# Tannaka formalism and the étale fundamental group

For quite a while, I have been wondering if there is a general principle/theory that has both Tannaka fundamental groups and étale fundamental groups as a special case.

To elaborate: The theory of the étale fundamental group (more generally of Grothendieck's Galois categories from SGA1, or similarly of the fundamental group of a topos) works like this: Take a set valued functor from the category of finite étale coverings of a scheme satisfying certain axioms, let $\pi_1$ be its automorphism group and you will get an equivalence of categories ( (pro-)finite étale coverings) <-> ( (pro-)finite cont. $\pi_1$-sets).

The Tannaka formalism goes like this: Take a $k$-linear abelian tensor category $\mathbb{T}$ satisfying certain axioms (e.g. the category of finite dim. $k$-representations of an abstract group), and a $k$-Vector space valued tensor functor $F$ (the category with this functor is then called neutral Tannakian), and let $Aut$ be its tensor-automorphism functor, that is the functor that assigns to a $k$-algebra $R$ the set of $R$-linear tensor-automorphisms $F(-)\otimes R$. This will of course be a group-valued functor, and the theory says it's representable by a group scheme $\Pi_1$, such that there is a tensor equivalence of categories $Rep_k(\Pi_1)\cong \mathbb{T}$.

Both theories "describe" under which conditions a given category is the (tensor) category of representations of a group scheme (considering $\pi_1$-sets as "representations on sets" and $\pi_1$ as constant group scheme). Hence the question:

Are both theories special cases of some general concept? (Maybe, inspired by recent questions, the first theory can be thought of as "Tannaka formalism for $k=\mathbb{F}_1$"? :-))

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I took the liberty of adding the F_1 tag. I also had already thought that we have one of the typical analogies here... –  Peter Arndt May 7 '10 at 16:57
Oh, yeah, thanks Pit, I forgot that. I think we briefly discussed the F_1 analogy a few months back, when Wave was still new :) –  Lars May 7 '10 at 17:26
Maybe arxiv.org/abs/0911.0977 is an answer to your question. –  Xandi Tuni May 25 '10 at 17:38

Besides that the theories (étale fundamental group and Tannakian formalism) just formally look alike, there exist actual comparison results between certain étale and Tannakian fundamental groups.

Namely: there is Nori's fundamental group scheme $\pi_1^N(S,s)$, where $S$ is a proper and integral scheme over a field $k$ having a $k$--rational point. It is defined to be the fundamental group of some Tannakian category associated with $S$ (to be precise: The full $\otimes$-subcategory of the category of locally free sheaves on $S$ spanned by the essentially finite sheaves). In the case $k$ is algebraically closed, there is a canonical comparison morphism from Nori's fundamental group to Grothendieck's étale fundamental group, and if moreover $k$ is of characteristic zero this morphism is an isomorphism.

One more comment: The classical Tannaka-Krein duality theorem for compact topological groups (see e.g. Hewitt&Ross vol. II) should presumably be another realisation of the common generalisation you seek.

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