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As mentioned, it’s the pyknotic approach that actually gives rise to toposes, so perhaps pyknotic cohesion would be more appropriate. But I wouldn’t be surprised if ’condensed’ prevails.
Now what is that word ’essentially’ doing in Scholze’s comment?
there are such sites that are (essentially) cohesive over condensed sets; for example, the big pro-etale site on all schemes over a separably closed field $k$.
Can one consider the idea of homotopy dimension relative to a condensed base?
Enough for now, but this MO question and answer probably contain something useful.
As Scholze has suggested many times, I wouldn’t emphasise the distinction between condensed and pyknotic; it is just a question of size. In some ways it might be best with some neutral name that is neither of the two! But if one has to make a choice, I’d stick with condensed, as the overall story into which condensed sets fits (where this name is used) is broader at the time of writing as far as I see.
I think the essentially refers for example to the non-existence of the right adjoint to $f_*$, i.e. co-Disc. I agree with Scholze, however, that this right adjoint does not seem reasonable to ask for in general; when I was thinking about this kind of axiomatics long ago, the right adjoint was not part of the picture, this was suggested by David Carchedi and Urs I think (following Lawvere? Or possibly they came up with it and only later discovered Lawvere’s work).
Also the failure of the $\pi_\infty$ functor to preserve finite products, which will not be able to be corrected in general, though it holds sometimes.
Thanks, David, for starting this.
But if in the condensed setup the right adjoint does not exist…
(right, we started looking at that when considering embedding of concrete objects and diffeological spaces with Dave Carchedi – it’s presence is needed for some constructions but not for others, notably the differential cohomology hexagon does not need it)
… then we shouldn’t be speaking of “cohesion”.
I am wondering then how much more it really says than that there is always a pro-left adjoint?
But I don’t have the leisure now to look into this myself. I am hoping you can make a clean writeup of what is known, for me to come back to once I feel I have the time… :-)
In that vein, I’d suggest to keep it concrete and down to earth, plain info like “Given XYZ, there is an adjunction ABC, which satisfies UVW.”
Once we see that really clearly, the more visionary outlook will write itself. As long as we don’t have that, the visionary outlook risks to be hallucination…
In the condensed setting, what is true is no more and no less that there is an actual left adjoint with condensed sets as base. This is stronger than having a pro-left adjoint, and will turn out to be very conceptually important I believe. In the pyknotic setting, apparently the right adjoint does exist, but this seems a bit of a red herring. In neither setting does the left adjoint necessarily preserve finite products.
Thanks, Richard. You seem to have a good grasp of the situation. Maybe you could add that to the entry?
If the extra right adjoint is there in the pyknotic setting, we should record that in the entry to justify the entry’s title!
What I mean about pro-left adjoint is that changing the base topos from $Sh(\ast)$ to $Sh(ProFinSet)$ is the kind of move we’d expect for fitting in a pro-left adjoint with codomain $Pro(Sh(\ast))$. But I haven’t looked into it and will shut up now.
What I mean about pro-left adjoint is that changing the base topos from $Sh(\ast)$ to $Sh(ProFinSet)$ is the kind of move we’d expect for fitting in a pro-left adjoint with codomain $Pro(Sh(\ast))$.
I agree, but it is of course non-trivial to find a topology on profinite sets (and more generally on schemes other than the point) which works, which is where the work of Scholze/Bhatt/Clausen comes in. Yes, I can try to tweak the entry when I get a chance.
I would suggest that the entry could be called ’pro-étale cohesion’ to cover both the condensed and pyknotic cases. I believe that the same story works not only in algebraic geometry, but also for topological spaces, i.e. one can get cohesion in that case with a pro-étale-like topology without restricting to locally connected spaces. I began adding a proof of this to the nLab last year, but got distracted and didn’t finish it.
to cover both the condensed and pyknotic cases.
Except that the condensed case is not actually cohesive, lacking the extra right adjoint? Maybe “algebraic cohesion”.
Regarding the (non-)preservation of finite products: Maybe the condition here needs to be adjusted to account for the non-trivial base topos?
Or, if it turns out that the idea of actual cohesion cannot be salvaged here, after all, probably the creative energy that gave us pyknotic condensed anima can be channeled to make up a new term altogether ;-)
Re #12, that sounds like a good idea to change title. Given that this is all about pairs of $(\infty, 1)$-toposes, it surely makes sense to follow Barwick-Haine’s construction who, red herrings or no, actually employ such things, and as a bonus give us the right adjoint too.
So the only issue is the left adjoint not preserving finite products.
Looking at things in the large, as we scan the $(\infty, 2)$-category of $(\infty, 1)$-toposes looking for cohesively related pairs, is it possible that such lesser forms of cohesion (i.e., not preserving finite products) are the norm?
In the nCafé conversation, when Scholze argued that finite products aren’t preserved:
This is actually a nontrivial question, nontrivial approximations to which are true (related to Künneth formulas in etale cohomology)
he was speaking about schemes in general. Why in this MO answer did he restrict to schemes over a separably closed field $k$?
I think this is simply so that the pro-étale topos of the point is condensed sets.
That makes sense.
Back to cohesion in general, there are further possible conditions. What determines that ’finite-product preservation’ is part of the standard package and these are add-ons is, I imagine, decided by what’s met in nature and their consequences.
It probably needs some tunnelling back and forth from the constructions that already exist to the abstract general work in cohesion. If the differential cohomology diagram doesn’t require finite-product preservation, that might be a good starting point.
What determines that ’finite-product preservation’ is part of the standard package and these are add-ons is, I imagine, decided by what’s met in nature and their consequences.
Indeed. In the pro-étale setting, one does have things akin to finite product preservation (Künneth theorems in étale cohomology), just not the full strength of it in general.
I am slowly building up to editing the entry. I wish to be able to prove the existence of the left adjoint; this is what my edit to finite set yesterday is the start of.
Just as a marker for a possible target:
Cohesion does not seem to have been applied in algebraic or $p$-adic contexts. However, I realized recently (before this nCatCafe discussion), in my project with Laurent Fargues on the geometrization of the local Langlands correspondence, that the existence of the left adjoint to pullback (“relative homology”) is a really useful structure in the pro-etale setting. I’m still somewhat confused about many things, but to some extent it can be used as a replacement to the functor $f_!$ of compactly supported cohomology, and has the advantage that its definition is completely canonical and it exists and has good properties even without any assumptions on $f$ (like being of finite dimension), at least after passing to “solid $\ell$-adic sheaves”. So it may be that the existence of this left adjoint, which I believe is a main part of cohesion, may play some important role. (Scholze, MO)
Apologies for the lack of updates, for the last couple of evenings I have been thwarted in my wish to work on this by nLab software matters! Hopefully I’ll find some time tomorrow evening.
So we now have some information to feed in from Scholze, here. Seems to think that the fracturing of the differential cohomology diagram isn’t very instructive.
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