Compatible germs and the espace étalé

Question

The following is a confusion I'm having that I cannot find answers to anywhere. If this question has already been asked, I apologise, but I couldn't find any answers after some pretty extensive searching. I know this is four questions, but I think really it's just one (i.e. how do I understand these two concepts in light of each other).

After starting to read Vakil's The Rising Sea (which is fantastic, by the way), I have one big confusion. There is the concept of compatible germs and also the concept of the étalé space. They seem very linked, but I can't quite pin down how.

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Edit: question. In the comments and answers there has been plenty of help with the first and last question, so it's really just the two questions in bold that I'm left with now :)

Here's what I've come up with after thinking about this some more, as a more concrete version of the remaining questions (hopefully): we know that taking sections of $p\colon\sqcup_{x\in X}\mathcal{F}_x\to X$ gives us the sheafification of $\mathcal{F}$, as does taking compatible germs. So is there an association between compatible germs and sections $\sigma$ of $p$, e.g. a bijection between the two?

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Let $\mathcal{F}$ be a sheaf (of sets) on a topological space $X$, and $U$ an open set of $X$. Here are some facts/definitions (largely from Vakil's The Rising Sea):

1. The natural map $\varphi\colon\mathcal{F}(U)\to\prod_{x\in U}\mathcal{F}_x$ is injective.
2. An element $(s_x)_{x\in U}\in\prod_{x\in U}\mathcal{F}_x$ is a collection of compatible germs if any of the following equivalent properties hold:
   1. for all $x\in U$ there exists a neighbourhood $U_x\subset U$ and a section $f\in\mathcal{F}(U_x)$ such that for all $y\in U$ we have $s_y=f_y$ (where $f_y$ is the germ of $f$ at $y$);
   2. $(s_x)_{x\in U}$ is the image of a section $f$ under the map $\varphi$ (i.e. the above condition holds but with $U_x=U$ for all $x$).
3. The espace étalé $\Lambda(\mathcal{F})$ associated to $\mathcal{F}$ (or more generally any presheaf) is constructed as follows:
   1. as a set, $\Lambda(\mathcal{F})=\coprod_{x\in X}\mathcal{F}_x$;
   2. as a topological space, the basis for the open sets of $\Lambda(\mathcal{F})$ is given by the $\{V_{U,\,f}\mid U\in\mathsf{Op}(X), f\in\mathcal{F}(U)\}$ where $V_{U,\,f}=(f_x)_{x\in U}$;
   3. as an étalé space, the local homeomorphism is given by projection, i.e. $p\colon\Lambda(\mathcal{F})\to X$ acts as $f_x\mapsto x$.
4. The sheaf $\Gamma(p\colon E\to X)$ associated to a continuous map $p\colon E\to X$ acts on open sets as follows: $\Gamma(p\colon E\to X)(U)=\{\sigma\colon U\to E \mid p\circ\sigma=\mathrm{id}_U\}$.
5. Sheafification, which can be constructed by taking only compatible germs, is just $\Gamma\Lambda$.

(The last fact is emphasised because it seems to me like it should be the thing that ties everything together.)

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Questions:

1. Is all of the above correct?
2. How can we think of compatible germs in terms of the étalé space of a (pre)sheaf? I am almost certain that I have read somewhere it is equivalent to the continuity of the sections $\sigma$ or something similar, but I can't find this anywhere. It seems like a collection of germs is compatible if and only if it is open in $\Lambda(\mathcal{F})$, but this doesn't sound right to me (or at least not the whole picture), especially when you ask...
3. ...why does the germ map $\varphi$ use the product of sets while the étalé space uses the coproduct? Does this mean we can't link the two concepts?
4. Is there a less confusing notation for elements of $\prod_{x\in U}\mathcal{F}(U)$? Writing $(s_x)_{x\in U}$ always looks to me like we take one section $s$ and look at all of its germs (i.e. compatibility!), but writing something like $(s_x^{(x)})_{x\in U}$ (trying to emphasise that the section that we take the germ of varies with the point we're taking the germ at) seems quite cumbersome (and also something I've never seen!).

Answer 1

As about 2, of course with these definitions a basis for the étalé space topology is exactly made up of compatible germs (they just satisfy the second definition).

About 3, I think it has already been cleared up: the elements of $\prod_{x\in X} \mathcal{F}_x$ are sequences of germs $(g_x)_{x\in S}$, while elements in $\coprod_{x\in X}\mathcal{F}_x$ are just germs $g_x$, with a label attached to remind the point they come from.

I think there is no way to avoid the confusion in 4, except maybe to reserve a special notation for compatible germs, but notations are already quite heavy in these topics. In practice, the nature of the object you are considering is made clear from the context.

What is the use of this strange object? Well, it is an older point of view over sheaf theory which retains somewhat more geometric intuition than the current modern definition. You can view a sheaf $\mathcal{F}$ on a topological space $X$ as the triple $(X,\Lambda,p)$, where $\Lambda$ is a topological space and $p:\Lambda \longrightarrow X$ is a local homemorphism; in fact, just define $\Lambda$ as the étalé space and $p$ as above.

It gives some advantages also in topoi theory: see for instance this MO thread.