What is "pullback"

Pullback \Pull"back`\, n.

1. That which holds back, or causes to recede; a drawback; a hindrance.

2. (Arch) The iron hook fixed to a casement to pull it shut, or to hold it party open at a fixed point.

n. 1 The act or result of pulling back; a withdrawal. 2 (context film English) The act of drawing a camera back to broaden the visible scene. 3 That which holds back, or causes to recede; a drawback; a hindrance. 4 (context architecture English) The iron hook fixed to a casement to pull it shut, or to hold it partly open at a fixed point. 5 (context finance English) A reduction in the price of a financial instrument after reaching a peak 6 (context category theory English) Given a pair of morphisms $f:Xrightarrow\; Z$ and $g:\; Yrightarrow\; Z$ with a common codomain, ''Z'', their '''pullback''' is a pair of morphisms $p\_1:Prightarrow\; X$ and $p\_2:Prightarrow\; Y$ as well as their common domain, ''P'', such that the equation $fcirc\; p\_1\; =\; gcirc\; p\_2$ is satisfied, and for which there is the ''universal property'' that for any other object ''Q'' for which there are also morphisms $q\_1:\; Qrightarrow\; X$, $q\_2:\; Qrightarrow\; Y$; there is a unique morphism $u:\; Qrightarrow\; P$ such that $p\_1circ\; u\; =\; q\_1$ and $p\_2\; circ\; u\; =\; q\_2$.

1. n. a device (as a decorative loop of cord or fabric) for holding or drawing something back; "the draperies were drawn to the sides by pullbacks" [syn: tieback]

2. (military) the act of pulling back (especially an orderly withdrawal of troops); "the pullback is expected to be over 25,000 troops"

Suppose that φ:M→ N is a smooth map between smooth manifolds M and N; then there is an associated linear map from the space of 1-forms on N (the linear space of sections of the cotangent bundle) to the space of 1-forms on M. This linear map is known as the pullback (by φ), and is frequently denoted by φ. More generally, any covariant tensor field – in particular any differential form – on N may be pulled back to M using φ.

When the map φ is a diffeomorphism, then the pullback, together with the pushforward, can be used to transform any tensor field from N to M or vice versa. In particular, if φ is a diffeomorphism between open subsets of R and R, viewed as a change of coordinates (perhaps between different charts on a manifold M), then the pullback and pushforward describe the transformation properties of covariant and contravariant tensors used in more traditional (coordinate dependent) approaches to the subject.

The idea behind the pullback is essentially the notion of precomposition of one function with another. However, by combining this idea in several different contexts, quite elaborate pullback operations can be constructed. This article begins with the simplest operations, then uses them to construct more sophisticated ones. Roughly speaking, the pullback mechanism (using precomposition) turns several constructions in differential geometry into contravariant functors.

In category theory, a branch of mathematics, a pullback (also called a fiber product, fibre product, fibered product or Cartesian square) is the limit of a diagram consisting of two morphisms and with a common codomain. The pullback is often written

and comes equipped with two natural morphisms and . The pullback of two morphisms and need not exist, but if it does, it is essentially uniquely defined by the two morphisms. In many situtations, may intuitively be thought off as consisting of pairs of elements with and and . For the general definition, a universal property is used, which essentially expresses the fact that the pullback is the "most general" way to complete the two given morphisms to a commutative square.

The dual concept of the pullback is the pushout.

In mathematics, a pullback is either of two different, but related processes: precomposition and fibre-product. Its "dual" is pushforward.

In algebraic topology, given a continuous map f: X → Y of topological spaces and a ring R, the pullback along f on cohomology theory is a grade-preserving R-algebra homomorphism:

f : H(Y; R) → H(X; R)
from the cohomology ring of Y with coefficients in R to that of X. The use of the superscript is meant to indicate its contravariant nature: it reverses the direction of the map. For example, if X, Y are manifolds, R the field of real numbers, and the cohomology is de Rham cohomology, then the pullback is induced by the pullback of differential forms.

The homotopy invariance of cohomology states that if two maps f, g: X → Y are homotopic to each other, then they determine the same pullback: f = g.