Pullback

Pullback

The notion of pullback in mathematics is a fundamental one. It refers to two different, but related processes: precomposition and fiber-product.

Precomposition

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"). 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.

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 formsand their cohomology classes.

See:
*Pullback (differential geometry)
*Pullback (cohomology)

Fibre-product

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.

See:
*Pullback (category theory)
*Inverse image sheaf
*Pullback bundle
*Fibred category

Relationship

The relation between the two notions of pullback can perhaps best be illustrated by sections of fibre bundles: 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) f^* s=scirc f of "s" with "f" is a section of the pullback (fibre-product) bundle "f"*"E" over "M".


Wikimedia Foundation. 2010.

Игры ⚽ Поможем решить контрольную работу

Look at other dictionaries:

  • Pullback — Saltar a navegación, búsqueda Se considera a esta teoria como la mejoración de la teoría de la relatividad... En matemática, el concepto de la teoria de la Larreateguidad o teoría de Pullback, tiene diferentes significados según sea el contexto.… …   Wikipedia Español

  • Pullback — Pull back , n. 1. That which holds back, or causes to recede; a drawback; a hindrance. [1913 Webster] 2. (Arch) The iron hook fixed to a casement to pull it shut, or to hold it party open at a fixed point. [1913 Webster] …   The Collaborative International Dictionary of English

  • pullback — [pool′bak΄] n. 1. a pulling back; esp., a planned military withdrawal 2. something that retards or hinders 3. a device for pulling something back …   English World dictionary

  • pullback — The downward reversal of a prolonged upward price trend. Bloomberg Financial Dictionary * * * pullback pull‧back [ˈpʊlbæk] noun [countable, uncountable] FINANCE a situation in which the price suddenly changes and goes back down or back up : • The …   Financial and business terms

  • Pullback — Dieser Artikel wurde auf der Qualitätssicherungsseite des Portals Mathematik eingetragen. Dies geschieht, um die Qualität der Artikel aus dem Themengebiet Mathematik auf ein akzeptables Niveau zu bringen. Dabei werden Artikel gelöscht, die nicht… …   Deutsch Wikipedia

  • pullback — pull|back [ˈpulbæk] n 1.) the act of moving soldiers away from the area where they were fighting pullback from ▪ The government is planning to implement a second pullback from the area. 2.) a reduction in the value, amount, or level of something… …   Dictionary of contemporary English

  • Pullback — A falling back of a price from its peak. This type of price movement might be seen as a brief reversal of the prevailing upward trend, signaling a slight pause in upward momentum. Often pullbacks are seen as buying opportunities after a security… …   Investment dictionary

  • pullback — noun 1. 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 • Hypernyms: ↑holding device 2. (military) the act of pulling back (especially an… …   Useful english dictionary

  • Pullback (differential geometry) — 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… …   Wikipedia

  • Pullback (category theory) — In category theory, a branch of mathematics, a pullback (also called a fibered product or Cartesian square) is the limit of a diagram consisting of two morphisms f : X → Z and g : Y → Z with a common codomain. The pullback is often written: P = X …   Wikipedia

Share the article and excerpts

Direct link
… Do a right-click on the link above
and select “Copy Link”