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

Question

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 ?

Thank you for your time.

Answer 1

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 $\mathcal{F}$.

Answer 2

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.