Galois categories are introduced (for the first time?) in SGA1, but here's an English introduction that's available online: http://www.math.uchicago.edu/~may/VIGRE/VIGRE2009/REUPapers/Lynn.pdf

It seems that Galois Categories are a way of axiomatizing all the Galois correspondences in the various fields: Galois theory for fields, Galois theory for covers, Galois theory tame covers and so forth.

What is the benefit, if at all, of this formalism? Is it just to outline the commonalities of these seemingly different topics, or is there some applicable virtue to this language?

  • $\begingroup$ The linked paper is full of errors and "empty assertions". $\endgroup$ – Martin Brandenburg Jan 4 '11 at 20:46
  • $\begingroup$ I haven't read it through. I should perhaps put SGA1 chapter V as the main reference, 4.1 being the main theorem. $\endgroup$ – James D. Taylor Jan 4 '11 at 20:46
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    $\begingroup$ Or perhaps Ricky's link: websites.math.leidenuniv.nl/algebra/GSchemes.pdf $\endgroup$ – James D. Taylor Jan 4 '11 at 20:47
  • $\begingroup$ Galois categories inspired the Tannakian categories formalism that reconstructs an affine group scheme from its finite-dimensional representations. $\endgroup$ – Mozibur Ullah Dec 25 '12 at 19:08

(I only just saw this one year on!) I find your question very strange. Grothendieck gives a simple categorical formulation of a situation that encompasses the three main examples of Galois theoretic machines. That means he shows what makes things really tick... isn't that good enough for you! He does this with the clearly stated aim of developing a fundamental group for schemes, and the theory gives that and a lot more. If you go to the slightly wider results on the fundamental groupoid of categories of locally finite sheaves, that is a first step towards his Pursuing Stacks, the letters to Larry Breen, and enroute for his Longue Marche.

In another direction it provides a first step towards the Joyal-Tierney theory of locales etc. and their relation with toposes. It provides a background for all of Jacob Lurie's work on higher toposes, and I could go on with fundamental groups of toposes, homotopy theory of toposes. SGA1 is the key for understanding a large part of modern mathematics.

Grothendieck's methodology was always to seek the clarity that came from abstraction and generalisation. His aim was not only to solve problems (say in algebraic geometry) but to understand as fully as possible their solution and why they worked.


At the end, a Galois category is equivalent to the category $\pi$-sets, of finite sets with a continuous action of a profinite group $\pi$, so once you know this is easy to study Galois categories. The interesting part is that sometimes you a have a category, you can prove that it is Galois, so you have your $\pi$, but this is the only way you have to define the group. This is the method used by Grothendieck to define the fundamental group of a scheme (w.r.t. to a geometric point. used to define the fibre functor). See here link text for the details.

  • $\begingroup$ Correct me if I'm wrong, but can't we also define it simply as the automorphisms group of the geometric fiber functor as is done in Szamuely's book? Do we really need the power of Galois categories to do this? $\endgroup$ – James D. Taylor Jan 4 '11 at 19:39
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    $\begingroup$ I don't know the book you are talking about, but of course, the $\pi_1$ is the group of automorphisms of the fibre functor. But using the theory of Galos category you can really understand what the category of finite étale covering is (for example what a connected or Galois object is). To my taste, this allows you to better understand the relation to other Galois theory, in particular why classical Galois theory is a particular case of the one of Grothendieck. $\endgroup$ – Ricky Jan 4 '11 at 19:49

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