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We had discussed here at some length the formalization of formally etale morphisms in a differential cohesive (infinity,1)-topos. But there is an immediate slight reformulation which I never made explicit before, but which is interesting to make explicit:
namely I used to characterize formal étaleness in terms of the canonical morphism $\phi : i_! \to i_*$ between the components of the geometric morphism $i : \mathbf{H} \hookrightarrow \mathbf{H}_{th}$ that defines the differential cohesion – because that formulation made close contact to the way Kontsevich and Rosenberg formulate formal étaleness.
But there is a more suggestive/transparent but equivalent (in fact more general, since it works in all of $\mathbf{H}_{th}$ not just in $\mathbf{H}$) formulation in terms of the $\mathbf{\Pi}_{inf}$-modality, the “fundamental infinitesimal path $\infty$-groupoid” operator:
a morphism $f : X \to Y$ in $\mathbf{H}_{th}$ is formally étale precisely if the canonical diagram
$\array{ X &\to & \mathbf{\Pi}_{inf}(X) \\ \downarrow^{\mathrlap{f}} && \downarrow^{\mathrlap{\mathbf{\Pi}_{inf}(f)}} \\ Y &\to& \mathbf{\Pi}_{inf}(Y) }$is an $\infty$-pullback.
(It’s immediate that this is equivalent to the previous definition, using that $i_!$ is fully faithful, by definition.)
This is nice, because it makes the relation to general abstract Galois theory manifest: if we just replace in the above the infinitesimal modality $\mathbf{\Pi}_{inf}$ with the finite path $\infty$-groupoid modality $\mathbf{\Pi}$, then the above pullback characterizes the “$\mathbf{\Pi}$-closed morphisms” which precisely constitute the total space projections of locally constant $\infty$-stacks over $Y$. Here we now characterize general $\infty$-stacks over $Y$.
And for instance in direct analogy with the corresponding proof for the $\mathbf{\Pi}$-modality, one finds for the $\mathbf{\Pi}_{inf}$-modality that, for instance, we have an orthogonal factorization system
$(\mathbf{\Pi}_{inf}-equivalences\;,\; formally\;etale\;morphisms) \,.$I’ll spell out more on this at Differential cohesion – Structures a little later (that’s why this here is under “latest changes”), for the moment more details are in section 3.7.3 of differential cohomology in a cohesive topos (schreiber).
To have this relation between “formally étale“ and “étale“ would be great, but isn’t $\Pi_{inf}=i_* i^*$?
Yes, it is. Let me know why this makes you say “but”.
Probably not relevant, but whatever happen to Picard–Vessiot theory? If I remember correctly, it was a differential version of Galois theory.
whatever happen to Picard–Vessiot theory?
I haven’t looked into this. Will try to do so as soon as I have some other things out of the way. If you have a preferred pointer to the literature, please let me know.
Yes, ok. I am convinced. This is really great! (You omitted a subscript $inf$ in the diagram…)
Okay, good. On the other hand, I just realized that I had a typo in the first post above: there was an ${}_{inf}$-subscript missing. Maybe that led to a misunderstanding, sorry. I have fixed it now.
Will you add a pointer to this fact from pi-closed morphism?
I just realized that I had a typo in the first post
I tried to find a morphism as proposed in your first post, so I had doubts in first place.
Here are some slides of The origins of differential Galois theory. The basic idea is to add solutions of a linear differential equation to a differential field and see what the Galois group tells you about the solutions. Wikipedia.
Michael Singer has written a big book on this with van der Put, item 2 of his Books.
It seems there’s an approach via Tannakian categories.
Will you add a pointer to this fact from Pi-closed morphism?
Ah, I had forgotten about the existence of this entry. Looking at it now, I have edited it a bit and added a brief remark on the formall étale morphisms.
But I don’t really have more time for this right now. If you have time and feel like moving some of the stuff that I have in the pdf to the relevant $n$Lab pages, please do, that would be helpful.
David,
thanks! I have collected some of this stuff now at differential Galois theory.
The articles by dos Santos now listed there formulate the theory in terms of D-geometry. This is likely the most natural way to connect it to what is being discussed above, since $D$-geometry is (as discussed there) essentially the geometry over de Rham spaces $\mathbf{\Pi}_{inf}(X)$.
But beyond that vague remark I don’t have anything interesting to say right now. But I agree that eventually it would be nice to connect this further. It would help if there were something on “categorical differential Galois theory” out there, being the pairing of categorical Galois theory with differential Galois theory. If you come across anything along these lines, please let me know.
What I am really after, though, is this:
let $(\mathbf{H}_th)_{/X}^{fet} \hookrightarrow (\mathbf{H}_{th})_{/X}$ be the inclusion on the formally étale morphisms into $X$. This will have have reflections.
This is important for the following: if $A$ is a group object in $\mathbf{H}_{th}$ and $exp(i S) \colon X \to A$ is a function, with differential $\mathbf{d} \exp(i S) \colon X \to A \stackrel{\theta_A}{\to} \flat_{dR} \mathbf{B}A$, then the critical locus of $\exp(i S)$ is the vanishing locus of $\mathbf{d} \exp(i S)$ in $\mathbf{H}_{th}$ and one might naively think that this is just its homotopy fiber. But it’s not: it should be the homotopy fiber after reflecting to the étale slice over $X$, only.
(For example for $X$ a manifold, then $X \times \Omega^1(-)$ regarded in $\mathbf{H}_{/X}$ is not quite the sheaf of sections of the 1-form bundle $\Gamma(T^* X)$: it is so only over étale maps into $X$).
Picard-Vessiot theory/differential Galois from the Tannakian/categorical point of view is properly treated in
Unfortunately, these are ODE-s: for $D$-modules in several variables there is no Galois theory developed at all, and the lack is felt in applications.
Hm, yeah, I am not sure: can we say anything about $(\mathbf{H}_th)_{/X}^{fet} \hookrightarrow (\mathbf{H}_{th})_{/X}$ having a right adjoint?
Re 12, this article, Universal Covers and Category Theory in Polynomial and Differential Galois Theory, by Magid has a running header, Categorical Differential Galois Theory, and has all the right literature in section 1.1.
16: it says nothing new. It iterates that the reference is Deligne’s article, repeats the standard reference on Grothendieck Galois theory in lectures on Murre, and that the general (not differential case) categorical Galois theory is in the book of Janelidze and Borceaux. And then quotes too books from this same noncategorical community as that over there they among the rest reexplain the Deligne’s approach. So, lots of waving around and no new reference on categorical approach than Deligne’s. For model theorists, there is something on differential Tannaka in Kamensky’s work. And I get irritated when I see Magid and others quoting again and again Kolchin’s book as a good reference. It is a reference in 19th century style, working with 1 derivation, and the community took some efforts in building some sort of differential varieties at the level of algebraic geomety of 18th century, no D-modules, no schemes, no sheaves, it is waste of time to learn it that way nowdays.
Zoran,
okay, I haven’t read Deligne’s text yet, but since you seem to have, maybe you can just tell me:
what I am after is the refinement to differential Galois theory of the standard topos-theoretic reformulation of standard Galois theory. That standard theory says that for $k$ a field and $Et(k)$ its small étale site, we consider the topos $\mathcal{E} \coloneqq Sh(Et(k))$ and then
Galois extensions of $k$ correspond to the locally constant objects in $\mathcal{E}$;
The Galois group of $k$ is the fundamental group of the topos $\mathcal{E}$.
The question that I am after is: how does this story refine to differential Galois theory? Does Deligne discuss this? Or some other text?
I have sent you the file, the relevant are the past 5 pages of Deligne’s Grothendieck’s Festschrift article.
No, this story does not refine to differential Galois theory yet, and I think it is even a wrong question. Namely it is highly unnatural to limit to one derivatation. It is asking do you refine the forestry science to treat a forest together with an iron sword ? (Quite odd :)) It makes more sense to to treat a forest together with all iron artifacts simultaneously. So I think the right question would be to add all infinitesimals at once, hence work with D-schemes instead of ordinary schemes, or even I daresay instead of only with D-modules/synthetic topos (I recently changed the idea section to D-geometry – it is not only to add the synthetic point of view, hence infinitesimals and differential operators, hence D-modules, but also the nonlinear part, the crystals of schemes, or D-schemes), or only with rings plus one single derivation in characteristic $0$. The last thing is called differential Galois theory, and I think it does not even deserve that name. Plus if you want a natural topos then I am sure it will be an extension of topos of etale site to etale site of D-schemes. This is not done: even no Galois theorem whatsoever is known for the case of D-modules where at least two derivations appear. Hence a good place for you to start something (but not by reading Magid and trying to rephrase it in fancy topos terms, but rather by direct study of the etale site of D-schemes itself, I think).
then I am sure it will be an extension of topos of etale site to etale site of D-schemes
That's exactly what I am hoping, since that's what will match with the beginning of this thread here. Should you come across literature that tries to do this, drop me a note.
So there are 3 levels: D-modules with only one derivation; then the 2nd case of arbitrary D-modules and finally 3rd of D-schemes. The Galois theory is known only for the first case of Picard-Vessiot, none even for D-modules in general, and no even a gist in the nonlinear case of D-schemes. The second and third stage might be worth to do in topos language. For the third, the noncommutative geometry point of view might help as well.
Should I repeat ? Nobody is working even on stage 2, not to mention stage 3. So do not expect me ’to come across literature’ on it. I have spent enough time looking for any work touching the stage 2 to claim safely that there is no work in the literature (unless something changed within last year or so).
P.S. I have changed some details in the post 19 above, in the case that you responded to an earlier version.
Let us ask a concrete question. What would be an appropriate analogue of the (etale) fundamental groupoid of a D-scheme ? Or, if we do not want to work with etale topology and schemes, we can ask in the differential geometric language of diffieties; hence trying to extend the fundamental group construction internally from manifolds to diffieties.
Zoran,
not sure what you are so worked up about. I think there is some talking past each other here. For instance when you say "wrong question" and "unnatural to restrict to one derivation" I don't feel that you are reacting to anything I actually said.
What actually happened here is this: I said that in order to connect the starting point of this thread in #1 to differential Galois theory it would help to have the categorical formulation of the latter. In #14 you claimed that's in Deligne's article, and so I asked you what it is explicitly that is in Deligne's article.
It is a wrong question in the sense that you want to formulate an unnaturally narrow question called “differential Galois theory” in general topos language. Differential Galois theory is by the definition about differetial rings and differential fidls, and as you wrote in $n$Lab it is about the fied equipped with ONE derivation. The natural question geometrically is to attack a much wider question, as I outlined above. ANY natural topos formulation (extending etale Galois theory of schemes) would I guess at least include a general D-module if not general D-scheme. What is I repeat, not being done.
Deligne is using the Tannaka theorem. Tannaka theorem is a linearization of Galois theory to some category of linear representations. Thus out of all the geometry on the level of fundamental group and so on, one is left only with the linear part at the level of monoidal categories. So one has just the fiber functor at the level of monoidal categories and wants to do the reconstruction from the automorphism of this linearized fiber. This is done very categorically, using some sort of generalized linear algebra. It is about adjunctions as the general Galois theory of Janelidze is, if you want. But it is not topos theoretic. The topos theoretic question would something starting with a etale fundamental groupoid of a D-scheme. Deligne has monoidal categories, gerbes, fibre functors, comonads, descent, groupoids, (bi)torsors. This is incomparably more categorical than one derivation terminology of Kolchin’s school.But it is not involving full geometry of any topos. It is Tannaka, hence linearized; plus it deals with one derivation case (but it is just application, there is nothing in the previous part of the Deligne’s article specific to one derivation; but it is not clear if the case of several derivations even in this context can fit into general Tannaka formalism as in the rest of Deligne’s article.
I do not think I misunderstood what you said. I was very attentive to all its content. I would say, other way around, you ignore the distinctions between what I present as the existent state of the science from what you want to do. So you should either try to attack the question yourself along the lines I suggest or at least give up expecting to find it in Magid and other literature. Still one may expect having an intermediate answer from some extension of work of Deligne (not topos theoretic Galois theory but some sort of linearized version but for several derivations), though I doubt, as it would be already done if that straightfoward.
“unnatural to restrict to one derivation” I don’t feel that you are reacting to anything I actually said
You wrote $n$Lab entry explaining that differential ring is a ring equipped with ONE derivation. Differential Galois theory is about extensions of differential fields/rings. So, asking if differential Galois theory extends etale, means asking if there is (in literature) an extension of Grothendieck’s etale Galois theory by the case with ONE derivation.
Instead, one should first dismiss the differential ring theory of Kolchin’s school first (modulo the interesting and essential step forward contained in the 5 pages of Deligne’s article), and not give it big expectations, IMHO. Discussing existing literature on this, what you asked about, is tautologicallly asking about the case of one derivation only.
Instead of talking differential Galois theory we should try to think how to attack the baby stage D-module Galois theory and, finally, D-scheme Galois theory (and, slightly more general, crystals of anything Galois theory).
I expect the relation to D-geometry, as I already said in #12. But elsewhere, in the $n$Lab entry on “differential Galois theory”, I have to write about its understood definition, not about my expected generalization of it. That doesn’t imply by some induction that I am asking a wrong question here.
I’ll look at Deligne’s text now. If he formulates it in terms of Janelidze-style Galois theory, that’s the best I can hope for, for that’s what #1 is alluding to (as discussed around DCCT, remark 3.5.31 ).
on “differential Galois theory”, I have to write about its understood definition
IMHO, I think its definition should stay the same. I think it is better to call D-module Galois theory simply D-module Galois theory and etale Galois theory of D-schemes, the etale Galois theory of D-schemes, rather than unconventionally calling any of those differential Galois theory. Let the differential Galois stays for differential field/ring people, and D-geometry terminology for D-geometry people.
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