Fiber (mathematics)

Let f : X → Y be a map. The fiber of an element ${\displaystyle y\in Y,}$ commonly denoted by ${\displaystyle f^{-1}(y),}$ is defined as

$${\displaystyle f^{-1}(y):=\{x\in X\mid f(x)=y\}.}$$

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 ${\displaystyle f^{-1}(A)}$ generalizes the concept of the fiber to subsets ${\displaystyle A\subseteq Y}$ of the codomain. The notation ${\displaystyle f^{-1}(y)}$ is still used to refer to the fiber, as the fiber of an element y is the preimage of the singleton set ${\displaystyle \{y\}}$, as in ${\displaystyle f^{-1}(\{y\})}$. That is, the fiber can be treated as a function from the codomain to the powerset of the domain: ${\displaystyle f^{-1}:Y\to {\mathcal {P}}(X),}$ while the preimage generalizes this to a function between powersets: ${\displaystyle f^{-1}:{\mathcal {P}}(Y)\to {\mathcal {P}}(X).}$

If f maps into the real numbers, so ${\displaystyle y\in \mathbb {R} }$ is simply a number, then the fiber ${\displaystyle f^{-1}(y)}$ is also called the level set of y under f: ${\displaystyle L_{y}(f).}$ 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 d − 1.

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

${\displaystyle X\times _{Y}\operatorname {Spec} k(p)}$

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