Fiber (mathematics)


In mathematics, the term fiber 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

Fiber in naive set theory

Let be a map. The fiber of an element commonly denoted by is defined as That is, the fiber of y under f is the set of elements in the domain of f that are mapped to y.
The inverse image or preimage generalizes the concept of the fiber to subsets of the codomain. The notation is still used to refer to the fiber, as the fiber of an element y is the preimage of the singleton set, as in. That is, the fiber can be treated as a function from the codomain to the powerset of the domain: while the preimage generalizes this to a function between powersets:
If f maps into the real numbers, so is simply a number, then the fiber is also called the level set of y under f: If f is a continuous function and 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

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 fiber product of schemes
where k is the residue field at p.