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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.


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?


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.)


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!).
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  • $\begingroup$ 3.2 : The open sets you mention are only a basis of the topology 3.3 doesn't even make sens. The correct formula is just $s_x\mapsto x$. (I haven't checked all of the other points). $\endgroup$ Dec 3, 2016 at 21:31
  • $\begingroup$ @GeorgesElencwajg woops, forgot the word 'basis, thank you! but with regards to 3.3, here's partially where I'm confused then, are we assuming that every element is formed of compatible germs? i.e. for some general $(s_x)_{x\in X}$ what do we map it to if there isn't one $f$ such that $f_x=s_x$ for all $x$? $\endgroup$
    – Tim
    Dec 3, 2016 at 21:34
  • $\begingroup$ About 3.3: An element of $\Lambda(\mathcal{F})$, the source of $p$, is just one germ $s_x=f_x$ , not a family of germs. And an element of the target $X$ of $p$ is just a point $x$, not some set of non zero elements. $\endgroup$ Dec 3, 2016 at 21:40
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    $\begingroup$ If you'd like to see that these definitions of sheafification are equivalent, see this post. Alternatively, you could show that they both satisfy the universal property. As for 4), I prefer to write $s|_x$ or $\text{res}^U_x(s)$ for the germ at $x$ of the section $s$ (defined on $U$). $\endgroup$ Dec 4, 2016 at 0:49
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    $\begingroup$ Also, as Caligula says in his answer, the original definition of a sheaf was similar to that of a vector bundle from differential topology. For instance, vector fields on a smooth manifold are sections of the tangent bundle. $\endgroup$ Dec 4, 2016 at 0:51

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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.

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  • $\begingroup$ "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" <- for some reason this just hadn't clicked, but now you've said it it's so obvious! I was getting confused, thinking that the coproduct was always a cofinitely-zero version of the product (like with abelian groups, for example) $\endgroup$
    – Tim
    Dec 4, 2016 at 1:49
  • $\begingroup$ I think most of the confusion arises when you regard the coproduct as a direct sum: the direct sum in his tamest definition is actually a biproduct, that is both product and coprodut, but its elements are usually treated in a product-like fashion. The key point about coproducts (of set-like objects) is that the starting elements are put together but in a way that they don't mix. $\endgroup$
    – Caligula
    Dec 4, 2016 at 11:37
  • $\begingroup$ thank you, very helpful comments! any ideas on the final part (summarised in the edit)? $\endgroup$
    – Tim
    Dec 6, 2016 at 1:58

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