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I am interested in knowing which natural categories of its representations the étale fundamental group of a scheme can be recovered from.

Suppose $X$ is a scheme. Let $\pi_1^\text{ét}(X)$ be its étale fundamental group. Thinking along the lines of Tannakian reconstruction of a pro-algebraic group from the category of its representations over some field $k$ as the automorphism group of the forgetful functor to the category of $k$-vector spaces, I am wondering if there is a natural category of objects associated to $X$ (e.g., sheaves) on which $\pi_1^\text{ét}(X)$ acts and from which it can be reconstructed in a manner analogous to Tannakian reconstruction.

Also, what role does the choice of the field $k$ play?

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    $\begingroup$ You can consider the Tannakian category of "finite vector bundles" in the sense of Nori, see his paper "On the representations of the fundamental group" in Compositio. His results have been extended in many directions, you can look at the Mathscinet references to this paper (or Google Scholar). $\endgroup$
    – naf
    Commented Jul 22, 2022 at 3:44
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    $\begingroup$ I’m a little confused by the question — isn’t the definition of the étale fundamental group an answer? I.e. the étale fundamental group is “the thing that acts on geometric fibers of finite étale covers,” exactly analogous to the Tannakian setting. $\endgroup$
    – Daniel Litt
    Commented Jul 22, 2022 at 3:55
  • $\begingroup$ @naf you should convert your comment into an answer $\endgroup$
    – Niels
    Commented Jul 22, 2022 at 16:20

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Yes, this is possible. Since finite groups are algebraic groups, pro-finite groups are pro-algebraic groups. So one can recover $\pi_1$ in exactly the way you say from the category of algebraic representations of $\pi_1$, i.e. finite-dimensional representations that are continuous for the discrete topology of $k$. (These all factor through a finite group).

In geometric terms, these are locally-constant finite-rank sheaves of $k$-vector spaces in the étale topology, if $X$ is normal - if $X$ is not normal there may be locally constant sheaves where the $\pi_1$-action is not continuous.

For these purposes, the choice of field $k$ doesn't matter very much.

But it's usually more interesting to study a different construction, where you consider finite-dimensional representations of $\pi_1$ that are continuous for the $\ell$-adic topology on $k$ for some $\ell$-adic field $k$. These form a Tannakian category, and the resulting pro-algebraic group is usually not pro-finite, and thus much larger than $\pi_1$. But its component group recovers $\pi_1$, basically because the representations of the component group are the category we discussed above.

Geometrically, this is (at least for $X$ normal) the category of lisse $\ell$-adic sheaves defined the usual (slightly complicated) way.

For this construction, the field matters a lot - it must be $\ell$-adic, and we get very different groups for different primes $\ell$ (though there are known to be some relationships between their representations, at least for schemes over finite fields).

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  • $\begingroup$ I am not sure how this algebraic structure relates to the "algebraic structure" on X. It seems to be constructed out of the (pro)étale topos of X, or the Galois category à la Barwick–Haine, which seems to contain only "topological" data. $\endgroup$
    – Z. M
    Commented Jul 22, 2022 at 3:41
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    $\begingroup$ @Z.M One doesn't have to go to topos theory or pyknotic stuff - it's just the category of finite étale covers. Yes, as long as one considers finite étale covers to be purely topological (they're not always determined by the Zariski topology!), the étale fundamental group is a purely topological notion. When I say "algebraic", I literally mean as an algebraic group. $\endgroup$
    – Will Sawin
    Commented Jul 22, 2022 at 10:46
  • $\begingroup$ in your second paragraph when you mention locally constant sheaves you should specify the topology else it doesn't really make sense $\endgroup$
    – Niels
    Commented Jul 22, 2022 at 16:12
  • $\begingroup$ @WillSawin, What is a good reference for the reconstruction from lisse $l$-adic sheaves? In the last paragraph, do you mean that the Tannakian pro-algebraic groups for different $l$ are different, but their component groups are always isomorphic to $\pi_1$? Thanks! $\endgroup$
    – Vite
    Commented Jul 22, 2022 at 18:28
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    $\begingroup$ @Vite I don't know a reference (maybe the Nori paper naf mentions discusses it), but it's not hard to prove. I'm not saying the Tannakian groups are necessarily different, but more that it would be very difficult to prove they are isomorphic (while it's easy to prove the component groups are isomorphic.) Conjecturally, a quotient of the $\ell$-adic Tannakian group should be the base change to $\mathbb Q_\ell$ of the motivic Galois group of the category of locally constant motives, whatever that means. $\endgroup$
    – Will Sawin
    Commented Jul 22, 2022 at 18:55

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