Fiber (mathematics)

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

  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 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.

Definitions[edit]

Fiber in naive set theory[edit]

Let f : X → Y be a map. The fiber of an element , commonly denoted by , is defined as

In various applications, this is also called:

  • the inverse image of under the map f
  • the preimage of 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[edit]

In algebraic geometry, if f : X → Y is a morphism of schemes, the fiber of a point p in Y is the fiber product of schemes

where k(p) is the residue field at p.

Terminological variance[edit]

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

  • the fiber of the element y under the map f
  • the inverse image of the set under the map f
  • the preimage of the set 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:[citation needed]

  • 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.

See also[edit]