sheaf of sections
0.1 Presheaf Definition
Consider a rank $r$ vector bundle^{} $E\to M$, whose typical fibre is defined with respect to a field $k$. Let $\{{U}_{\alpha}\}$ constitute a cover for $M$. Then, sections^{} of the bundle over some $U\subset M$ are defined as continuous functions $U\to E$, which commute with the natural projection^{} map $\pi :E\to M$; $\pi \circ s=i{d}_{M}$. Denote the space of sections of the bundle over U to be $\mathrm{\Gamma}(U,E)$. The space of sections is a vector space^{} over the field $k$ by defining addition and scalar multiplication pointwise: for $s,t\in \mathrm{\Gamma}(U,E)$, $p\in U$ and $a\in k$
$$(s+t)(p)\equiv s(p)+t(p)\mathit{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}(a\cdot s)(p)\equiv a\cdot s(p).$$ 
Then, this forms a presheaf $\mathcal{E}$, a functor from $(({\text{top}}_{M}))$ to the category of vector spaces, with restriction^{} maps the natural restriction of functions.
0.2 Sheaf Axioms
It is easy to see that it satisfies the sheaf axioms: for $U$ open and $\{{V}_{i}\}$ a cover of $U$,

1.
if $s\in \mathcal{E}(U)$ and ${s}_{{V}_{i}}=0$ for all $i$, then $s=0$.

2.
if ${s}_{i}\in \mathcal{E}({V}_{i})$ for all $i$, such that for each $i,j$ with ${V}_{i}\cap {V}_{j}\ne \mathrm{\varnothing}$, ${{s}_{i}}_{{V}_{i}\cap {V}_{j}}={{s}_{j}}_{{V}_{i}\cap {V}_{j}}$, then there is an $s\in \mathcal{E}(U)$ with ${s}_{{V}_{i}}={s}_{i}$ for all $i$.
The first follows from the fact that for any $U$, there is always at least
one element of $\mathcal{E}(U)$, the zero section, and that the transition
functions^{} of the bundle are linear maps. The second follows by the construction of the bundle.
1 Sheafification
We may also see the vector bundle by applying associated sheaf construction to the presheaf $U\mapsto \mathrm{\Gamma}(U,E)$. First though, we show that the stalk of the sheaf $\mathcal{E}$ at a point is isomorphic^{} to the fibre of the bundle $E$ at the point. Let $[s,U]$ be a germ at $p\in M$ $(p\in U\subset M)$, and define a map $\psi :{\mathcal{E}}_{p}\to {E}_{p}$ by
$$\psi :[s,U]\mapsto {s}_{p}.$$ 
First, we show that the map is a vector space homomorphism. Consider two germs $[s,U]$ and $[t,V]$ in ${\mathcal{E}}_{p}$. These map to ${s}_{p}$ and ${t}_{p}$ respectively. We add the germs by finding an open set $W\in U\cap V$ and adding the restrictions of the sections;
$$[s,U]+[t,V]\equiv [{s}_{W}+{t}_{W},W].$$ 
Of course, $p\in W$, so we have $\psi ({s}_{W}+{t}_{W})={s}_{p}+{t}_{p}$, since the restriction maps are simply restriction of functions. Now, it is easy to show that $\psi $ is injective^{}. Assume $\psi ([t,V])=\psi ([s,U])={s}_{p}$. Then
$\psi ([t,V])\psi ([s,U])$  $={s}_{p}{s}_{p}$  
$\psi ([t,V][s,U])$  $=0$  
$=[s,U]$ 
Now, we show that $\psi $ is surjective^{}. For ${s}_{p}\in {E}_{p}$, let $U\subset M$ open be isomorphic to some subset ${U}_{\mathbb{R}}$ of ${\mathbb{R}}^{m}$. Then, $\mathrm{\Gamma}(U,E)$ is the set of continuous maps $U\to {V}_{E}$, where ${V}_{E}$ is the typical fibre of $E$;
$$\mathrm{\Gamma}(U,E)=\underset{i=1}{\overset{r}{\oplus}}{\mathcal{C}}_{{U}_{\mathbb{R}}}^{\mathrm{\infty}}.$$ 
Then let $[s,U]$ be the constant function $s:{U}_{\mathbb{R}}\mapsto {s}_{x}$, and we
have constructed an isomorphism^{} $\psi $ between ${\mathcal{E}}_{p}$ and ${E}_{p}$.
To construct the Étalé space, take the disjoint union^{} of stalks, $\text{Sp\xe9}(\mathcal{E})={\coprod}_{p\in M}{\mathcal{E}}_{p}$, and endow it with the following topology: the open sets shall be of the form
$${U}_{s}=\{{s}_{p}s\in \mathrm{\Gamma}(U,\mathcal{E}),p\in U\subset M\},$$ 
collection^{} of germs of sections at points in $U\subset M$.
Then, the associated sheaf to $\mathcal{E}$ is the presheaf which assigns continuous maps $\mathrm{\Gamma}(U,\text{Sp\xe9}(\mathcal{E}))$ to each open $U$. These are maps where the preimage^{} of ${U}_{s}$ is open. Clearly, this implies that $\mathrm{\Gamma}(U,E)\subset \mathrm{\Gamma}(U,\text{Sp\xe9}(\mathcal{E}))$. To go the other way, note that open sets of $\text{Sp\xe9}(\mathcal{E})$ are the images of continuous maps $U\to E$. An open subset of $\text{Sp\xe9}(\mathcal{E})$ may be written as a union of ${U}_{t}$; ${U}_{ts}\equiv \{{t}_{p},{s}_{p}p\in U\}$. Then, by singlevaluedness of maps, a continuous map $U\to \text{Sp\xe9}(\mathcal{E})$ must map to ${U}_{t}$ for some $t\in \mathrm{\Gamma}(U,E)$, so we have $\mathrm{\Gamma}(U,E)\supset \mathrm{\Gamma}(U,\text{Sp\xe9}(\mathcal{E}))$.
Title  sheaf of sections 

Canonical name  SheafOfSections 
Date of creation  20130322 15:46:36 
Last modified on  20130322 15:46:36 
Owner  guffin (12505) 
Last modified by  guffin (12505) 
Numerical id  7 
Author  guffin (12505) 
Entry type  Definition 
Classification  msc 55R25 
Related topic  VectorBundle 
Defines  Sheaf of Sections 