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started Galois cohomology
Is there a nice homotopy type theoretic description?
In cohesive homotopy type theory there is.
The delooping of the absolute Galois group $G_{Galois}$ of $K$ is $\mathbf{B}G_{Galois} \simeq \Pi(X)$ for $X \coloneqq Spec K$;
group cohomology of $G_{Galois}$ is the dependent product of function types between types dependent on $\mathbf{B}G_{Galois}$.
I made a note on this at Galois cohomology – In terms of cohesive homotopy type theory .
Is anything interesting going on there concerning the difference between fundamental groupoid and fundamental $\infty$-groupoid of $Spec K$? I seem to remember Minhyong dampening any such thought, his own interests being to get at the full noncommutativeness of $\Pi_1$.
By the way, first of all we can of course get $\Pi_1(X)$ in the type theory, too, simply by truncating the type $\Pi(X)$ to h-level 3. But there is a canonical projection $\Pi(X) \to \Pi_1(X)$ and so the representations of $\Pi(X)$ subsume those of $\Pi_1(X)$. So looking at $\Pi(X)$ retains all information of traditional 1-Galois theory but allows to refine it, if desired.
I seem to remember Minhyong dampening any such thought,
Could you remind me which thought this is about, more precisely?
his own interests being to get at the full noncommutativeness of $\Pi_1$
Here I am not following, maybe there is a misunderstanding. We are not tampering with the “non-commutativeness” of $\Pi_1$ by looking at it in homotopy theory.
Maybe I am missing your point, sorry. Maybe you could expand a bit more.
A generalization in the setup of corings:
It would be nice to see if the higher category theory reproduces this somehow. I left the reference at Galois cohomology.
It would be nice to see if the higher category theory reproduces this somehow.
For all these algebraic geometries, everything goes through in cohesive homotopy type theory if/as soon as the geometry is faithfully modeled over a site of geometrically contractible objects (that’s not quite a necessary condition, maybe, but a sufficient one).
(Here “geometrically contractible” means: contractible as seen by the corresponding étale homotopy theory.)
That’s why there is a model of cohesive HoTT in differential geometry, because there the $\mathbb{R}^n$-s constitute a good enough site of geometrically contractible objects.
So all of algebraic geometric, commutative or non-commutative, cannot be a model of cohesion. But a fragment of it can be, that of those “algebraic spaces” that are faithfully tested on geometrically contractible ones. If then in your application you are concerned with such spaces, then we are in business.
Re Urs #5, obviously the full fundamental $\infty$-groupoid contains the information of the fundamental groupoid. Minhyong’s point is that number theorists haven’t extracted everything from the latter about Diophantine equations.
There was a Non-Abelian Fundamental Groups in Arithmetic Geometry program at the Newton Institute. I attended a couple of talks at the final workshop, including Minhyong’s.
I can’t find, but I’m sure he said somewhere, that higher homotopy was not so interesting.
Thanks, David.
There are not maybe pdf slides, are there? Videos have such a low information/minute ratio.
he said somewhere, that higher homotopy was not so interesting.
I think we have to be careful about some distinctions here:
What Minhyong Kim probably means is that the higher homotopy groups and/or their representations of a space such as $Spec(K)$ for $K$ a field are not so interesting.
But the point of formulating Galois theory first entirely topos-theoretically, then higher-topos-theoretically and then homotopy-type theoretically is the observation that traditional Galois theory is just a tiny special subcase of a grand and general theory.
For instance elsewhere in homotopy theory, representations of $\Pi_1(X)$ are known as local systems or similar and these are already happily and fruitfully generalized, notably in derived algebraic geometry, to representations of $\Pi_\infty(X)$. There is a long list of literature on this for just the case of the $\infty$-toposes over a topological space at geometric homotopy groups in an (infinity,1)-topos. If anyone doesn’t trust me and wants to see big names, there you find them.
And this is still just a tiny subcase of full (higher) topos-theoretic Galois theory.
I find (I may have said that before) there is a curious mismatch in the community between the voiced appreciation for abstract sheaf theory and the active adoption of it. Most discussions are very much specific to specific sites and even very specific objects in the toposes over these sites (such as field extensions, etc. ). But there is a whole universe of phenomena beyond such isolated concrete particulars.
It’s interesting that with Mochizuki some algebraic/number-theory person is now, apparently, trying to apply genuine topos theory more systematically. Maybe this will lead to a change.
Preprint of NIkolai Durov (in Russian) is looking at more general philosophy of “good” embeddings (“realizations”) of some category of varieties (schemes etc.) to spaces (simplicial sets, infinity topoi etc.). Voevodsky-Morel motives and anabelian geometry being just examples. But he lists lots of requirements for good hypothetical realizations; to fullfill various axioms is difficult but desirable and itself an open problem. So the philosophy does not consider that varieties live in ONE infinity topos but that for various purposes of their analysis we realize them in different topological embeddings while trying to keep those properties which are of main geometrical or number-theoretic interest. Analysis of which properties to preserve and how to fullfill that is the main objective of further research. Hopefully there will be a longer English preprint in near future. Abstract below has an English version included.
I think we have to be careful about some distinctions here:
What Minhyong Kim probably means is that the higher homotopy groups and/or their representations of a space such as $Spec(K)$ for $K$ a field are not so interesting.
Yes, that’s why I said
Is anything interesting going on there concerning the difference between fundamental groupoid and fundamental ∞-groupoid of $Spec K$? I seem to remember Minhyong dampening any such thought…
But looking back, e.g., at this fruitless call, I see there’s a suggestion of something higher-categorical in the air.
Maybe the best place to start is the discussion we had with Minhyong at the Cafe.
By the way, I’ve also been chatting with Michael Harris, another number theorists, who’s developed an interest in homotopy type theory. But he’s been wondering whether it could yield anything in his own areas. For example, he writes
The Sato-Tate conjecture illustrates the principle that in number theory whatever is not governed by structure (like the purity of the eigenvalues of Frobenius) is governed by randomness (the Frobenius conjugacy classes are equidistributed in the Mumford-Tate group). Or equivalently, any departure from randomness is due to the presence of algebraic cycles that cut down the Mumford-Tate group. However, the notion of randomness presupposes the notion of Dirichlet density, and thus the pole of the L-function. If that has a categorical interpretation I don’t know what it is.
@David
Yes, that’s why I said
Oh, I see. Sorry, i did miss that.
Concerning the quote by Harris:
[…] the pole of the L-function. If that has a categorical interpretation I don’t know what it is.
I don’t expect that it is useful to look at any given complicated structure and ask if it has a categorical interpretation right away. As in that famous analogy, one instead has to start filling the valley with water. One day it may reach up to that complicated structure.
I am asking myself for years what the categorica interpretation of some complicated structure is. It seems that after many years I know now which valley to fill with water. Now we let the water flow. It’s already washing away lots of things, but it still needs to rise a good bit to get where I want it to be.
I expect that for Michael Harris and others it will be similar.
@Zoran,
that sounds good. So if you ever see Durov talk about looking at varieties in over a site of étale-contractible objects, let me know!
Wait a second, why would you like this property phrased in the source (it even can’t as one starts typically with 1-category) ? You want this property in the target, topological (eventually infinite categorical) setup ! So it is not about varieties of this type, but some category of varieties, which by some functor device lands in a topological setup where you have this (or some other) property.
Not sure what you mean by source here.
I am saying that the $\infty$-topos of sheaves $\mathbf{H} = Sh_\infty(C)$ over a site $C$of étale contractible objects has a good chance of being cohesive. So if you want to apply cohesive homotopy type theory reasoning to spaces in some category $MySpaces$, then you want to construct something like a faithful embedding $MySpaces \hookrightarrow \mathbf{H}$.
Perhaps on the same general theme, I have been looking at some of the ideas of realisation of simplicially based étale homotopy types and what is called étale realisation, e.g. in the paper by Quick (Profinite Homotopy Theory). Unfortunately although it is clear that that paper sketches out the right sort of track to take, there are some shortcomings in the exposition for instance if $X$ is a simplicial profinite space, the paper says it constructs a profinite fundamental groupoid by using a variant of the SGA method. This looks fine and seductive but fails to say how to make the profinite fundamental groupoid into a profinite groupoid. It is easy to apply the SGA machine to give it a profinitely enriched groupoid structure but I cannot see how that together with the topology on $X_0$ gives everything we need. There are ways around some of this using a profinite loop groupoid construction but then to show the two approaches give the same answer is not clear.
The homotopy theory he defines uses cohomology with continuous coefficients in a continuous local system but sometimes the details subtly fail to match what is needed, or so it seems to me.
The étale realisation stuff has been looked at by Isaksen (2004).
(What I mean by source here.)
Urs, varieties make typically some 1-category to start with. This source category is embedded in some nice target infinity category of ’spaces’; the embedding is required to have some good properties itself (like commuting with certain limits, for example). In this philosophy one has no reason to require that the varieties be over contractible site, or that this even makes sense to start with, but rather that, the contractibility be tested in the target infinity category. For different problems one will embed the varieties in different categories of spaces, sometimes one may have cohesiveness, sometimes some other useful property, with the same categories of varieties to start with. Of course, for some categories of varieties cohesiveness is likely easier to get, in that sense it “exists” in the source as you point out. Cf. 10 and the abstract (scroll for English version).
I am not sure, are we disagreeing about anything? You need an embedding $MySpaces \hookrightarrow Sh_\infty(C)$ with $C$ consisting of etale contractibles.
Well, I just thought that your saying “So if you ever see Durov talk about looking at varieties in over a site of étale-contractible objects” implies that such a source category of varieties can be recognized to start with, what I think would be too much to ask for. Instead one has tools to get some topological embeddings (a LONG list of other subtle requirements present in his paper) and then one can fiddle with various additional functors to extract this or that information, including possibly introducing such functors which you mention. It is a different philosophy.
In #10 you said
NIkolai Durov is looking at more general philosophy of “good” embeddings (“realizations”) of some category of varieties
That’s precisely what I am talking about, in reply to your desire in #6 to see cohesive homotopy type theory be able to speak about these varieties: the good embedding one needs is into a category of sheaves over a site of etale-contractible objects.
This is not about philosophy. This is the technical reply to your question! I would say.
Is the infinity category of motives of Voevodsky-Morel such ?
Is the infinity category of motives of Voevodsky-Morel such ?
The short answer is “no”, but let’s look at how this comes about to what would make it a “yes”:
So I suppose “the infinity category of motives of Voevodsky-Morel” is
4) The stabilization…
3) …of the A1-localization…
2) …of the (infinity,1)-topos $\mathbf{H}$…
1) … over the Nisnevich site $Alg^{op}_{Nis}$.
So at stages 4) and 3) we do not have an $\infty$-topos, and hence there is no point in asking whether it is cohesive. But of course both 3) and 4) can be thought of as taking place in or relative to the $\infty$-topos $\mathbf{H}$ in stage 2), and so we’d be happy if that were cohesive (it is connected by reflections to stages 3) and 4), so that’s okay).
Now, as I keep saying, $\mathbf{H} = Sh_\infty(Alg^{op}_{Nis})$ won’t be cohesive, because the smooth schemes that are the objects of $Alg^{op}_{Nis}$ are not locally étale contractible, and hence there is no dense subsite of objects which are étale contractible.
We need a different site $C$ than $Alg^{op}_{Nis}$ whose objects are locally étale contractible!
You may remember from earlier discussion that I keep saying a good guess seems to be the site of smooth Berkovich analytic spaces. Because these are locally contractible by a celebrated result of Bekovich’s.
So I am guessing that the answer to your question becomes a yes if you replace (motivic) geometry modeled over smooth schemes by geometry modeled over smooth analytic spaces.
But, as I kept saying elsewhere, I haven’t worked this out. It’s just an educated guess. Hopefully somebody will work it out. Maybe Durov already knows this, in slightly different language, maybe.
The subtlety is that what Berkovich actually shows is that the underlying topological space of a smooth Berkovich analytic space is locally contractible. But we need that the analytic space itself is locally étale-contractible. This is potentially a slightly different question. You may remember that I once emailed Berkovich to ask this question. But it remains open.
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