nLab
fiber
Redirected from "fibre".
Contents
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
fibered limit
2-Categorical
(∞,1)-Categorical
Model-categorical
Contents
Idea
The fiber of a morphism or bundle f : E → B f : E \to B over a point of B B is the collection of elements of E E that are mapped by f f to this point.
Definition
Fibers make sense in any category with a terminal object * * and pullbacks. For f : A → B f : A \to B a morphism in such a category and B B equipped with the structure of a pointed object pt : * → B pt : * \to B , the fiber of f f is the fiber product of f f with pt pt , hence the pullback
A × B * → * ↓ ↓ pt A → f B
\array{
A \times_B * &\to& *
\\
\downarrow && \downarrow_pt
\\
A &\stackrel{f}{\to}& B
}
Examples
Bundles
Given a bundle p : E → B p: E \to B and a global element x : 1 → B x: 1 \to B , the fibre or fiber E x E_x of ( E , p ) (E,p) over x x is the pullback
E x → E ↓ ↓ p 1 → x B \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 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 ℰ \mathcal{E} of 𝒪 \mathcal{O} -modules over a locally ringed space ( X , 𝒪 ) (X,\mathcal{O}) at a point x ∈ X x \in X is defined as the vector space ℰ ( x ) ≔ ℰ x ⊗ 𝒪 x k ( x ) \mathcal{E}(x) \coloneqq \mathcal{E}_x \otimes_{\mathcal{O}_x} k(x) over the residue field k ( x ) k(x) . If ℰ \mathcal{E} is quasicoherent , the associated vector bundle of the fiber is the pullback of the associated vector bundle of ℰ \mathcal{E} :
V ( ℰ ( x ) ) = Spec Sym ℰ ( x ) → Spec ̲ X Sym ℰ = V ( ℰ ) ↓ ↓ Spec k ( x ) → X \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
}
Last revised on April 29, 2023 at 21:05:32.
See the history of this page for a list of all contributions to it.