I am a graduate student working on category theory. I am familiar with categorical Galois theory, in the way developed by Janelidze - as described for example in "Galois Theories". I am now trying to comprehend the paper by Joyal and Tierney "An extension of the Galois theory of Grothendieck". It feels like there is a connection between Galois-theoretic results in category theory and Etale Cohomology, which is not clear to me. I do not have a very solid background on algebraic geometry. Can someone describe the connection between the two? I am quite sure that descent theory plays some role but as I said it is not clear to me. If someone could provide a way to look at this correctly, or some material that describes the connection, I would be very grateful - I have found some papers but the ones describing Etale Cohomology are using a very algebraic-geometric language that I can not really follow. There must be a way to look at etale cohomology from a pure category-theoretic point of view. Thanks for any help.
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1$\begingroup$ I don't believe that cohomology really plays any role in the cited work of Joyal and Tierney. I tried to give an expository talk on that subject recently; you can see the slides here. $\endgroup$– Zhen LinCommented Aug 23, 2013 at 18:36
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$\begingroup$ Hello Zhen Lin. Thanks for the reply. About the slides, I can not open them properly - something wrong with the symbols after a certain slide. Is this a problem with my reader?! $\endgroup$– Franz NemeyerCommented Aug 25, 2013 at 13:26
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$\begingroup$ Works for me – try a different application? $\endgroup$– Zhen LinCommented Aug 25, 2013 at 13:44
1 Answer
The relation between Galois theory and etale cohomology is simple : what Galois theory of a Field $K$ said is that the etale topos of $Spec K$ is the topos of (the category of) continuous $G$ set where $G$ is the Galois group of the separable closure of $K$ (over $K$) endowed with its profinite topology. Hence the etale cohomology of $Spec K$ is related to the group cohomology of the absolute galois goup of $K$.
Now the title of the paper you mentioned refer to a result of Grothendieck which give a characterization of categories which are the category of continuous $G$-set for some (topological) group $G$. (I don't remember the precise formulation of this theorem, but a more modern form of this result is the fact that a connected atomic topos with a point is the topos of continuous $G$-set for $G$ the localic group of automorphism of the point.)
Now the main result of the paper of Joyal and Tierney is that not only connected atomic topos, but any topos can be represented (relatively explicitly) as the topos of equivariant sheaf over a localic groupoid. Which in some sense generalize the result of grothendieck.
This can be used for example to represent the Etale topos of an arbitrary scheme by something geometrical, involving a space of geometric point of the scheme on which some galois groups acts locally.
