, the pushforward of a sheaf by a map
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Fiberwise integral, the direct image of a differential form or cohomology by a smooth map, defined by "integration on the fibres"
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Transfer operator
Transfer may refer to:
Arts and media
* ''Transfer'' (2010 film), a German science-fiction movie directed by Damir Lukacevic and starring Zana Marjanović
* ''Transfer'' (1966 film), a short film
* ''Transfer'' (journal), in management studies
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, the pushforward on the space of measurable functions; its adjoint, the pull-back, is the composition or Koopman operator
{{mathematical disambiguation
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