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We know that the category of the (quasi-)coherent sheaves on a smooth projective variety $X$ determine the variety (aka. Gabriel–Rosenberg reconstruction theorem) and the derived category of coherent sheaves on a smooth projective variety $X$ with $\omega_X$ ample or anti-ample determine the variety (aka. Bondal–Orlov reconstruction theorem).

Do we have any known results about whether the category of algebraic vector bundles (or locally free sheaves) $\mathbf{Vect}_a(X)$ can determine a variety? $$\mathbf{Vect}_a(X)\cong \mathbf{Vect}_a(Y)\implies X\cong Y$$ Does it work when the equivalence is between categories, $k$-linear categories or a $k$-linear tensor categories? In particular, we are interested in

Edit: refer to Martin Brandenburg in the comment, the affine case is algebraic and one can show it by corresponding results in algebra.

We know that the category of the (quasi-)coherent sheaves on a smooth projective variety $X$ determine the variety (aka. Gabriel–Rosenberg reconstruction theorem) and the derived category of coherent sheaves on a smooth projective variety $X$ with $\omega_X$ ample or anti-ample determine the variety (aka. Bondal–Orlov reconstruction theorem).

Do we have any known results about whether the category of algebraic vector bundles (or locally free sheaves) $\mathbf{Vect}_a(X)$ can determine a variety? $$\mathbf{Vect}_a(X)\cong \mathbf{Vect}_a(Y)\implies X\cong Y$$ Does it work when the equivalence is between categories, $k$-linear categories or a $k$-linear tensor categories? In particular, we are interested in

Edit: refer to Martin Brandenburg in the comment, the affine case is algebraic and one can show it by corresponding results in algebra.

We know that the category of the (quasi-)coherent sheaves on a smooth projective variety $X$ determine the variety (aka. Gabriel–Rosenberg reconstruction theorem) and the derived category of coherent sheaves on a smooth projective variety $X$ with $\omega_X$ ample or anti-ample determine the variety (aka. Bondal–Orlov reconstruction theorem).

Do we have any known results about whether the category of algebraic vector bundles (or locally free sheaves) $\mathbf{Vect}_a(X)$ can determine a variety? $$\mathbf{Vect}_a(X)\cong \mathbf{Vect}_a(Y)\implies X\cong Y$$ Does it work when the equivalence is between categories, $k$-linear categories or a $k$-linear tensor categories?

Edit: refer to Martin Brandenburg in the comment, the affine case is algebraic and one can show it by corresponding results in algebra.

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reconstruct Reconstruct a variety from the category of locally free sheaves

We know that the category of the (quasi-)coherent sheaves on a smooth projective variety $X$ determine the variety (aka. Gabriel–Rosenberg reconstruction theorem) and the derived category of cherentcoherent sheaves on a smooth projective variety $X$ with $\omega_X$ ample or anti-ample determine the variety (aka. Bondal–Orlov reconstruction theorem).

Do we have any known results about whether the category of algebraic vector bundles (or locally free sheaves) $\textbf{Vect}_a(X)$$\mathbf{Vect}_a(X)$ can determine a variety? $$\textbf{Vect}_a(X)\cong \textbf{Vect}_a(Y)\Rightarrow X\cong Y$$$$\mathbf{Vect}_a(X)\cong \mathbf{Vect}_a(Y)\implies X\cong Y$$ Does it work when the equivalence is between categories, $k$-linear categories or a $k$-linear tensor categories? In particular, we are interested in

Edit: refer to Martin Brandenburg in the comment, the affine case is algebraic and one can show it by corresponding results in algebra.

reconstruct a variety from the category of locally free sheaves

We know that the category of the (quasi-)coherent sheaves on a smooth projective variety $X$ determine the variety (aka. Gabriel–Rosenberg reconstruction theorem) and the derived category of cherent sheaves on a smooth projective variety $X$ with $\omega_X$ ample or anti-ample determine the variety (aka. Bondal–Orlov reconstruction theorem).

Do we have any known results about whether the category of algebraic vector bundles (or locally free sheaves) $\textbf{Vect}_a(X)$ can determine a variety? $$\textbf{Vect}_a(X)\cong \textbf{Vect}_a(Y)\Rightarrow X\cong Y$$ Does it work when the equivalence is between categories, $k$-linear categories or a $k$-linear tensor categories? In particular, we are interested in

Edit: refer to Martin Brandenburg in the comment, the affine case is algebraic and one can show it by corresponding results in algebra.

Reconstruct a variety from the category of locally free sheaves

We know that the category of the (quasi-)coherent sheaves on a smooth projective variety $X$ determine the variety (aka. Gabriel–Rosenberg reconstruction theorem) and the derived category of coherent sheaves on a smooth projective variety $X$ with $\omega_X$ ample or anti-ample determine the variety (aka. Bondal–Orlov reconstruction theorem).

Do we have any known results about whether the category of algebraic vector bundles (or locally free sheaves) $\mathbf{Vect}_a(X)$ can determine a variety? $$\mathbf{Vect}_a(X)\cong \mathbf{Vect}_a(Y)\implies X\cong Y$$ Does it work when the equivalence is between categories, $k$-linear categories or a $k$-linear tensor categories? In particular, we are interested in

Edit: refer to Martin Brandenburg in the comment, the affine case is algebraic and one can show it by corresponding results in algebra.

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We know that the category of the (quasi-)coherent sheaves on a smooth projective variety $X$ determine the variety (aka. Gabriel–Rosenberg reconstruction theorem) and the derived category of cherent sheaves on a smooth projective variety $X$ with $\omega_X$ ample or anti-ample determine the variety (aka. Bondal–Orlov reconstruction theorem).

Do we have any known results about whether the category of algebraic vector bundles (or locally free sheaves) $\textbf{Vect}_a(X)$ can determine a variety? $$\textbf{Vect}_a(X)\cong \textbf{Vect}_a(Y)\Rightarrow X\cong Y$$ Does it work when the equivalence is between categories,  $k$-linear categories or a $k$-linear tensor categories? In particular, we are interested in

Edit: refer to Martin Brandenburg in the comment, the affine case is algebraic and one can show it by corresponding results in algebra.

We know that the category of the (quasi-)coherent sheaves on a smooth projective variety $X$ determine the variety (aka. Gabriel–Rosenberg reconstruction theorem) and the derived category of cherent sheaves on a smooth projective variety $X$ with $\omega_X$ ample or anti-ample determine the variety (aka. Bondal–Orlov reconstruction theorem).

Do we have any known results about whether the category of algebraic vector bundles (or locally free sheaves) $\textbf{Vect}_a(X)$ can determine a variety? $$\textbf{Vect}_a(X)\cong \textbf{Vect}_a(Y)\Rightarrow X\cong Y$$ Does it work when the equivalence is between categories,$k$-linear categories or a $k$-linear tensor categories?

We know that the category of the (quasi-)coherent sheaves on a smooth projective variety $X$ determine the variety (aka. Gabriel–Rosenberg reconstruction theorem) and the derived category of cherent sheaves on a smooth projective variety $X$ with $\omega_X$ ample or anti-ample determine the variety (aka. Bondal–Orlov reconstruction theorem).

Do we have any known results about whether the category of algebraic vector bundles (or locally free sheaves) $\textbf{Vect}_a(X)$ can determine a variety? $$\textbf{Vect}_a(X)\cong \textbf{Vect}_a(Y)\Rightarrow X\cong Y$$ Does it work when the equivalence is between categories,  $k$-linear categories or a $k$-linear tensor categories? In particular, we are interested in

Edit: refer to Martin Brandenburg in the comment, the affine case is algebraic and one can show it by corresponding results in algebra.

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