If $\mathfrak{C}$ is a $k$-linear rigid abelian tensor category with End(1)=$k$(strictly speaking is isomorphic to $k$ as a $k$-algebra), and $k=\bar{k}$, and if $\omega_1$ and $\omega_2$ are two fibre functors (i.e.exact faithful $k$-linear tensor functor to $\text{Vec}_{k}$), then can I say that $\omega_1\cong \omega_2$?

In Deligne's article "catégories tannakiennes" in The Grothendieck Festschrift Volume II, he explained a similar question in a more general context:

$\mathfrak{C}$ is a $k$-linear rigid abelian tensor category with End(1)=$k$, $S$ is a $k$-scheme. $\omega_1$ and $\omega_2$ are two fibre functors over $S$ (with values in the category of locally free sheaves of finite ranks over $S$). Then there is an fpqc covering $T\to S$ such that $\omega_1\cong \omega_2$ over $T$.

This is an easy corollary of the main theorem of Tannakian category (see remarques 1.13 in Deligne's article). From this point of view my question is equivalent to finding a $k$-rational point on some specific component of the representing groupoid. But since the representing groupoid is in general not of finite type, this might be difficult, but should still be possible since this is a component of a groupoid not an arbitrary affine scheme.

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