TITLE: Is there a ``path'' between any two fiber functors over the same field in Tannakian formalism?
QUESTION [8 upvotes]: I will take the approach of this question: Tannaka formalism and the étale fundamental group
and think of the etale fundamental group as Tannakian formalism for $\mathbb{F}_1$. Then our "Tannakian category" is the category of finite etale covers, and each fiber functor is a functor from this category to $Sets$ (thought of as finite dimensional spaces over $\mathbb{F}_1$).
For the etale fundamental group, it is true that for any two fiber functors (given by two different geometric points) there is a ``path'' between them. Meaning: there is a natural isomorphism between these two functors.
My question is whether this ``independence of the basepoint'' result applies to Tannakian formalism as well:
Is it true that for any two fiber functors $H_1, H_2: \mathcal{C}\rightarrow Vec_K$, there is a natural isomorphism $H_1 \cong H_2$?

REPLY [8 votes]: The obstruction to the existence of such an isomorphism is a (bi)torsor, that has been studied in various real-life situations. An example extracted from 
On the relation between Nori Motives and Kontsevich Periods
Annette Huber, Stefan Müller-Stach
http://arxiv.org/abs/1105.0865
"As already explained by Kontsevich, singular cohomology and algebraic de Rham
cohomology are both fiber functors on the same Tannaka category of motives. By
general Tannaka formalism, there is a pro-algebraic torsor of isomorphisms between them. The period pairing is nothing but a complex point of this torsor."
Basically, by tannaka duality, you can build a counter-example out of any couple of non-isomorphic objects of a gerbe.