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: ...
, 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 (formulating manifold) on tangent spaces. Suppose that \varphi\colon M\to N is a smooth map between smooth manifolds; then the differential of \varphi at a point x, ...
, 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 functor ...
, 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 mea ...
, 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'' & ...
, 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
In mathematics, the transfer operator encodes information about an iterated map and is frequently used to study the behavior of dynamical systems, statistical mechanics, quantum chaos and fractals. In all usual cases, the largest eigenvalue is 1 ...
, the pushforward on the space of measurable functions; its adjoint, the pull-back, is the composition or Koopman operator
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