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## 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 2010 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 2010 at 17:26
Maybe arxiv.org/abs/0911.0977 is an answer to your question. – Xandi Tuni May 25 2010 at 17:38

I think that the answer to your question is precisely the topic of the paper "The fundamental groupoid scheme and applications" by H. Esnault and P. H. Hai, Annales de l'Institut Fourier 2008, see here. I quote from the introduction, on the fourth page : "The purpose of this article is to reconcile the two viewpoints"... The key is Deligne's formalism of nonneutral tannakian categories.

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 It's not quite an answer: They come up with a Tannakian category that contains a lot of information about coverings and then show that the resulting Tannaka group relates to the usual etale fun. group (which is not surprising as both constructions use similar data). My question rather aimed at the underlying theories (Galois categories a la SGA1 vs Tannaka). As far as I remember the nonneutral cats show up because they are interested in nonclosed bases. It certainly is a very nice paper! – Lars May 7 2010 at 20:25 OK, I get it. I remember Niels Borne asking to me once about the possibility to cook up a theory including Galois categories and Tannakian categories under the same formalism. I have to admit that I have no idea how to do that. – Matthieu Romagny May 7 2010 at 20:40

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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