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Let $X$ be a smooth affine scheme over a finite field $k$. Denote its Witt ring by $W(k)$, and the fraction field of its Witt ring by $K$. Let $F\text{-Isoc}(X)$ denote the category of convergent $F$-isocrystals on $X$. If $X_K$ is a smooth lift of $X$, then there is a forgetful functor $F\text{-Isoc}(X)\longrightarrow \operatorname{VIC}(X_K)$, with values in the category of vector bundles with an integrable connection.

A $K$-rational point $b$ of $X_K$ induces a fibre functor on $\operatorname{VIC}(X_K)$. This works by sending $(\mathcal{E},\nabla)$ to $\mathcal{E}_b$. When people speak about fibre functors for the category $F\text{-Isoc}(X)$, they typically pick $b$ to be a point of the special fibre, and identify the value of the associated fibre functor on an object $(\mathcal{E},\nabla,\Phi)\in F\text{-Isoc}(X)$ to $\mathopen]b\mathclose[^*\mathcal{E}$.

What is the reasoning behind these different conventions? Is it still possible to neutralize the Tannakian category $F\text{-Isoc}(X)$ using a $K$-rational point as in $\operatorname{VIC}(X_K)$? I suspect this is a stylistic preference that has to do with the fact that locally constant sheaves trivialize on contractible neighborhoods (such as a residue disk).

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    $\begingroup$ the I think you have explained it perfectly yourself: the functors you define at two $K$-rational points that reduce to the same residue field point are canonically isomorphic via integrating the connection, so a fiber functor can be specified by giving only the residue point, and in a stylistic convention people prefer this approach. $\endgroup$
    – Will Sawin
    Feb 12 at 16:20
  • $\begingroup$ Thank you very much. $\endgroup$
    – kindasorta
    Feb 12 at 16:28

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