Could not include topos theory - contents
Recall that a point $x$ of a topos $E$ is a geometric morphism
The stalk at $x$ of an object $e \in E$ is the image of $e$ under the corresponding inverse image morphism
i.e.
If $E$ is the category of sheaves on the category of open subsets $Op(X)$ of a topological space $X$
then the topos points of $E$ come precisely from the ordinary points
of the space $X$, where the direct image morphism
sends every set to the sheaf which is the constant functor on that set. By the general Kan extension formula for the inverse image (see there) one finds in this case for any sheaf $F \in Sh(X)$ the stalk
So for sheaves on (open subsets of) topological spaces the stalk at a given point is the colimit over all values of the sheaf on open subsets containing this point.
By the general definition of colimits in Set described at limits and colimits by example, the elements in this colimit can in turn be described as equivalence classes represented pairs $(z, V)$ with $x \in V$ $z \in F(V)$, where the equivalence relation says that two such pairs $(z_1, V_1)$ and $(z_2, V_2)$ coincide if there is a third pair $(z,U)$ with $U \subset V_1$ and $U \subset V_2$ such that $z = z_1|_U = z_2|_U$.
for $F = C(-)$ a sheaf of functions on $X$, such an equivalence class, hence such an element in a stalk of $F$ is called a function germ.
For $E$ a topos with enough points, the behaviour of morphisms $f : A \to B$ in $E$ can be tested on stalks:
A morphism $f : A \to B$ of sheaves on $X$ is a
resp. epimorphism
resp. isomorphism
if and only if every induced map of stalk sets $stalk_x(f) : stalk_x(A) \to stalk_x(B)$ is, for all $x \in X$
The statement for isomorphisms follows from the identification of sheaves with etale spaces (e.g. section II, 6, corollary 3 in MacLane-Moerdijk, Sheaves in Geometry and Logic). The statement for epimorphisms/monomorphisms is proposition 6 there.
Let $X$ be a smooth manifold and let $\Omega^n(X)$ and $Z^{n+1}(X)$ be the sheaves of differential $n$-forms and that of closed differential $(n+1)$-forms on $X$, respectively, for some $n \in \mathbb{N}$. Let
be the morphism of sheaves that is given on each open subset by the deRham differential.
Then:
for $U \subset X$ the map $d_U : \Omega^n(U) \to Z^{n+1}(U)$ need not be epi, since not every closed form is exact;
but by the Poincare lemma every closed form is locally exact, so that for each $x \in X$ the map of stalks $d_x : stalk_x(\Omega^n(X)) \to stalk_x(Z^{n+1}(X))$ is an epimorphism.
Accordingly, the morphism $d : \Omega^n(X) \to Z^{n+1}(X)$ is an epimorphism of sheaves.
This kind of example plays a crucial role in the computation of abelian sheaf cohomology, see the examples listed there.
For a locally ringed topos with structure sheaf $\mathcal{O}$, the stalk of the multiplicative group $\mathbb{G}_m$ at a point $x$ is the multiplicative group $\mathcal{O}_x^\times$ in the stalk local ring of the structure sheaf. (e.g. Milne, example 6.13)
Examples of sequences of local structures
geometry | point | first order infinitesimal | $\subset$ | formal = arbitrary order infinitesimal | $\subset$ | local = stalkwise | $\subset$ | finite |
---|---|---|---|---|---|---|---|---|
$\leftarrow$ differentiation | integration $\to$ | |||||||
smooth functions | derivative | Taylor series | germ | smooth function | ||||
curve (path) | tangent vector | jet | germ of curve | curve | ||||
smooth space | infinitesimal neighbourhood | formal neighbourhood | open neighbourhood | |||||
function algebra | square-0 ring extension | nilpotent ring extension/formal completion | ring extension | |||||
arithmetic geometry | $\mathbb{F}_p$ finite field | $\mathbb{Z}_p$ p-adic integers | $\mathbb{Z}_{(p)}$ localization at (p) | $\mathbb{Z}$ integers | ||||
Lie theory | Lie algebra | formal group | local Lie group | Lie group | ||||
symplectic geometry | Poisson manifold | formal deformation quantization | local strict deformation quantization | strict deformation quantization |