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Noah Schweber said here the following:

Why would you want a notion of sheaf theory for objects more general than topological spaces? Well, the original motivation (to my understanding) was to develop a notion of etale cohomology for schemes; so if you care about schemes, you should care about sites.

Question : In what sense does Grothendieck topologies are in relation with Etale cohomology of Schemes?

Any other explanation for "why would you want a notion of sheaf theory for objects more general than topological spaces?" is also welcome.

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    $\begingroup$ You're implicitly asking several questions: Why care about etale cohomology? Why care about the etale topology independent of etale cohomology? Why care about Grothendieck topologies independently...? A short answer to the first 2 is that for many purposes the Zariski topology, which is a true topology, is too coarse. For example, Zariski gives the "wrong" answer, when you take cohomology with constant coefficients. Grothendieck realized that by generalizing the notion of topology, he could define the etale topology that behaved closer to the analytic topology over $\mathbb{C}$ ... $\endgroup$
    – Donu Arapura
    Commented Jul 26, 2019 at 11:43
  • $\begingroup$ ... for many purposes. But this is very big topic. Let me suggest looking at the book or notes by Milne for more info. $\endgroup$
    – Donu Arapura
    Commented Jul 26, 2019 at 11:45
  • $\begingroup$ @DonuArapura thank you. I will see Milne’s notes.. $\endgroup$
    – Praphulla Koushik
    Commented Jul 26, 2019 at 11:57
  • $\begingroup$ 4 upvotes.. 4 downvotes... If you have any comments to make, please leave comments here :) $\endgroup$
    – Praphulla Koushik
    Commented Jul 26, 2019 at 13:15
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    $\begingroup$ If you're just trying to get some intuition for Grothendieck topologies, perhaps you can start by getting an intuition for locales. This MO answer might help in that regard. There is also an MO question about making the leap from locales to Grothendieck topologies. $\endgroup$
    – Timothy Chow
    Commented Jul 26, 2019 at 14:55

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Etale topology, required to define etale cohomology, is not a topology in the usual sense. It is Grothendieck topology only.

In the category of topological manifolds, an etale cover of $X$ is a surjective local homeomorphism $Y\rightarrow X$. In the etale topology on varieties, this is "morally true" as well with local homeomorphism, replaced by a surjective morphism, defining isomorphism of tangent cones. In the schemes the cones are further replaced by henselisations of local rings...

Note that there are numerous further examples of useful Grothenidieck topologies that are not topologies...

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  • $\begingroup$ I understand that Etale topology is used to define Etale cohomology (I only know this much before posting the question)... I was expecting little more information :)... I could not understand "In the etale topology on varieties, this is "morally true" as well with local homeomorphism, replaced by a surjective morphism, defining isomorphism of tangent cones." $\endgroup$
    – Praphulla Koushik
    Commented Jul 26, 2019 at 10:43
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    $\begingroup$ Doc, if you want more information, ask a question. You should not expect others to read your mind! $\endgroup$
    – Bugs Bunny
    Commented Jul 26, 2019 at 13:40
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    $\begingroup$ The founding fathers knew that the Weil conjecture was "morally correct" as soon as you can create cohomology theory of schemes where Frobenius map behaves like a continuous map of topological spaces. For this they introduced etale maps that are replacements of local homeomorphisms. A map of varieties over alg. cl. field $f:X\rightarrow Y$ is etale if and only if for each $x\in X$ the differential $d_x f$ is an isomorphism. The trick is that at singular points $d_x f$ is defined on tangent cones, not on tangent spaces. $\endgroup$
    – Bugs Bunny
    Commented Jul 26, 2019 at 13:48
  • $\begingroup$ I thank you for the explanation... I agree that my question and above comment are vague... As suggested by Donu Arapura, I will see Milne's notes.. I will come back and ask if I have any specific question :) $\endgroup$
    – Praphulla Koushik
    Commented Jul 26, 2019 at 14:07

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