# nLab fiber

### Context

#### Limits and colimits

limits and colimits

# 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

###### Lemma

For $f:A\to B$ a morphism in a category and $B$ equipped with the structure of a pointed object $\mathrm{pt}:*\to B$, the fiber of $f$ is the fiber product of $f$ with $p$, hence the pullback

$\begin{array}{ccc}A{×}_{B}*& \to & *\\ ↓& & ↓\\ A& \stackrel{f}{\to }& B\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ A \times_B * &\to& * \\ \downarrow && \downarrow \\ 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 $\left(E,p\right)$ over $x$ is the pullback

$\begin{array}{ccc}{E}_{x}& \to & E\\ ↓& & {↓}_{p}\\ 1& \stackrel{x}{\to }& B\end{array}$\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 $ℰ$ of $𝒪$-modules over a locally ringed space $\left(X,𝒪\right)$ at a point $x\in X$ is defined as the vector space $ℰ\left(x\right)≔{ℰ}_{x}{\otimes }_{{𝒪}_{x}}k\left(x\right)$ over the residue field $k\left(x\right)$. If $ℰ$ is quasicoherent, the associated vector bundle of the fiber is the pullback of the associated vector bundle of $ℰ$:

$\begin{array}{ccc}V\left(ℰ\left(x\right)\right)=\mathrm{Spec}\mathrm{Sym}ℰ\left(x\right)& \to & {\underline{\mathrm{Spec}}}_{X}\mathrm{Sym}ℰ=V\left(ℰ\right)\\ ↓& & ↓\\ \mathrm{Spec}k\left(x\right)& \to & X\end{array}$\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 }

Revised on February 9, 2013 02:55:13 by Ingo Blechschmidt (79.219.164.73)