nLab
fiber
Context
Limits and colimits
limits and colimits

1-Categorical
limit and colimit

limits and colimits by example

commutativity of limits and colimits

small limit

filtered colimit

sifted colimit

connected limit , wide pullback

preserved limit , reflected limit , created limit

product , fiber product , base change , coproduct , pullback , pushout , cobase change , equalizer , coequalizer , join , meet , terminal object , initial object , direct product , direct sum

finite limit

Kan extension

weighted limit

end and coend

2-Categorical
(∞,1)-Categorical
Model-categorical
Contents
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
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 $\mathcal{E}$ :

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