I learned the definition of a sheaf from Hartshorne---that is, as a (co-)functor from the category of open sets of a topological space (with morphisms given by inclusions) to, say, the category of sets. While fairly abstract at the outset, this seems to be (to me) an intuitive view; in particular, all of the manipulations and constructions with sheaves fit nicely into this schema.

I know that the older view of a sheaf on $X$ was to consider it as a triple $$ (E, X, \pi) $$ where $\pi : E \to X$ is a local homeomorphism, and so that the "sheaf of sections" of this map $\pi$ is the sheaf in the functorial sense described above.

This view makes much less sense to me, but I have to wonder if that is simply due to my having learned it second. However, it also makes me wonder if I am missing something, and so my question is as follows.

What are some (edit:) specific benefits of viewing a sheaf in this sense? What is gained by considering a sheaf as the espace étalé over $X$?

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    $\begingroup$ I guess it's more geometric, I think it was the original definition, it's similar to sections of bundles. If I recall correctly pullbacks are easier to define in this setup. But I agree with you, I much prefer the algebraic way of dealing with sheaves. $\endgroup$ May 7 '12 at 21:09
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    $\begingroup$ Isn't this a special case of the more general phenomenon that it's best to know as many ways as possible of thinking about a concept, not least because some questions become trivial when thought about using one picture and they're less clear with another. For example if you were to be asking "I understand matrices so what is the point of linear maps?" or "I understand linear maps so what is the point of matrices?" then in both cases I'm sure you can see a good answer. Why not just extend this logic to the sheaf situation? $\endgroup$ May 7 '12 at 23:08
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    $\begingroup$ It's also a special case of a phenomenon less general than Kevin's: in many contexts, there is a correspondence between "objects over $X$" and "families indexed by $X$". The simplest instance is in set theory: a set over $X$ (that is, a function $E \to X$) is essentially the same thing as a family $(E_x)_{x \in X}$ of sets. Another example involves fibrations of categories vs. Cat-valued functors. $\endgroup$ May 7 '12 at 23:54
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    $\begingroup$ Fair enough, I wasn't really quite as specific in my phrasing: I want to know specific benefits. e.g. for linear maps instead of matrices, we get motivation for the formula for matrix multiplication, while for matrices we are given an efficient computation method. $\endgroup$
    – Simon Rose
    May 8 '12 at 0:41
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    $\begingroup$ This seems closely related: mathoverflow.net/questions/6477/… $\endgroup$ May 8 '12 at 9:55

Let me expand on Yosemite Sam's comment. Pullbacks are indeed easier to define if you view a sheaf as a local homeomorphism. On the other hand, pushforwards are easier to define if you view a sheaf as a set-valued functor.

Suppose we have a continuous map $f: X \to Y$ of topological spaces.

Given a sheaf $F$ on $Y$, viewed as a local homeomorphism $\pi: F \to Y$, we can simply pull $\pi$ back along $f$ to obtain a map into $X$; it is easily shown to be a local homeomorphism too. This is the pullback sheaf $f^* F$.

On the other hand, given a sheaf $E$ on $X$, viewed as a functor $\mathrm{Open}(X)^{op} \to \mathbf{Set}$ (where $\mathrm{Open}(X)$ is the poset of open subsets of $X$), we can simply compose $E$ with the functor $\mathrm{Open}(Y) \to \mathrm{Open}(X)$ that takes inverse images along $f$. This gives a set-valued functor on $\mathrm{Open}(Y)$; it is easily shown to be a sheaf too. This is the pushforward sheaf $f_* F$.

So, there are advantages to proving the equivalence between the two definitions early on.

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    $\begingroup$ As an exercise I recently proved the equivalence of categories between ètalè spaces over X and sheaves of sets over X, which essentially also entails proving the sheafification functor and its properties. My question for you is, for sheaves of groups, rings, modules etc. is there always a ‘modified’ ètalè space for which the equivalence of categories holds? I have seen Serre’s treatment of the case of groups in FAC, but do we always have the sheaf/ètalè correspondence for sheaves that takes values in an abelian category? $\endgroup$
    – Prince M
    May 28 '18 at 3:15

To me the obvious answer involves sheafification of a presheaf. If you look at the construction of the associated sheaf to a presheaf in, say, Hartshorne it goes through the étalé space construction without specifically telling you, and to me it makes the construction somewhat unmotivated.

Namely, if $P$ is a presheaf on $X$, then taking the stalk $P_x$ at each point of $X$ gives you an $X$-indexed set, or as Tom would say above, a set over $X$. One can then define a topology on $\biguplus_{x\in X}P_x$ so that the natural projection $\biguplus_{x\in X}P_x\to X$ is a local homeomorphism in the obvious way namely if $U$ is a neighborhood in $X$ and $s\in P(U)$, then $(s,U) = \lbrace germ_x(s)\mid x\in U\rbrace$ is a basic neighborhood. This topology immediately makes $(s,U)$ homeomorphic to $U$ and makes $s$ a section over $U$ via $x\mapsto germ_x(s)$ for $x\in U$. The sheaf of sections of $p$ is the associated sheaf of $P$. I find this construction completely unmotivated without going through étalé spaces.

Added. Another good reason is it is convenient for defining actions of a topological groupoid on a sheaf. If $G=(G_0,G_1)$ is a groupoid, a $G$-sheaf is an étalé space $p:X\to G_0$ over $G_0$ together with an action map $G_1\times_{d,p} X\to X$ satisfying obvious axioms. This is more difficult to phrase in the sheaf as a functor language. A theorem of Joyal and Tierny says that every Grothendieck topos is equivalent to the topos of sheaves on a localic groupoid.

Additional additions From the étalé space point of view it is clear that covering spaces are indeed elements of the topos $Sh(X)$ of sheaves on $X$ and that the fundamental group of $Sh(X)$ (in the sense of Barr and Diaconescu) is the usual fundamental group of $X$ if $X$ is locally simply connected.

Of course it is not hard to see that covering spaces correspond to locally constant sheaves but I don't think this is the way people think about covering spaces.

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    $\begingroup$ Oh, that's a good point (about the sheafification functor). I hadn't really thought of the details of that construction, but it does make it more clear, doesn't it? $\endgroup$
    – Simon Rose
    May 8 '12 at 0:46

One advantage is that it gives you a geometric representation for slice topoi of sheaves over a space:

Given a topos $E$, $E$ is equivalent to the full-subcategory of $Top/E$, the category of topoi over $E$ consisting of etale morphisms of topoi. The equivalence sends an element $e \in E$ to the morphism $E/e \to E$ where $E/e$ is the slice topos.

Topoi are generalizations of spaces, and to view a space as a topos, we send a space $X$ to its topos of sheaves $Sh(X)$. Etale geometric morphisms $Sh(X) \to Sh(Y)$ are in bijection with local homeomorphisms $X \to Y$ (when Y is sober.). So, this means that if $F \in Sh(X)$, the local homeomorphism $E(F) \to X$ which corresponds to $F,$ viewed as a map of topoi is nothing but the etale geometric morphism $Sh(X)/F \to Sh(X)$. In particular, this implies that $Sh(X)/F \cong Sh(E(F)).$


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