Suppose we have a presheaf [tex]\mathcal F[/tex] of abelian groups over
[tex]M[/tex] and pick a point [tex]x[/tex]. On the collection of all
abelian groups defined over some neighbourhood of [tex]x[/tex] (disjoint
union) we put an equivalence relation which identifies
[tex]f\in\mathcal F(U)[/tex] and [tex]g\in\mathcal F(V)[/tex]
precisely if there is some open [tex]W[/tex] in the intersection where
[tex]f[/tex] and [tex]g[/tex] coincide. (or more precisely, their
restrictions coincide). The set of equivalence classes turns out to be
an Abelian group [tex]\mathcal F_x[/tex] called the *stalk* of the
presheaf [tex]\mathcal F[/tex] at [tex]x[/tex].

So, with more fluff introduced, the stalk is all the elements in the
presheaf that are defined above any neighbourhood of the point, and
counted as the same if they seem to be.

For an open set [tex]U[/tex] and a point [tex]x\in U[/tex] there is a
canonical group morphism [tex]\rho_x:\mathcal F(U)\to\mathcal
F_x[/tex] which sends an element [tex]f\in\mathcal F(U)[/tex] to its
equivalence class. This image is the germ of [tex]f[/tex] at
[tex]x[/tex].

## Example

Let [tex]\mathcal O[/tex] be the presheaf of holomorphic function on
[tex]\mathbb C[/tex]. Then [tex]\mathcal O_{z_0}[/tex] is precisely
the set of all convergent power series of the form
[tex]\sum_{n=0}^\infty c_n(z-z_0)^n[/tex] for complex
[tex]c_n[/tex]. The germ of some [tex]f[/tex] defined on a
neighbourhood of [tex]z_0[/tex] is precisely the Taylor series around
that point.

## Example

Take [tex]\mathbb Z[/tex] the constant presheaf where over each open
set, the additive group of all integers hovers. Then we have for the
construction of the stalk a disjoint union of copies of [tex]\mathbb
Z[/tex] indiced by all the neighbourhoods of the point. Since all the
neighbourhoods contain [tex]x[/tex] they all have a non-empty
intersection, on which elements agree if they have the same value. So
[tex]\mathcal F_x=\mathbb Z[/tex] for any point [tex]x[/tex]. The
germ of [tex]i\in\mathbb Z=\mathcal F(U)[/tex] at [tex]x[/tex] is
[tex]i\in\mathbb Z=\mathcal F_x[/tex].

We define exact sequences by looking at the induced maps of the stalks.
A sequence of sheave morphisms is exact if for every point on the
underlying space, the corresponding induced stalk map sequences are
exact.

## Example

We can construct a morphism of sheaves of abelian groups from
[tex]\mathbb C[/tex] to [tex]\mathcal O[/tex] by sending
[tex]c\in\mathbb C(U)[/tex] to the function [tex]f(z)=c[/tex] in
[tex]\mathcal O(U)[/tex]. For some point [tex]z_0[/tex], this induces
a stalk map that takes [tex]c[/tex] in [tex]\mathbb C_{z_0}[/tex] and
sends it to the function defined by the convergent power series
[tex]c(z-z_0)^0=c[/tex]. Regardless of [tex]z_0[/tex], this map is
obviously an injection, and so this map is a monomorphism.

### Space étalé and sheafifications

Given a presheaf [tex]\mathcal F[/tex], we construct the set
[tex]|\mathcal F|[/tex] as the disjoint union of all stalks
[tex]\mathcal F_x[/tex]. There's an natural projection
[tex]\pi:f_x\mapsto x[/tex] down to the underlying topological space.
We can introduce a topology on [tex]|\mathcal F|[/tex] by
constructing, for each open set [tex]U\subseteq M[/tex] and each
element [tex]f\in\mathcal F(U)[/tex], the set
[tex][U,f]=\{\rho_x(f)\mid x\inU\}\subseteq|\mathcal F|[/tex]
and take these sets to be a basis of our topology. So an open set is
given from the [tex][U,f][/tex] by a sequence of (possibly infinite)
unions and finite intersections.

This turns out to be a covering of [tex]\mathcal F[/tex], i.e. each
[tex]e\in|\mathcal F|[/tex] has an open neighbourhood which is
homeomorphic to its image under the projection [tex]\pi[/tex]. We call
the topological space [tex]|\mathcal F|[/tex] with this topology the
space étalé of the presheaf [tex]\mathcal F[/tex].

Using the space étalé, then, we can construct a canonical sheaf. A
continuous section of a covering space [tex]\pi\colon\mathcal F\to
M[/tex] over a subset [tex]U\subseteq M[/tex] is a continuous map
[tex]\sigma\colon U\to|\mathcal F|[/tex] such that
[tex]\pi\circ\sigma=\mathbb 1[/tex].

## Example

[tex]\mathbb R[/tex] is a covering space of the circle (viewed as the
interval [0,1] with 0 identified with 1) with the projection
[tex]x\mapsto x\pmod1[/tex]. A continuous section of the upper open
halfcircle is a map [tex]\sigma_n\colon x\mapsto x+n[/tex]. Indeed,
[tex]\pi\circ\sigma_n[/tex] takes some point [tex]x[/tex] to
[tex]x+n[/tex] and then to [tex]x+n\pmod1[/tex]

Now, let [tex]\Gamma(U,|\mathcal F|)[/tex] denote the set of all
continuous sections of [tex]|\mathcal F|[/tex] over [tex]U[/tex].
This ends up in the same category that [tex]\mathcal F[/tex] goes to.
From this, we then define a sheaf [tex]\hat{\mathcal F}[/tex] by
setting [tex]\hat{\mathcal F}(U)=\Gamma(U,|\mathcal F|)[/tex], and
letting restriction be the usual restriction of maps. This gives us a
functor from presheaves to sheaves called *sheafification*.

## Example

We already saw that [tex]\mathbb Z_{\text{const}}[/tex] assigning the
group [tex]\mathbb Z[/tex] to each open subset [tex]U[/tex] of a
manifold [tex]M[/tex] and with all restrictions being identity morphisms
is not a sheaf. What happens if we sheafify? First, we need to construct
our space étalé. This is the disjoint union of all stalks. A stalk over
a point [tex]x[/tex] is the quotient of the disjoint union of all
[tex]\mathbb Z_{\text{const}}(U)[/tex] for [tex]x\in U[/tex] with
the equivalence relation that identifies [tex](n,U)\equiv(m,V)[/tex] if
there is some [tex]W\subset U\cap V[/tex] such that
[tex]n|_W=m|_W[/tex]. Now, all neighbourhoods of [tex]x[/tex]
intersect in some open subset, and since all restrictions are
identities, we identify [tex](n,U)[/tex] with [tex](n,V)[/tex] for all
pairs of open neighbourhoods [tex]U[/tex],[tex]V[/tex]; so in end-effect
all stalks are isomorphic, as groups, to [tex]\mathbb Z[/tex].

The space étalé is thus the disjoint union for all points [tex]x\in
M[/tex] of copies of [tex]\mathbb Z[/tex]; i.e. the set of ordered
pairs on the form [tex](x,n)[/tex] for [tex]x\in M[/tex] and
[tex]n\in\mathbb Z[/tex]. Finally, our sheafified sheaf assigns to
each open subset [tex]U\subseteq M[/tex] the group of sections
[tex]\Gamma(U,|\mathbb Z_{\text{const}}|)[/tex], thus a point
there is a function [tex]U\to U\times\mathbb Z[/tex] such that
[tex]x\mapsto n_x[/tex] for some [tex]n_x\in\mathbb Z[/tex]. Since
this map has to be continuous, the map is constant on neighbourhoods.
And thus we recover the sheaf of locally constant maps for the constant
sheaf; just as exhibited earlier.

### Kernels and quotients

Let [tex]\tau:\mathcal A\to\mathcal B[/tex] be a map of sheaves. For
any open [tex]U\subseteq M[/tex], we define [tex]\mathcal
K(U)=\ker\tau_U\colon\mathcal A(U)\to\mathcal B(U)[/tex]. The
sheaf [tex]\mathcal K[/tex] formed by these groups together with the
induced morphisms from is called the kernel of the sheaf map.

The quotient of a sheaf map is formed almost the same way - pointwise
components are formed as expected, but the presheaf thus formed need not
be a sheaf. So we sheafify it.