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We have a bit of a mess of closely related entries related to étale homotopy groups which existed more or less in parallel without seeming to know much of each other. I have tried to do some minimum of cross-linking and cleaning up, but this needs more attention.
There is more even, there is Grothendieck’s Galois theory and what not. (Maybe we need to wait until somebody gives a course on this and uses the occasion to clean it all up and harmonize it.)
Janelidze’s categorical Galois theory is supposedly the cleanest general abstract machinery for doing Galois theory, much beyond the special cases like the Galois theory in topos context (Grothendieck Galois theory); the axioms are in terms of adjunctions. (There is also an enriched generalization, but it is not published in full generality.)
That is indeed yet another entry we have: categorical Galois theory. There are still more like this floating around on the Lab
I am a little reluctant so say that there is anything except the idea of étale homotopy theory which ought to take the place of “cleanest abstract machinery” for Galois theory. I think Hoyois 13 precisely nails it down.
The paper you quote starts abstract with “The etale homotopy type of a scheme X is a pro-space which contains enough information…”. You see the pro-space topology appears in classical Galois theory in something what is in this business called the Pierce spectrum. The functor which realizes the Pierce spectrum is just a mere special concrete example of a functor which appear in general clean axiomatization of categorical Galois theory. Imagine a functor which does anything you like, just that it fits with the other functors in the axiomatization the same way as Pierce spectrum fits with other characters in Galois theory. Isn’t that far more abstract than the special cases pro- and etale- constructions ?? Having higher versions to , etc. can be of course done both for usual etale topoi as well as for categorical Galois theory, this is another direction of generalization and not affecting our discussion.
Also take into account that various special things like separable algebras etc. in the entry have their categorical counterparts.
(Unfortunately, I just feel the gist and am not competent to write a good entry on this yet. Anybody whom I talked to and who understood this theory I found it very deep (e.g. Igor who studied it far more than me).
Having said that I have one doubt: does the categorical Galois theory have a way to specialize to Hopf-Galois phenomena in noncommutative geometry. If I try to do few steps I see that I need to go beyond the usual formulation; it seems that the enriched version could help, but one should try hard.
The general abstract idea that I am referring to is simply that one produces a left adjoint to the constant stack functor, and if that does not exist on the nose then one produces the canonically next best thing – which is the pro-left adjoint.
That’s why there is pro-business in algebraic fundamental groups: the algebraic site is not cohesive and so the left adjoint does not exist on the nose, but its pro-version always does, and that’s what produces the etale homotopy type.
The rest is details :-)
There is also Galois theories which is in the form of a book review at present.
A point to be made is that fundamental groupoids are still defined in cases where Galois theory does not work. For example, a scheme which is not locally connected has a meaningful étale fundamental groupoid , but -sets are not equivalent to locally constant étale sheaves. From the point of view of topos theory, the fundamental groupoid of a topos is defined so as to classify cohomology with constant coefficients, but only in good cases can it also be characterized by a Galois correspondence. From what little I’ve seen of categorical Galois theory, it aims to find general settings in which Galois correspondences arise, so it does not obviously apply to this notion of fundamental groupoid.
Marc, if you ever feel bored (this is a joke) maybe you’d enjoy adding a brief paragraph to, say, Galois theory, with the lighning idea of what you lay out in Hoyois 13, specifically where it concerns traditional Galois theory.
(Or else I’ll do it. But then it will be more cumbersome for you to get me to do it the way you deem appropriate ;-)
The general abstract idea that I am referring to is simply that one produces a left adjoint Π\Pi to the constant stack functor, and if that does not exist on the nose then one produces the canonically next best thing – which is the pro-left adjoint.
You see, but isn’t it better if there is a categorical characterization/axiomatization what is the next best thing and in the same time satisfactory thing, rather than just improvizing in this special case which does not go to the bottom of the matter.
It is very strange that you oppose an axiomatics which to Grothendieck Galois theory is a generalization similar to comparing cohomology in some classical context with cohesive topos axiomatics.
I think pro-left adjoints are as “categorical” as it gets.
Its dual is ind-right adjoint, presumably.
I see this paper speaks of them, and mentions SGA4-III, Expose XVII, beg. of 1.2 as a source of some connection to a calculus of fractions.
Are there other names for them – they don’t seem to be used very often?
We seem also to be calling pro-left adjoint proadjoint, a page that Zoran started.
And profinite completion of a group has ’left pro-adjoint’.
Oh, sorry, that is my mistake then, wasn’t aware of that other entry.
So then these two entries must be merged. (But I won’t do it right now, need to run…)
I think pro-left adjoints are as “categorical” as it gets.
Yes, if you can afford it. You can afford it in a special cases, when you can make sense of “constant stack functor” and then this characterization suffices. The categorical Galois framework has other examples beyond those in which you have constant stack functor and proadjoint. Finding essential features which make this possible and still agree with your favorite story in the etale/pro case looks like one step deeper into the theory. I will show you once I have time to delve into it (hopefully real soon) and I hope you will change your mind.
BTW I often use the proadjoint terminology and ind-adjoints when needed. The link with calculus of fractions is classical Verdier derived functor theory. Grothendieck’s Galois theory was considered to be very categorical when it first appeared. As category theory has been absorbed by those who initially said it was too abstract, what was very categorical does not seem that categorical now!!!!!
have now added to proadjoint a paragraph Examples – Shape theory, etale homtopy and Galois theory.
Zoran,
re #15: this is not an issue which we may objectively clarify, having too much of a subjective aspect to it. Given everything we know about the shape modality, of course you may see why I believe this is the “preferred” perspective on Galois theory (whence the section titled “Galois theory” in dcct).
Of course, by adjunction, this has a dual interpretation in terms of what is known as “categorical Galois theory” and that is much of the point of the adjunction between the shape modality and the flat modality (or else of their pro-versions).
Just so that you won’t be too disappointed, let me warn you that when you produce an example of categorical Galois theory which involves “spaces” forming a category that is not a topos, then chances are that I will pass to the topos over that site and claim that this is the right context for applying Galois theory via pro-etale homotopy theory for the original category :-)
I added a brief discussion of higher topoi at Galois theory as requested by Urs #9.
Re #13, we never did merge pro-left adjoint and proadjoint. Is there a preferred name?
Would a page for ’ind-right adjoint’ or ’indadjoint’ be worth having?
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