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I have recently become aware of, and started to study in my free time (abundant in these summer months) Grothendieck's Galois Theory (GGT), as formulated in SGA 1 and later by Grothendieck's contemporaries. I understand there to have been a number of unresolved and open questions relating to GGT upon its formulation, some of which seem to persist. This is a truly gorgeous subject, and I wonder whether it is still studied rigorously or researched at all today? Where/Who produces interesting results regarding things such as Galois and Atomic topoi, applications of the Grothendieck fundamental group, etc., today?

Is there a great deal of utility to GGT beyond the foundational work in algebraic geometry later formulated by Grothendieck?

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If by "GGT" you mean any mathematics involving finite etale covers of schemes, then the answer is yes - the theory is still studied intensely today, and is quite useful in non-foundational contexts. I should note that Grothendieck viewed Galois theory from several different perspectives during his career, and terminology such as "GGT" may be confused with, e.g., his work on dessins d'enfants in the 1980s. –  S. Carnahan Jul 18 '10 at 23:01
Thank you for that clarification, Scott. Indeed, I understand that my precise meaning might seem ambiguous here; I am referring to the only context in which I have encountered it (in my cursory study thereof), that is, in the development in SGA1 of Grothendieck's fundamental group and its applications. Clearly this is still a concept which is very much put to use today. I am also interested in whether considerations in foundational contexts, viz. topos theory (for instance, the development and study of atomic topoi). This is more closely related to the mathematics in which I am interested. –  lambdafunctor Jul 18 '10 at 23:14
The gigantic subject of $\ell$-adic representations of Galois groups of global fields uses \'etale fundamental groups of higher-dim. schemes all over the place (for constructions, to prove theorems, etc.). Deligne's proof of RH and generalizations uses the close link between lisse $\ell$-adic sheaves and representations of etale fundamental groups. One can say so much, but in time you'll learn about it if you make a deeper study of Galois representations and etale cohomology. Basically, the etale fundamental group is as important in arithmetic geometry as the usual fund. gp is in topology. –  BCnrd Jul 19 '10 at 6:57
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5 Answers

The basic Grothendieck's assumptions means we are dealing with an connected atomic site $\mathcal{C}$ with a point, whose inverse image is the fiber functor $F: \mathcal{C} \to \mathcal{S}et$:

(i) Every arrow $X \to Y$ in $\mathcal{C}$ is an strict epimorphism.

(ii) For every $X \in \mathcal{C}$ $F(X) \neq \emptyset$.

(iii) $F$ preseves strict epimorphisms.

(iv) The diagram of $F$, $\Gamma_F$ is a cofiltered category.

Let $G = Aut(F)$ be the localic group of automorphisms of $F$.

Let $F: \widetilde{\mathcal{C}} \to \mathcal{S}et$ the pointed atomic topos of sheaves for the canonical topology on $\mathcal{C}$. We can assume that $\mathcal{C}$ are the connected objects of $\widetilde{\mathcal{C}}$.

(i) means that the objects are connected, (ii) means that the topos is connected, (iii) that $F$ is continous, and (iv) that it is flat.

By considering stonger finite limit preserving conditions (iv) on $F$ (corresponding to stronger cofiltering conditions on $\Gamma_F$) we obtain different Grothendieck-Galois situations (for details and full proofs see [1]):

S1) F preserves all inverse limits in $\widetilde{\mathcal{C}}$ of objets in $\mathcal{C}$, that is $F$ is essential. In this case $\Gamma_F$ has an initial object $(a,A)$ (we have a "universal covering"), $F$ is representable, $a: [A, -] \cong F$, and $G = Aut(A)^{op}$ is a discrete group.

S2) F preserves arbritrary products in $\widetilde{\mathcal{C}}$ of a same $X \in \mathcal{C}$ (we introduce the name "proessential for such a point [1]). In this case there exists galois closures (which is a cofiltering-type property of $\Gamma_F)$, and $G$ is a prodiscrete localic group, inverse limit in the category of localic groups of the discrete groups $Aut(A)^{op}$, $A$ running over all the galois objects in $\mathcal{C}$.

S2-finite) F takes values on finite sets. This is the original situation in SGA1. In this case the condition "F preserves finite products in $\widetilde{\mathcal{C}}$ of a same $X \in \mathcal{C}$ holds automatically by condition (iv) ($F$ preserves finite limits), thus there exists galois closures, the groups $Aut(A)^{op}$ are finite, and $G$ is a profinite group, inverse limit in the category of topological groups of the finite groups $Aut(A)^{op}$.

NOTE. The projections of a inverse limit of finite groups are surjective. This is a key property. The projection of a inverse limit of groups are not necessarily surjective, but if the limit is taken in the category of localic groups, they are indeed surjective (proved by Joyal-Tierney). This is the reason we have to take a localic group in 2). Grothendieck follows an equivalent approach in SGA4 by taking the limit in the category of Pro-groups.

S3) No condition on $F$ other than preservation of finite limits (iv). This is the case of a general pointed atomic topos. The development of this case we call "Localic galois theory" see [2], its fundamental theorem first proved by Joyal-Tierney.

[1] "On the representation theory of Galois and Atomic topoi", JPAA 186 (2004)

[2] "Localic galois theory", Advances in mathematics", 175/1 (2003).

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I suggest that you read Deligne's wonderful paper “Le Groupe Fondamental de la Droite Projective Moins Trois Points”. I read a little bit of it and was astonished. Please do take it up and read it without further loss of time, notwithstanding the French.

Jordan Ellenberg's and Matthew Emerton's opinion is available here.

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There is a book on a more general approach to Galois theory by Borceux and Janelidze (called, imaginatively, 'Galois theories' Google books). A bit more concrete is 'Galois Theory in Symmetric Monoidal Categories' by Janelidze and Street, which uses a Galois theoretic approach to Tannaka duality (a very Grothendieckian study).

Marta Bunge has a bunch of stuff on topoi and Galois-type theories, for example 'Galois Groupoids and Covering Morphisms in Topos Theory', 'Constructive Theory of Galois Toposes' (joint with Eduardo Dubuc) and a bunch of others.

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I don't know much about this topic, but I was recently recommended the paper An extension of the Grothendieck Galois theory of Grothendieck by Joyal and Tierney as an enlightening abstract generalisation in the language of toposes. It seems that it predates some of the other references given above, but might be worth reading.

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This isn't my research group (though some of the people for whom it is are around), but here's a website devoted to this stuff, with lots of papers, names, etc. It's a VERY active area, with a lot of interesting problems.

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The question was not Galois theory ? but Grothendieck's version/extension of Galois theory... –  Zoran Skoda Mar 21 at 17:40
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