Tannaka formalism and the étale fundamental group

Question

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 $k=\mathbb{F}_1$"? :-))

Answer 1

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.

Answer 2

I stumbled upon this very old question and I had some observations to make, I hope they are not considered entirely unuseful after all this time. I want to highlight two points.

1. Even if we restrict to profinite groups, Tannakian categories contain strictly more information than Galois categories.

2. There is (almost) no reason to search for a concept more general than Tannakian categories.

If $k$ is not separably closed, the classical étale fundamental group should not be thought as a group scheme: the reason is that if we want to work with group schemes over $k$, everything should be relative to the base $\text{Spec}~k$, while the classical étale fundamental group is absolute. For example, the only reasonable fundamental group scheme of $\text{Spec}~k$ is is the trivial one, while the étale fundamental group is $\text{Gal}(k_s/k)$.

If one has a group scheme $G$, the right way of constructing a group out of it is to take $G(k_s)\rtimes\text{Gal}(k_s/k)$: the group scheme contains more information, since it remembers the projection to $\text{Gal}(k_s/k)$. By the way, an usual profinite group should not be thought as a constant group scheme: there is a natural way of giving it an algebraic structure such that the associated topology is the profinite one, not the discrete one.

Let us now restrict ourselves to Tannakian categories whose associated group (or gerbe) is profinite, since Galois categories can only "see" profinite groups. By what we have said above, even in this case a Tannakian category contains more information than a Galois category: the Tannakian category determines a Galois category plus a Galois subcategory isomorphic to $\text{Ét}~k$. The Galois category associated to a Tannakian category $T$ can be constructed by taking the étale covers of the gerbe associated to $T$.

As already said by others, one can compare the two concepts if the base field is algebraically closed. In this case, Galois categories and "profinite" Tannakian categories are essentially the same thing, i.e. Tannakian categories do generalize Galois categories.

One could try to take categories with fibre functors in other categories different from $\text{Set}$ or $\text{Vect}_k$. But the point now is that there is (almost) no reason to do this: if, in the end, one wants to obtain an affine group scheme, Tannakian categories already detect them all. Hence, one may try to do this only to get groups which are not affine.

Answer 3

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.