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Tannakian-type reconstruction of etaleétale fundamental group

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

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

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

Suppose $X$ is a scheme. Let $\pi_1^{et}(X)$ be its etale 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 $\pi_1^{et}(X)$ 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

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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Tannakian-type reconstruction of etale fundamental group

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

Suppose $X$ is a scheme. Let $\pi_1^{et}(X)$ be its etale 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 $\pi_1^{et}(X)$ 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?