nLab fiber

Idea

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.

Kernels

In an additive category fibers over the zero object are called kernels.

Fibers of a sheaf of modules

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 $\mathcal{E}$:

$$\array { V(\mathcal{E}(x)) = \mathrm{Spec} \mathrm{Sym} \mathcal{E}(x) & \to & \underline{Spec}_X \mathrm{Sym} \mathcal{E} = V(\mathcal{E}) \\ \downarrow & & \downarrow \\ \mathrm{Spec} k(x) & \to & X }$$

Related concepts

• essential fiber

• homotopy fiber, fiber sequence

• cofiber, homotopy cofiber

• kernel, cokernel

• fiber type