Fiber (mathematics)

Let ${\displaystyle f:X\to Y}$ be a function between sets.

The fiber of an element ${\displaystyle y\in Y}$ (or fiber over ${\displaystyle y}$) under the map ${\displaystyle f}$ is the set

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

that is, the set of elements that get mapped to ${\displaystyle y}$ by the function. It is the preimage of the singleton ${\displaystyle \{y\}.}$ (One usually takes ${\displaystyle y}$ in the image of ${\displaystyle f}$ to avoid ${\displaystyle f^{-1}(y)}$ being the empty set.)

The collection of all fibers for the function ${\displaystyle f}$ forms a partition of the domain ${\displaystyle X.}$ The fiber containing an element ${\displaystyle x\in X}$ is the set ${\displaystyle f^{-1}(f(x)).}$ For example, the fibers of the projection map ${\displaystyle \mathbb {R} ^{2}\to \mathbb {R} }$ that sends ${\displaystyle (x,y)}$ to ${\displaystyle x}$ are the vertical lines, which form a partition of the plane.

If ${\displaystyle f}$ is a real-valued function of several real variables, the fibers of the function are the level sets of ${\displaystyle f}$. If ${\displaystyle f}$ is also a continuous function and ${\displaystyle y\in \mathbb {R} }$ is in the image of ${\displaystyle f,}$ the level set ${\displaystyle f^{-1}(y)}$ will typically be a curve in 2D, a surface in 3D, and, more generally, a hypersurface in the domain of ${\displaystyle f.}$

In algebraic geometry, if ${\displaystyle f:X\to Y}$ is a morphism of schemes, the fiber of a point ${\displaystyle p}$ in ${\displaystyle Y}$ is the fiber product of schemes

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

where ${\displaystyle k(p)}$ is the residue field at ${\displaystyle p.}$