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My definition of a site is a pair $(\mathcal{C},\DeclareMathOperator{\Cov}{Cov}\Cov (\mathcal{C}))$ where $\mathcal{C}$ is a category and $\Cov (\mathcal{C})$ is a set consisting of elements of the form $\{U_i\rightarrow U\}_{i\in I}$, and $\Cov (\mathcal{C})$ satisfies three axioms which I gonna omit since this is not the problem.

I have searched a few books about the definition of an big/small étale/fppf/... site.

For example, a small étale site of a scheme $X$, $X_{ét}$, has the underlying category whose objects are étale morphisms $U\rightarrow X$, and the covering set $\Cov (X_{ét})$ consists of all surjective families of étale morphisms $\{f_i:U_i\rightarrow U\}_{i\in I}$, i.e. $f_i$ is étale and $U=\cup_i f_i(U_i)$.

But those books don't mention that why the "covering set" is a set.

"Introduction to étale cohomology" by Gunter Tamme, Manfred Kolster did that.

"Étale Cohomology" by James Milne did not mention the definition of an arbitrary site, he just define a big/small $E$-site for a class of $E$-morphisms of schemes.

"Lecture notes on étale cohomology" by James Milne on the website see this link, he defines an arbitrary site s.t. for each object $U$, we have a distinguished set of covering families, and this is a system of coverings, didn't say the whole thing is a set.

Stacks project, uses a confusing way to construct the underlying scheme of a small/big étale/fppf/... site, so I have to ignore it.

SGA is in French which I have an incredibly slow reading speed, so ignored.

Which definition of site (arbitrarily) and small/big étale/fppf/... site should I accept?

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Let $X$ be a scheme. If you take the underlying category of $X_{\acute{e}t}$ to be the category of étale morphisms $U \to X$, then you have already lost because then the objects of $X_{\acute{e}t}$ form a proper class and not a set, hence a fortiori the coverings of $X_{\acute{e}t}$ form a proper class. (Exercise.)

This is why in the Stacks project, a large enough collection of schemes is chosen a priori and then you work with that. Something similar is done in SGA 4. In SGA 4 the authors allow themselves to use universes; using universes what you do can be succinctly formulated as follows: choose the a universe $V$ containing $X$. Then work only with $U$ which are in $V$ and with coverings which are in $V$ as well (a covering is itself a set, hence you can ask that it be in $V$).

Of course the site $X_{\acute{e}t}$ depends on the choice of $V$, but it is very easy to see that the topos doesn't depend on the choice of $V$. This topos is called the small étale topos of $X$.

None of this is confusing. Only a small amount of set theory is needed to make this rigorous; if you ever ended up talking to somebody who works in set theory, they'd laugh at the pitiably small sets you need to define the small étale topos of $X$. Cheers!

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    $\begingroup$ This is wrong. The category of étale morphisms over a fixed scheme is a subcategory of the category of (locally) finitely presented morphisms, which is essentially small by a cardinality argument. The rest of the answer is also wrong, but for more subtle reasons. $\endgroup$ – Harry Gindi May 17 at 21:59
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    $\begingroup$ Just to give an explicit example, if $X$ is a field in char 0, then the etale site is basically just finite field extensions and finitely many copies of them. This is certainly bounded by the cardinality of the field itself. $\endgroup$ – Asvin May 17 at 22:01
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    $\begingroup$ The comments objecting to this answer seem to overlook that the question is not about essential smallness but about actual smallness. Of course, an essentially small category might as well be replaced by an equivalent small category, and that's what this answer is doing. There is, however, useful information in the comments, namely that one doesn't need inaccessible cardinals (Grothendieck universes) to produce equivalent small sites; that might be important for non-set-theorists who think inaccessibles are a big deal. $\endgroup$ – Andreas Blass May 17 at 23:39
  • $\begingroup$ @AndreasBlass I think, and let me know if I'm mistaken, it is still important for more than just the non-set-theorists. Even if you use universes, if you don't first replace the essentially small site with a small site, it will be of proper class size (since up to equality, there is a proper class even of finitely generated polynomial rings over a fixed base ring, while up to isomorphism, it is a countable set). I don't see how using universes can circumvent this particular feature, at least from the POV of a material set theory. $\endgroup$ – Harry Gindi May 19 at 0:02
  • $\begingroup$ @HarryGindi Universes provide a general method for replacing essentially small sites like the ones in the question with small ones. Once you've fixed a universe containing the data of the problem (e.g., a ring), all the objects in your large but essentially small site (e.g., finitely generated algebras ovr that ring) will be isomorphic to ones in that universe. So the objects in that universe constitute a small site equivalent to the original essentially small one. $\endgroup$ – Andreas Blass May 19 at 1:12
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This is an explanation of the answer by dario above, but I don't have the reputation points to leave a comment. Being an etale morphism is local on the source. Given a scheme $X$ and a set $I$ the morphism $\coprod_{i \in I} X \to X$ is an etale morphism. This is the definition in all standard references (even Wikipedia has this definition).

If you want to do etale cohomology of non-separated and non-Noetherian schemes, you have to allow non-quasi-compact etale morphisms of schemes otherwise you may not have enough coverings. For example consider the infinite dimensional affine space with origin doubled (which is an interesting thing to consider even for "Noetherian" people) and try to make an interesting etale covering when you require the morphisms to be quasi-compact.

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