Kalpana Kalpana (Editor)

Fiber (mathematics)

Updated on
Edit
Like
Comment
Share on FacebookTweet on TwitterShare on LinkedInShare on Reddit

In mathematics, the term fiber (or fibre in British English) can have two meanings, depending on the context:

Contents

  1. In naive set theory, the fiber of the element y in the set Y under a map f : X → Y is the inverse image of the singleton { y } under f.
  2. In algebraic geometry, the notion of a fiber of a morphism of schemes must be defined more carefully because, in general, not every point is closed.

Fiber in naive set theory

Let f : X → Y be a map. The fiber of an element y ∈ Y , commonly denoted by f − 1 ( y ) , is defined as

f − 1 ( { y } ) = { x ∈ X | f ( x ) = y } .

In various applications, this is also called:

  • the inverse image of { y } under the map f
  • the preimage of { y } under the map f
  • the level set of the function f at the point y.

    • The term level set is only used if f maps into the real numbers and so y is simply a number. If f is a continuous function and if y is in the image of f, then the level set of y under f is a curve in 2D, a surface in 3D, and more generally a hypersurface of dimension d-1.

    Fiber in algebraic geometry

    In algebraic geometry, if f : X → Y is a morphism of schemes, the fiber of a point p in Y is the fibered product X × Y S p e c k ( p ) where k(p) is the residue field at p.

    Terminological variance

    The recommended practice is to use the terms fiber, inverse image, preimage, and level set as follows:

  • the fiber of the element y under the map f
  • the inverse image of the set { y } under the map f
  • the preimage of the set { y } under the map f
  • the level set of the function f at the point y.

    • By abuse of language, the following terminology is sometimes used but should be avoided:

  • the fiber of the map f at the element y
  • the inverse image of the map f at the element y
  • the preimage of the map f at the element y
  • the level set of the point y under the map f.

    • References

    Fiber (mathematics) Wikipedia
    (Text) CC BY-SA


    Similar Topics