# What is the difference between the stalk of a sheaf at a point and the section of said sheaf over that point?

I'm asking this question with a particular regard to the geometric aspect(s) of the difference(s).

Let there be a sheaf $$\mathcal{F}: Top(X)^{op} \rightarrow \mathfrak{Ab}$$, with $$Top(X)$$ being the category whose objects are open subsets of X, and whose morphisms are inclusions, and $$\mathfrak{Ab}$$ is the usual category of abelian groups. What then, is the difference between the section $$\mathcal{F}(P)$$ and the stalk $$\mathcal{F}_P$$, where $$P$$ is some point in the topological space $$X$$. I suspect that they might be the same, perhaps up to isomorphism, but I don't have a good way of confirming this. Would I be correct in suspecting this, and even if not, why ?

• F(P) does not make any sense, your functor is defined on open sets not points in these. Commented Jul 21, 2019 at 17:19
• I'm rather new to sheaves but aren't points also open sets ? Commented Jul 21, 2019 at 17:20
• @DatMinhHa In most topological spaces one come across, one-point subsets aren't open. Commented Jul 21, 2019 at 17:22
• you even wrote that the objects in your category are open sets of some topological space. Are you sure you understand the definition of a functor? For every open subset $U$ of $X$, $F(U)$ is an abelian group. The stalk at a point $p$ in $X$ is obtained by taking the dircet limit over all open sets containing $p$.en.wikipedia.org/wiki/Stalk_(sheaf) Commented Jul 21, 2019 at 17:25
• @Riquelme Ah right I see where I was wrong now. Thank you. Commented Jul 21, 2019 at 17:27

Singletons are usually not open and therefore we cannot plug these into our sheaf. This means that we sort of have to approximate $$\mathcal{F}$$ at a singleton $$\lbrace x \rbrace$$. This is exatly what the stalk is. The stalk is given by $$\mathcal{F}_x = \varinjlim \mathcal{F}(U),$$ where the limit runs over $$x \in U$$. That means we approximate the sheaf $$\mathcal{F}$$ at the point $$x$$ by looking at smaller and smaller open neighborhoods of $$x$$ under $$\mathcal{F}$$.
As is already explained in the comments: It usually doesn't make sense to speak about $$\mathcal F(P)$$ for a point $$P$$ - that is: whenever $$\{P\}$$ is $$\textit{not}$$ open in $$X$$.
Note however that there always is an inclusion $$\mathcal F(U) \longrightarrow \prod\limits_{P \in U}\mathcal F_P, s \longmapsto (s_P)_{P \in U}$$ for any open set $$U \subseteq X$$. So arbitrary sections of a sheaf are still determined by their stalks and one uses these properties all the time when working with sheaves.
Related things you could want to take a look into are $$\textit{sheafification}$$ of presheaves and the $$\textit{espace étalé}$$ way of viewing sheaves.
Also note that $$\mathcal F(P) \overset{\mathrm{def}}{=} \kappa_{X}(P) \otimes_{\mathcal O_{P}} \mathcal F_{P}$$ is a well established notation for the $$\textit{geometric fibre}$$ of a sheaf of modules over a locally ringed space.