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the entry Galois theory used to be a stub with only some links. I have now added plenty of details.
A terminological remark: any theorem about Galois theory even of fields which states an equivalence of categories is from Grothendieck or after that. I mean even in the case of fields, this was not known before Grothendieck.
Thank you! I fixed a few typos. A couple questions/comments:
In the theorem under “Galois theory for separable algebras” it would probably make more sense to say $Gal(K_S/K)$-sets at this point, since we haven’t introduced any schemes yet? Also, do you mean finite $Gal(K_S/K)$-sets?
It seems that not every etale morphism should be called an etale covering, only the surjective ones—right?
Zoran,
thanks a lot for the book by Borceux and Janelidze!
It seems that not every etale morphism should be called an etale covering, only the surjective ones—right?
Yes. I have now created etale cover.
ad 5 What is $p(U_i)$ ? Are we talking unions of underlying topological spaces or coproducts in the category of schemes ?
ad 4 You are welcome, I hope you can compare to the remarks in the emails in the afternoon to nail down the classical case in terms of Janelidze’s formalism. Unfortunately my free online time before the train tomorrow is counting in minutes, and I can be of very little help before about Sunday or so.
ad 5
I am in a rush. Was just copying that notion from de Jong, but thought myself that this needs to be said in more detail.
Maybe you can do it. I have to interrupt now and make a phone call.
I can not get into a new reference like Jong before Monday or so, after that I would be glad to discuss. But so far I do not understand what it says.
There is a very good treatment of Galois theory in Douady and Douady, Algebre et theories galoisiennes (1977). I think it exists in translation. The book you mentioned by Borceux and Janelidze is also excellent.
George Janelidze’s theory of Galois theories always looked to me almost ripe for categorification. I think the nPOV version is more or less dotted around the Lab (not explicitly), but I am not enough of an expert in it to be able to collect it up and get links.
I have added a section on ’La Longue Marche à travers la Théorie de Galois’ to Galois theory. I intend to creat a new entry where this will be handled in more depth. Grothendieck’s anabelian dream although as yet unattained seems to my inexpert eyes to be the dimension one version of a sort of homotopy hypothesis.
Yes - anabelian schemes are basically $K(G,1)$s. The conjectured higher version I imagine to be a version of this for other homotopy types, which is going to need new machinery, given that apart from curves and some homogeneous spaces, there aren’t many $K(A,n)$s for $n>1$ in the smooth realm, let alone the algebraic as traditionally conceived.
there aren’t many $K(A,n)$s for $n \gt 1$ in the smooth realm, let alone the algebraic as traditionally conceived.
You are all waiting for me to say it, so I’ll go ahead and indeed say it: of course in the corresponding $\infty$-topos there are!
Recall for all this discussion that as soon as we have locally $\infty$-connectivity, then we have a beautiful $\infty$-Galois theory that applies to Eilenberg-Maclane objects and whatnot. This is described at geometric homotopy groups in an (infinity,1)-topos.
The whole discussion here and elsewhere is all about the subtleties of generalizing away from high local connectivity. That’s why the discussion has been focusing on the 1-categorical case so much. Because since we already know $\infty$-Galois theory for the locally $\infty$-connected case, it seems that one just needs to carefully look at Galois theory for the non locally connected 1-case to also have it for the $\infty$-case.
There are a number of problems, however, with how to define ’homotopy types’. There are at least four or five different definitions of homotopy groups (in all dimensions) in algebraic geometry (schematic, etale, l-adic, motivic, crystalline), and they don’t all agree all the time, as far as I know. All of these come equipped with various actions of Galois groups, so the situation is I think even richer than the ’geometric homotopy groups’ description. There are also Grothendieck’s schematic homotopy types, treated in HAG I.
Ah good. I started a nice hare with my comment! My thought was that the Anabelian dreamland relates to analogues of $K(G,1)$ in the first place, and being a naive hopeful character I thought what about analogues of MacLane-Whitehead classifications of 2-types in this context. What would the question look like. Any more detailed thought would be welcome. (I got some referee’s remarks on my monoprof book that said what I had written was not relevant to the anabelian problem. I felt that that was a bit to simplistic as a remark so am seeking to find out more! I was also aggrieved since I have known about the anabelian problem since 1984, although not knowledgeable enough in variety theory to understand its full complexity.)
I also wish I understood HAG and Toen’s other stuff better.:-(
David,
I’d think the different notions of homotopy groups are all induced from choosing different sites and hence applying the definition of geometric homotopy groups in an $\infty$-topos in different $\infty$-toposes. But I haven’t checked all the cases that you mention. We should.
Concerning schematic homomotoy type: this has been pretty much clarified as being given by the $(\mathcal{O} \dashv Spec)$-adjunction on the $\infty$-topos over $Alg^{op}$ in Toën’s “Champs affine”, summary of which is at function algebras on infinity-stacks.
I made a half-hearted attempt to further expand on the story of classical Galois theory as a story in topos theory, Galois theory in topos theory, but I don’t have the energy to do this justice now. It’s too late at night.
Anything new from Marc Hoyois’s Higher Galois theory to add? What happened since that conversation with Cisinski?
What’s the origin of the observation that Galois theory is all driven by the constant sheaf/stack functor participating in an adjunction? At the time that I observed this in dcct it seemed to be far from commonly understood.
Perhaps http://arxiv.org/abs/math/0407507? At least for a special case?
When was the first dcct? Perhaps you should include dates in it. And is it time for an arXiv update of 1310.7930?
So what is the result of that conversation I linked to?
since we do not have this adjunction (yet?) on (∞,1)Topos, Galois theory with general ∞-toposes is not (yet?) tautologized, yes.
The relevant references that I was aware of at that time are listed on p. 358 of arXiv:1310.7930, including Toën00, Polesello-Waschkies05, and Shulman07 (all for topological spaces). Back then I enjoyed the realization that the key structure of the theory follows from just a left adjoint to the locally constant functor and I don’t remember that people told me this is an old hat. On the contrary, when I noticed that from this and applying the $\infty$-Yoneda lemma four times in a row the central fact of Galois theory, that the homotopy type is read off from the fiber functor, follows formally, I recall some senior person said publically that that such an abstract formulation of Galois theory would make many people very unhappy.
I think it’s a great add-on to observe that in the absence of cohesion, then the missing left adjoint still exists as a pro-left adjoint, so that then the formal theory still exists on pro-objects, and that this is what makes all shape be pro-thingies in algebraic geometry. I really love this. I am just wondering if this was well-appreciated all along and I missed it.
This is essentially the same paper that’s been on my webpage for a couple years now, so I’m afraid there’s nothing new.
The reason I felt comfortable speaking of “Galois theory” so openly is because of the Corollaries 2.14 and 2.15. As far as I understand, within ordinary topos theory, Galois theory has a rather small scope: it’s essentially the theory of Galois toposes. Nevertheless, people seem to have strong opinions when it comes to Galois theory: they say that it’s all about the recognition principle, or that it’s all about the Galois closure, or whatnot. But most of these things are consequences of the classification of locally constant sheaves, so I fully agree with Urs that this is the more fundamental statement. Although I don’t see any content in the existence of the left adjoint: this by itself says nothing about locally constant sheaves.
In the end, the local connectivity assumptions are somewhat disappointing, but I’ve become skeptical that anything nice can be said more generally. One might hope that the shape/classifying topos adjunction is idempotent and regard this as a very general abstract version of “Galois theory”, but I now suspect this is not true at all.
Marc, regarding
This is essentially the same paper that’s been on my webpage for a couple years now, so I’m afraid there’s nothing new.
that’s the thing. With an official preprint one will want to have the attributions sorted out more properly than in a private note, since, for better or worse, this is the currency with which people in the business ultimately need to feed their families. Here I am wondering if the observation that (higher) Galois theory abstractly is all controled by a (derived) left adjoint to the constant sheaf/stack functor is one worth attributing to precursors. If you feel it’s not, then that’ll be so. I was just wondering.
@Urs I actually added some references. In particular, to Toen-Vezzosi and to Lurie for the definition of the shape functor, which are the only references I know. But it looks like we have a fundamental disagreement on the importance of this functor. Although the classification of locally constant sheaves is formulated using this functor, the work is elsewhere. So I don’t think I agree with your observation.
By the way, the 1-topos analog of the whole story is in Moerdijk’s paper Prodiscrete groups and Galois toposes from 1989.
Marc: You might be interested in the folowwing. The origins of the shape terminology in both Toen and Lurie is probably in the categorical and other forms of (strong) shape theory way back in the 1980s. There the shape category was considered as being a quotient of Pro-sSets. The objects being studied did not necessarily correspond to ‘stable’ i.e. constant pro-simplicial sets. This corresponded in topological shape theory to lack of local connectedness. I did some earlier work on the use of Lubkin’s version of étale homotopy type within this context and in about 1978 came up with the idea of stacks as giving a higher form of Galois theory. Slightly later when I saw Pursuing stacks, I found that Grothendieck had been putting forward that sort of idea in letters to Breen in 1975. This gave me a lot of confidence that there was something there to say. I did not get funding for the projects that I put forward as part of this and had to go a long way around as a result.
Hey Marc,
there is no disagreement, since I am asking a question. Let me recall it: your note devotes some paragraphs to establishing an adjunction $(\Pi_\infty \dashv \iota)$. What I am asking is: since when is it understood that Galois theory is driven by an adjunction of this kind?
I don’t really see this preconfigured in “Prodiscrete groups and Galois toposes”, unless of course with hindsight.
Also, I don’t doubt that there is serious work involved elsewhere in your article, not at all. On the contrary, what I am after is the insights that reduce the amount of work necessary.
Tim: Thanks for the pointer to Grothendieck’s letters to Breen! I always thought they were the beginning of Pursuing Stacks rather than separate documents.
I am only partially aware of classical shape theory of topological spaces (many relevant papers are difficult to find online), and I can’t say much about the precise relation with the toposic story. Lurie claims in HTT, Remark 7.1.6.7, that his shape generalizes the strong shape for compact metric spaces, but he doesn’t expand any further. I did note in Example 1.7 in my paper that if $T$ is the inverse limit of a diagram of compact Hausdorff spaces with the homotopy types of CW complexes, then the toposic shape of $T$ is just that diagram viewed as a pro-space, and I assume strong shape has this property as well.
Urs, your question presupposes something with which I don’t agree, so I don’t know how to answer it. To my knowledge Lurie was the first to mention the adjunction $\Pi_\infty\dashv\iota$. Although I like this adjunction a lot, I wouldn’t say it’s one of the “insights” of Galois theory.
I don’t really see this preconfigured in “Prodiscrete groups and Galois toposes”, unless of course with hindsight.
Indeed, Moerdijk’s paper makes no use of such an adjunction: his Galois theory statement is that the subcategory of (colimits of) locally constant sheaves forms a topos which is equivalent to the classifying topos of a pro-group. Once you know that, it’s easy to deduce that this pro-group classifies torsors (this is the last remark in Moerdijk’s paper), so it’s the shape in Lurie’s sense.
I wouldn’t say it’s one of the “insights” of Galois theory.
Neither would I. What I said was that it “drives” Galois theory. It is an insight that saves some work when thinking about Galois theory. That is probably the reason why you talk about it in an article on Galois theory! :-)
But never mind. Thanks for the neat article.
Actually, maybe Grothendieck’s letter to Breen answers your question: statement (C) there is the classification of locally constant (n-1)-stacks in a locally n-connected topos in terms of the fundamental n-groupoid, which is almost the statement of the theorem in my paper (which extends this to n-stacks). I can’t believe I’ve never read this letter before…
Now that letter is interesting, thanks for pointing it out.
On page 2 the adjunction in question is almost stated – in the paragraph annotated “this seems suspect”. Interestingly, the problem he has is that he 1-truncates the $n$-categories for “fear” (previous paragraph, ending with “Brr!” :-) of $n$-toposes.
Too bad the “preceding letter” in which he expressed his fears doesn’t seem to be available.
It’s also interesting that he expects (at the end of the first page) that the local $n$-connectedness assumption can be dropped just by considering the pro-$n$-groupoid $\Pi_n X$. He cites the case $n=1$ and the etale fundamental group as evidence, but he seems to forget that local connectedness of the underlying scheme is still needed in the non-finite case. Unless he has in mind a fancier notion of “locally constant”…
… which seems to be the case. In his second letter to Breen, on page 14, he considers locally constant $n$-stacks as an ind-category, so that makes sense.
If you look at what may be the third letter (received in Bangor 8/4/1982), there is again a lot of relevant stuff. A quick look does not seem to reveal a date for that letter and I cannot remember its place in the correspondence. If Ronnie sees this or if someone can ask Larry Breen in person it might be good to sort out the context and the probable date.
My understanding of his letters was that SGA1 style Galois theory should extend to higher dimensions. This seemed to me to be much more central to things than the so called Homotopy Hypothesis, which was just one step on the way, and, in fact, was a test for a good candidate for infinity groupoids. The idea in the early part of PS (letter to Quillen) make most sense if read in conjunction with those letters to Breen.
Is there any chance that the “third” letter actually precedes the first one from 1975? It looks as though he talks about $n$-categories for the first time there, and he also expresses his fear of n-toposes:
On frémit à l’idée que les topos pourraient ne pas faire l’affaire, et qu’il y faille des “n-topos” !! (J’espère bien que ces animaux n’existent pas …)
“One shudders at the idea that toposes may not do the trick, and that one may need “n-toposes”!! (I hope that these animals do not exist…)”
My understanding of his letters was that SGA1 style Galois theory should extend to higher dimensions. This seemed to me to be much more central to things than the so called Homotopy Hypothesis, which was just one step on the way, and, in fact, was a test for a good candidate for infinity groupoids. The idea in the early part of PS (letter to Quillen) make most sense if read in conjunction with those letters to Breen.
I think this depends on exactly what “Galois theory” statements one looks at. Some of them extend but become more or less tautological, while others seem false without local connectivity assumptions (eg that the subtopos generated by locally constant $(n-1)$-stacks is the classifying topos of a pro-$n$-groupoid).
added pointer to:
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