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I learned the definition of sheaves from Algebraic Geometry by Hartshorne, while reading Serre's GAGA, I was wondering if there was another definition of sheaves. [Here is the link of the English translation of GAGA, my question originates from p.11 n.9], and I copy it down in the following:

Let $X$ be an variety over $\mathbb{C}$, and let $X^h$ be the analytic space associated to $X$. If $F$ is any sheaf on $X$, we will equip the set $F$ with a new topology which makes it into a sheaf on $X^h$; this topology is defined in the following manner: if $\pi : F \to X$ denotes the projection from $F$ to $X$, one injects $F$ into $X^h \times F$ by the map $f \mapsto (\pi(f),f)$, and the topology in question is that induced on $F$ by that of $X^h \times F$. One verifies immediately that one has equipped the set $F$ with the structure of a sheaf on $X^h$, a sheaf which we denote by $F'$. For each $x\in X$, one then has $F'_{x} =F_{x}$; the sheaves $F$ and $F'$ only differ in their topologies ($F'$ is nothing more than the inverse image of $F$ under the continuous map $X^h \to X$).

My questions are:

(1) Is the definition of sheaf here same as that in Hartshorne?

(2) What's the meaning of $ \pi: F \to X$ ?

(3) Why $F'$ is the same as inverse image of $F$ (I understand this also as inverse image sheaf in the sense of Hartshorne)?

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Serre is viewing sheaves as being identified with their associated «espaces étalés». Google should find information on this. This point of view is explained in detail, iirc, in Godement's book, among many other places. The answer to all of your questions will become clear ones you read up on that :) – Mariano Suárez-Alvarez May 25 '12 at 3:20
There is even an exercise in Hartshorne (In section II.1, most likely) that gives this alternative construction. – Dylan Moreland May 25 '12 at 3:22
I agree, but that exercise is about invertible sheaf (if I remember correctly), and also $X$ is a scheme (Here $X^h$ is not a scheme). I guess those should be corresponded in the similar way, but I want to know how do they exactly correspond. – Li Zhan May 25 '12 at 3:33
Oh, I just meant of the éspace etalé. It doesn't completely answer your question, but it's Ex. I.1.13. – Dylan Moreland May 25 '12 at 3:38
And I've goofed up the accents, as I always do. – Dylan Moreland May 25 '12 at 3:51
up vote 10 down vote accepted

1) The definition of "sheaf" as an étalé space was introduced by Cartan in his Seminar (following an original idea of Lazard) and developed by him and his students (Serre was one) in order to clarify Leray's original definition: see here for some context.
That definition was to be used by Bourbaki but since the corresponding chapter did not materialize, they gave Godement permission to use the preliminary draft he had written for an independent book (the one Mariano mentions).
The functor definition in Hartshorne's book was introduced by Grothendieck.

2) In the espace étalé vision, a sheaf over the topological space $X$ is by definition just a topological space $F$ endowed with a local homeomorphism $\pi: F\to X$.
This $\pi$ is the map used by Serre, which you were asking about.

3) If $Y\to X$ is an arbitrary continuous map and if $F$ is a sheaf on $X$, then the fiber product (=pull-back) $Y\times_XF\to Y$ in the category of all topological spaces endowed with its first projection to $Y$ is automayically a sheaf on $Y$, i.e. the map is demonstrably a local homeomorphism.
This sheaf is denoted by $f^{-1}(F)$.
Serre applies this general construction to the case $Y=X^h$, the holomorphization of the complex algebraic variety $X$ .

Of course there is an equivalence of categories, which respects pull-backs, between the espace étalé point of view and the functorial one.
If $\mathcal F$ is a functor-sheaf on $X$, the étalé-sheaf $F$ associated to it has as underlying set the pairs $(x,s_x)$ where $x$ is a point in $X$ and $s_x$ is the germ at $x$ of a section $s\in \mathcal F(U)$ of $\mathcal F$ over some open neighbourhood $U$ of $x$.
It is interesting to note that Hartshorne, who ostensibly eschews étalé spaces, actually uses them: his definition of the structure sheaf on an affine scheme is in terms of étalé spaces.
Since he may not use that concept, he unpacks the definition and translates the étalé terminology into an ad hoc construction .
The exact same analysis applies to his introduction of quasi-coherent sheaves on page 110.
The final irony is that Leray's definition was closer to the functor definition than to the étalé space definition (even though sections were defined on closed subsets of $X$ for Leray)

A mystery tale
In the first ever Séminaire Cartan (1948-49), Cartan gave a definition of sheaf in the exposés (12-17).
Cartan withdrew these 6 exposés in the second edition of that 1948-49 Séminaire and gave a new presentation of sheaf theory in the first exposé of the third, 1950-51, Séminaire: look here.
To my knowledge there is no trace in the world of the withdrawn exposés : rumour has it that Cartan destroyed all the copies of the first edition he could get hold of !

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I think the real final irony is that Grothendieck fibrations bring us back full circle to espaces étalés! – Zhen Lin May 25 '12 at 12:33
@Georges Elencwajg Thank you so much! And the links are very helpful. Just want to clarify one thing, the inverse image sheaf in your answer, as well as in the Serre's papar is $f^{-1}(F)$, rather than $f^{*}(F)$, because no module structure appears in the context. Is my understanding correct? – Li Zhan May 26 '12 at 2:40
Dear Li, yes, your understanding is absolutely correct. – Georges Elencwajg May 26 '12 at 7:26

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