Can one define a version of etale fundamental group which takes into account infinite etale covers? What properties of the usual etale fundamental group would fail for it?

P.S.: here one can find illuminating discussion:

As finite étale maps of complex varieties are equivalent to finite topological covering spaces, this definition begs the question: why have we restricted to finite covering spaces? There are at least two answers to this question, neither of which is new: the first is that the covering spaces of infinite degree may not be algebraic; it is the finite topological covering spaces of a complex analytic space corresponding to a variety that themselves correspond to varieties. The second is that Grothendieck’s étale $π_1$ classifies more than finite covers. It classifies inverse limits of finite étale covering spaces [SGA1, Exp. V.5, e.g., Prop. 5.2]. These inverse limits are the profinite-étale covering spaces we discuss in this paper (see Definition 2.3). Grothendieck’s enlarged fundamental group [SGA 3, Exp. X.6] even classifies some infinite covering spaces that are not profinite-étale.

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    $\begingroup$ What do you mean by infinite étale covers? Étale morphisms are always locally of finite presentation, so the only thing this would allow is certain rising unions (this does lead to a few interesting new covers, but not so many). A more interesting generalisation is to weakly étale morphisms, which leads to the pro-étale topology of Bhatt and Scholze (see also the relevant chapter in the stacks project). But you can never hope to have the exponential $\mathbb C\to\mathbb C^\times$ as a cover, because it is not algebraic. What is it that you're trying to do? $\endgroup$ – R. van Dobben de Bruyn Nov 25 '18 at 18:56
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    $\begingroup$ @R.vanDobbendeBruyn I thought of etale coverings that have fibers of countable cardinality. If I understand correctly, such covers would have to be non-quasi-compact. Are you saying that etale fundamental groups can be defined if we allow such covers and the result is the same? I am sorry if the last question is somewhat imprecise, I am only starting to learn this. $\endgroup$ – man Nov 25 '18 at 19:01
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    $\begingroup$ I think in many contexts there is a natural way to define fundamental groups, but it might not always be exactly the same recipe. For example, Bhatt and Scholze need to look at Noohi groups for their fundamental group. I think with the definition you propose the fundamental group of a projective nodal cubic should be $\mathbb Z$, rather than its usual étale fundamental group $\hat{\mathbb Z}$, as its universal cover is an infinite chain of $\mathbb P^1$s. I believe this has been studied somewhere, but I don't know a reference offhand. Maybe someone will post a genuine answer. $\endgroup$ – R. van Dobben de Bruyn Nov 25 '18 at 19:13
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    $\begingroup$ This is sometimes called the "enlarged fundamental group", see SGA3 Exp. X, 6. For a noetherian scheme which is geometrically unibranch (etale locally irreducible), this group agrees with the profinite fundamental group, and for the nodal rational curve it indeed (as expected by Remy) gives $\mathbf{Z}$ rather than $\hat{\mathbf{Z}}$. See also p. 122 in Artin, Mazur "Etale Homotopy". $\endgroup$ – Piotr Achinger Nov 25 '18 at 19:16

As mentioned in other comments, there is a "pro-étale fundamental group" considered by Bhatt and Scholze. It is introduced in Chapter 7. of their article "The pro-étale topology for schemes". It is a topological group that is a so-called "Noohi group". For a connected (locally topologically noetherian) scheme $X$, it parameterizes the schemes $Y \rightarrow X$ that are étale and satisfy the valuative criterion of properness - the authors call such $Y$ "geometric covers" of $X$. Because we do not assume $Y$ to be of finite type over $X$, the map is not necessarily proper and we get more than just finite covers.

Another words, there is an equivalence of categories between the category of (possibly infinite) discrete sets with a continuous action of $\pi_1^{\mathrm{pro\acute{e}t}}(X)$ and the category of geometric covers of $X$.

The pro-\'etale fundamental group generalizes both the usual étale fundamental group and the "SGA3 fundamental group". The group $\pi_1^{\mathrm{\acute{e}t}}(X)$ is the profinite completion of $\pi_1^{\mathrm{pro\acute{e}t}}(X)$ and $\pi_1^{\mathrm{SGA3}}(X)$ is the pro-discrete completion of $\pi_1^{\mathrm{pro\acute{e}t}}(X)$. The situation is as follows:

If the scheme is normal, all three groups match, i.e. every geometric cover is finite. In the case of the nodal curve there exists an infinite geometric cover. In that case one can show $\pi_1^{\mathrm{pro\acute{e}t}}(X)=\pi_1^{\mathrm{SGA3}}(X)=\mathbb{Z}$ (assuming the base field was algebraically closed). However, for more complicated non-normal schemes $\pi_1^{\mathrm{pro\acute{e}t}}(X)$ gives more than $\pi_1^{\mathrm{SGA3}}(X)$: see Example 7.4.9. of "The pro-étale topology for schemes".


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