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some trivial/stubby edits, announced here just in case anyone is wondering about edit activity:
added more references to étale (∞,1)-site
added a tad more text to the stubs Weil cohomology, étale cohomology
created stub l-adic cohomology
started splitting off formally étale morphism of schemes from formally étale morphism
I added a definition to l-adic cohomology plus a reference to Milne’s notes.
Thanks!
I added a few characters to make more explicit what your limit is over,
I have added this reference at etale site:
Got interested in that due to the claims there that the pro-étale sites are locally contractible (since on and off I am looking for useful categories of locally contractible schemes, for they would make a good context for algebraic cohesion). But I am not sure yet how the notion of locally contractible toposes in that articles relates to the notion that I have in mind.
Is this to look for something ’absolutely’ cohesive (i.e., relative to $\infty-Grpd$)? Does this give something beyond cohesion relative to another $\infty$-topos?
Maybe I am wrong, but my intuition is that if there is some flavor of a subcategory of schemes which makes a cohesive site, then that should be absolutely cohesive. By analogy with the case of manifolds.
Once that works, the corresponding formal schemes should then be cohesive over “fromal moduli problems”.
4: I mentioned recently this brilliant reference here. Wodzicki (who was delighted by appearance of this work) was saying that this gives also a light to how to repair the notion of smoothness of schemes, and remarked that pro-etale is not the best term as it is not directly about pro-objects in etale site, but maybe eventually the terminology is having right meaning. I am interested in your further insight.
Okay, thanks!
So that’s what I am asking in another thread: the text is vaguely suggestive of the statement that weakly étale maps are formally étale maps which are “locally pro-finitely presentable”. But in which sense is this so?
Yes, I agree it is suggestive/interesting to compare (but also non-obvious to decide), thanx.
If you have any further info on the pro-étale story, be it handwritten notes, personal communication or whatever, I’d be very interested.
Surely, I’ll try to discuss again with Mariusz.
Got interested in that due to the claims there that the pro-étale sites are locally contractible (since on and off I am looking for useful categories of locally contractible schemes, for they would make a good context for algebraic cohesion). But I am not sure yet how the notion of locally contractible toposes in that articles relates to the notion that I have in mind.
I’m pretty sure their pro-étale ∞-topos is locally contractible in the way you want, i.e., any object has a cover by objects of trivial shape. This is basically Lemma 2.4.9 in the paper. They allow non-finitely presented coverings though, which makes the result slightly less surprising.
That statement is what made me get interested, but I still have to think in detail about how that makes the pro-étale $\infty$-topos have an extra left adjoint to its inverse image functor. Maybe it’s obvious…
That statement is what made me get interested, but I still have to think in detail about how that makes the pro-étale $\infty$-topos have an extra left adjoint to its inverse image functor. Maybe it’s obvious…
This is Proposition A.1.8 in HA.
That prop. gives local contractibility from local contractibility of covering slices. So we first need that the slices over the w-contractibles are locally contractible.
But, yeah, I suppose that’s pretty immediate. For instance: consider the dense subsite of w-contractibles. This is an “infinity-connected (infinity,1)-site” by definition of w-contractibility, and that implies the statement.
What I find curious about the w-contractibles in the pro-étale topos is that they are not just geometrically contractible, but indeed are atomic. So they are really small. :-)
Also, with hindsight now I am a bit baffled that they haven’t been observed before in étale geometry…
The big question arising here is of course now whether the “big pro-étale topos” (of sheaves over the opposite category of w-contractible rings (not over any fixed scheme)) is cohesive…
@Urs Or more explicitly, if you have a site in which every object is covered by contractible objects, then to find the shape of $X$ you pick a hypercover by contractibles and the indexing simplicial set gives you the shape, doesn’t it?
Now I’m wondering what the pro-étale shape of a field is. It should be some homotopy type. Is it simply the classifying space of the Galois group as a discrete group???
What I find curious about the w-contractibles in the pro-étale topos is that they are not just geometrically contractible, but indeed are atomic. So they are really small. :-)
I might be confused here, but it seems to me that the contractible schemes for all of the standard topologies (more precisely, any coherent topology) are precisely the tiny objects, which are the points of the topos. E.g. the Zariski-contractible schemes are the local schemes, the étale-contractible schemes are the strictly Hensel local schemes, etc. The difference with the pro-étale site is that it is large enough to actually contain the points, which is not usually the case (e.g. the separable closure of a field $k$ is not of finite presentation so does not live in the étale site of $k$, but it is a genuine contractible cover of $k$ in the pro-étale site).
Concerning shape: sure!
Concerning contractible schemes: I have a real gap in my education here, and maybe you are now finally filling that gap..
So ever since cohesion hit the scene here we kept saying that we need to find a good category of schemes covered by contractibles. I had not been aware that such a something had been considered. To my mind the pro-étale site now is the first example that I see. But maybe I am missing something really basic here. Or maybe your last message just confirms this.
Given the title of this thread, another interesting question is now: is there an analog of this stament for $E_\infty$-rings with the pro-étale topology?
Is there a decent notion of “reduced $E_\infty$-ring” in the sense of “no non-trivial nilpotents? Maybe the fiber product with the reduced part of the $\pi_0$-ring? Is this discussed anywhere?
The big question arising here is of course now whether the “big pro-étale topos” (of sheaves over the opposite category of w-contractible rings (not over any fixed scheme)) is cohesive…
Let’s look more generally at the big pro-étale (∞,1)-topos of $B$-schemes for a given scheme $B$. If $hodim(B)\leq 0$ (what Bhatt and Scholze call “weakly contractible”), then $B$ is atomic which means the global section functor $\Gamma_B$ preserves colimits. If moreover $B$ is connected then it truly is contractible so the shape functor $\Pi$ will preserve $*$. It remains to check whether $\Pi$ preserves binary products. That would imply a Künneth formula for pro-étale cohomology with general constant coefficients. From section 5 in Bhatt-Scholze I gather this would imply Künneth in ordinary étale cohomology, which fails with $\mathbb{F}_p$ coefficients for schemes of characteristic $p$ (e.g. any affine scheme has $p$-cohomological dimension $\leq 1$)… This last problem is unlikely to go away with any étale-like topos, unless you assume $char(B)=0$. So at best you could hope for cohesion for schemes over a connected w-contractible ring of characteristic zero.
And if $hodim(B)> 0$, $\Gamma_B$ does not even preserve colimits, so in particular the “absolute” big topos, over $Spec(\mathbb{Z})$, is not even local. Is the sphere spectrum étale-contractible?
Hmm, now I’m wondering if this topos even is locally connected. As I noted above, “weakly contractible” in that paper only means “of homotopy dimension $\leq 0$“…
Concerning locality, of course we are not expect the slice over a “finitely extended” space to be local.
Concerning the thing about Künneth, thanks for pointing this out, that’s useful. (Cohesion without the product-preservation is still useful. The whole discussion here vaguely reminds me of the discussion here of how the slice of a cohesive topos over an atomic object is cohesive without possibly product-preserving shape.)
Concerning local connectivity: so Bhatt-Scholze’s weak contractibility is just “well-supported objects have points”, which is homotopy dimension $\leq 0$, yes. But existence of coverings by w-contractibles is stronger. w-Contractibles have contractible étale homotopy type, hence form a dense “locally $\infty$-connected site” (unless I am missiing something…) as mentioned in #15 and that gives the left adjoint.
I should have said more clearly what I meant by the “big site” above. I didn’t mean that big étale site over anything, but the one that gives what should be the “gros topos” in this context here:
some category of schemes, not equipped with étale maps into anything, with coverings the faithfully flat pro-étale morphisms. Some analog of the site of all smooth manifolds, not equipped with maps into anything.
In this context, one could consider the full sub site of only all w-contractible schemes (still not equipped with étale maps into anything). In the analogy that would be like the full subcategory of Cartesian manifolds inside all manifolds.
That seems to be a locally infinity-connected (infinity,1)-site. I still need to think about whether it is also an infinity-local site (though either way it may be obvious). If it were, that would make it an infinity-cohesive site, possibly up to the issue with products.
Whatever it is, once one has this “big” site (maybe I should say “gros” site or something else instead) then inside the gros $\infty$-topos of that site one finds all the (pro-)étale toposes over some base by the general construction here.
Do you see something that might be called ’dynamics’ then in this setting?
@Urs, My comments in #21 apply in particular to the big topos you describe (it’s the special case $B=Spec(\mathbb{Z})$) and show that it is not local for trivial reasons. Restricting the site to w-contractible instead of arbitrary schemes does not change the (∞,1)-topos of sheaves, since both are hypercomplete (in fact Postnikov towers are convergent). I was suggesting that the derived version may be local if the sphere spectrum is étale-contractible, but even then you’d need to pass to characteristic zero for any hope of product-preserving cohesion.
w-Contractibles have contractible étale homotopy type
That’s not true: a disjoint union of w-contractibles if w-contractible. I agree that a connected w-contractible is pro-étale contractible, but w-contractible schemes are not necessarily disjoint union of their connected components… In other words, the $\pi_0\Pi$ of a w-contractible can still be a non-constant pro-set, which means that the pro-étale topos (big or small) is not locally connected.
but w-contractible schemes are not necessarily disjoint union of their connected components…
Oh, right, sorry. That’s an evident trap i fell into there.
Hm, maybe it works after passing to the topos over profinite sets as the base topos… But I clearly should shut up for the moment and look at the details a bit more.
Oh, right, sorry. That’s an evident trap i fell into there.
So did I, that’s a very unusual thing for schematic topologies, and it doesn’t help that their choice of terminology is “wrong”. One might say that passing from étale to pro-étale is trading local connectedness for hypercompleteness…
Speaking of hypercompleteness, I don’t fully understand their proof (Prop. 3.2.3) that the pro-étale topos is hypercomplete. They seem to implicitly use the following implication for a 1-topos:
(?) If any object is covered by bananas, then the associated 1-localic (∞,1)-topos is generated by bananas under colimits.
Why is this the case, if we don’t already know hypercompleteness?
Speaking of hypercompleteness, I don’t fully understand their proof (Prop. 3.2.3) that the pro-étale topos is hypercomplete. They seem to implicitly use the following implication for a 1-topos:
(?) If any object is covered by bananas, then the associated 1-localic (∞,1)-topos is generated by bananas under colimits.
Why is this the case, if we don’t already know hypercompleteness?
Their whole argument only seems to make sense inside a hypercomplete (∞,1)-topos anyway (they use homotopy groups…), so I assume they are considering the hypercompletion in the first place, and that they are trying to prove Barwick’s version of hypercompleteness, which is that every object is the limit of its Postnikov tower. I still don’t understand their argument under these assumptions (I think they assume that homotopy groups commute with limits of Postnikov towers, which is not true as we’ve discussed before), but in any case the hypercompletion is locally of homotopy dimension $\leq 0$ so we know Postnikov towers converge there.
So here’s a summary of my current understanding of things:
I strongly suspect that:
These guesses are based on Example 4.1.10. They would follow if, for $A$ w-contractible, the pro-finite set $\pi_0(Spec(A))$ were actually the pro-étale shape of $\Spec(A)$. This pro-finite set is in any case the Zariski shape of $Spec(A)$, which receives a map, natural in $A$, from the pro-étale shape. So if the pro-étale shape is a point, then there has to be a point in $\pi_0(Spec(A))$ which is fixed by all endomorphisms of $A$, which already rules out many w-contractible rings.
ETA: The last two points are in fact true: it’s easy to show, by faithfully flat descent, that a constant Zariski sheaf of sets (on all schemes) is an fpqc sheaf, whence a pro-étale sheaf. So the map from the pro-étale shape to the Zariski shape is an iso on $\pi_0$.
Thanks, Marc, that’s great. I will have to postpone further thinking about this until I have some urgent other task out of the way.
But let me bounce one vague thought off you, again: wouldn’t it be very natural to consider contractibility and locality not over the base $Sh_\infty(\ast) \simeq \infty Grpd$, but over the base $Sh_\infty(ProFinSet)$?
@Urs, Yes, that’s an interesting idea. I think Bhatt and Scholze do something like that in section 4. In particular, they show in 4.1.9 that there is a “constant pro-finite set” functor to $X_{proet}$ which preserves limits. That sounds like local connectivity over pro-finite sets. For locality perhaps the big topos over a w-contractible ring is local over pro-finite sets, but $Spec(\mathbb{Z})$ is still going to have positive relative homotopy dimension.
Thanks. Or maybe one could/should use just extremally disconnected profinite sets, since only they appear underlying the w-contractibles (if that makes any difference for the sheaves, I don’t know.)
This makes me think again of an old idea: given any $\infty$-topos $\mathbf{H} \to \infty Grpd$, one could/should ask for a universal factorization $\mathbf{H} \to \mathbf{B} \to \infty Grpd$ such that $\mathbf{H} \to \mathbf{B}$ becomes locally contractible/local/cohesive. A refinement of the classical hyperconnected-localic factorization, maybe.
Here maybe $\mathbf{B} = Sh_\infty(ProFinSet^{extdisc})$, at least for local contractibility or something.
(?) If any object is covered by bananas, then the associated 1-localic (∞,1)-topos is generated by bananas under colimits.
For the record, this statement is false. See David Carchedi’s post here for a counter-example involving the infamous Hilbert cube. So it’s not clear at all whether w-contractible schemes generate the pro-étale (∞,1)-topos.
Here maybe $\mathbf{B} = Sh_\infty(ProFinSet^{extdisc})$, at least for local contractibility or something.
I would guess $\mathbf{B}=Sh_\infty(ProFinSet)$, since that is the pro–étale topos of a scheme with trivial pro-étale shape.
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