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?



Tannakian-type reconstruction of étale fundamental group
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).


