The fiber of a morphism or bundle$f : E \to B$ over a point of $B$ is the collection of elements of $E$ that are mapped by $f$ to this point.

Definition

Fibers make sense in any category with a terminal object $*$ and pullbacks. For $f : A \to B$ a morphism in such a category and $B$ equipped with the structure of a pointed object$pt : * \to B$, the fiber of $f$ is the fiber product of $f$ with $pt$, hence the pullback

$\array{
A \times_B * &\to& *
\\
\downarrow && \downarrow_pt
\\
A &\stackrel{f}{\to}& B
}$

Examples

Bundles

Given a bundle$p: E \to B$ and a global element$x: 1 \to B$, the fibre or fiber$E_x$ of $(E,p)$ over $x$ is the pullback

$\array {
E_x & \to & E \\
\downarrow & & \downarrow_p \\
1 & \stackrel{x}\to & B
}$

if it exists.

In a fiber bundle, all fibres are isomorphic to some standard fibre $F$ in a coherent way.

The fiber of a sheaf $\mathcal{E}$ of $\mathcal{O}$-modules over a locally ringed space$(X,\mathcal{O})$ at a point $x \in X$ is defined as the vector space $\mathcal{E}(x) \coloneqq \mathcal{E}_x \otimes_{\mathcal{O}_x} k(x)$ over the residue field$k(x)$. If $\mathcal{E}$ is quasicoherent, the associated vector bundle of the fiber is the pullback of the associated vector bundle of $\mathcal{E}$: