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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 with sets as base, 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.
The question arose in another thread as to whether this page is worth keeping.
We have Scholze saying
this relative notion of cohesiveness may be a convenient notion. In brief, there are no sites relevant in algebraic geometry that are cohesive over sets, but 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. So in this way the approach relative to condensed sets has benefits.
Now while people lined up to say the right adjoint to points is unimportant, it is part of the definition of cohesion. If it’s not available, the page must be renamed or removed. Is it that pyknoticity fixes the right adjoint issue? It does for pyknotic sets over sets.
Then, what is the status of the left adjoint to the discrete functor?
Thanks. Not wanting to be a pain, but I do feel most content of this entry remains out of place until the basic presumption is at least explicitly stated, if not referenced:
Are gros pro-étale $\infty$-toposes at least locally $\infty$-connected over condensed sets? I.e. do they have a geometric morphism to condensed sets with an extra left adjoint?
Until that is clearly stated and referenced, the entry reads like discussing the decoration of the panorama platform of Burj Khalifa before the foundation stone has even been laid.
I was under the impression that this was known to those who thought about it, and that the point of this entry would be to give this latent community knowledge a more explicit incarnation. That would be really useful!
So does that extra left adjoint exist? The MO-answer that the entry links to, MO:356836, is ambiguous in its wording: It starts out saying that “this has been observed long before” but points not to proof but to an old comment by myself raising the very question here.
But then the following sentence in the entry, starting with “Consider then…”, seems to suggest that the left adjoint does not preserve all products. This is a technical detail that must originate from somebody who knew the left adjoint does exist. Where is that statement from? This should be made the first statement in the entry.
We have Scholze saying
For a scheme $X$, one can consider the small pro-etale topos $X_{proet}$ built from the site of schemes pro-etale over $X$, and pro-etale coverings, and the big pro-etale topos $X_{Proet}$ built from the site of all schemes over $X$, and pro-etale coverings. There is a natural map of topoi $f: X_{Proet} \to X_{proet}$.
Proposition. $f^{\ast}$ has a left adjoint, and in the pyknotic formalization, $f_{\ast}$ has a right adjoint. Moreover, $f^{\ast}$ is fully faithful.
Then arguing that the left adjoint doesn’t preserve finite products.
After being alerted to your discussion with Marc Hoyois, he added about the left adjoint
I think the failure to be product-preserving should also be there in characteristic 0, although this is slightly more subtle.
That’s why I emerged from the discussions thinking that there is an adjoint quadruple (at least if pyknotic), but that the left adjoint doesn’t preserve finite products.
There were then questions of what still went through even if this is the case.
Thanks. This would be the proposition that the entry should state and reference!
Where is this quote from? I don’t see this in any of the four references given in the entry (I may have missed it as I have only scanned over them).
And is this claimed for 1-toposes or for $\infty$-toposes?
It’s a comment here.
There he’s only talking about 1-toposes, but doesn’t appear to make a difference responding to your discussion with Marc Hoyois.
Thanks. I see.
Okay, if you don’t mind, I”ll rework the entry a little now to bring out this statement and this reference.
Okay, I have edited the entry:
Expanded the Idea-section a fair bit (here), making it culminate in the main claim, now cited as Scholze 2020.
Also expanded the wording and hyperlinking of the application-sections on spectra, differential cohomology and on singular cohesion.
Moved the section on spectra to before that of differential cohomology, for flow of the logic. Renamed from “infinitesimal cohesion” to “stable cohesion” to give the reader better indication of what’s being conveyed here (as “infinitesimal” is ambiguous and not suggestive of the main point here).
Regarding the remark on pyknotic spectra: I haven’t touched it, but just to say that without following the pointer it is not clear (to me anyways) what the intended upshot of this remark is.
Maybe related to this: I have struggled a little with the wording regarding the base topos of the cohesion here: I understand that “condensed sets” refers only to the subcategory of small sheaves inside the full base topos over the pro-etale site of the point. But do we have a nice name for this full topos? At one point in the Idea-section I now made it read “topos-completion of condensed sets”, but that’s suboptimal.
Great. That looks much better.
Okay, thanks.
By the way, in one of the linked comments Peter S. voiced what seems to be a wide-spread opinion: That the concreteness-mode exhibited by the extra right adjoint does not seem to be so important, suggesting that the requirement of the existence of the extra right adjoint might just as well be dropped from the axioms of cohesion, with all the heavy lifting in cohesion done by the extra left adjoint. Therefore it’s curious to observe that for the singular cohesion over a smooth base, as in our “Proper Orbifold Cohomology” it’s the other way around: Here the main interest revolves around the rightmost “orbi-singular” modality $\prec$, while the leftmost modality $\lt$ seems to play a more modest role.
On the other hand, the failure of condensed cohesion to have a product-preserving shape seems a little sad, now that we have come so far: The key point of product-preserving shape is that it preserves deloopings of $\infty$-groups and hence sends moduli $\infty$-stacks to their underlying classifying spaces. That fact drives a lot of theory. For instance, together with the smooth Oka principle this fact implies the traditional classifying theory of principal bundles – and generalizes it to $\infty$-bundles and also to equivariant $\infty$-bundles (as I am busy to type out proof for as we speak). So the fact that condensed shape does not preserve finite products does not seem to be a minor failure. Maybe this should be double-checked, at least in the case of the base scheme being the point.
It seems odd when the choice of pyknoticity gives both a topos structure and the right adjoint not to prefer it over condensedness.
As for finite-product preservation, Richard had some ideas about that.
It is definitely true that the left adjoint cannot preserve products. The left adjoint can more or less be thought of as the étale homotopy type of a scheme, but with better technical properties. If it preserved products, one would immediately be able to deduce a Künneth theorem in étale cohomology, which does not hold in general.
But I think this may be solvable by adding a little more structure to the topoi involved. The kind of thing I had in mind in the comment David refers to is for instance discussed in the following paper, where one adds a bit of extra structure (the ’Weil-proétale site’). The introduction of v1 of the paper is worth a glance before turning to v3.
https://arxiv.org/abs/2012.02853
As for the right adjoint, I too am not convinced of its importance for the general picture and never have been, but this may be because my intuition is not coming from cohesion as such. In particular, I don’t see how indiscreteness/concreteness, etc, could possibly be interesting in situations like algebraic geometry which are a long way from being set-like; maybe this is a question of lacking the appropriate intuition, though.
Thanks!
Regarding terminology, since it was mentioned again: What’s good established terminology for the full topos over the pro-étale site of the point? i.e. for the base topos for the almost-cohesion that we are talking about, the one in which the condensed sets form just the subcategory of small sheaves. Is “pyknotic sets” the name for this full topos? If so, that would be good to openly admit in the entry pyknotic set.
Regarding the right adjoint: As I said above (following the discussion in Proper Orbifold Cohomology), in the cohesion of global homotopy theory $Glo(\mathbf{H})$ over a smooth cohesive base $\mathbf{H}$, it’s the two right adjoints $\sharp$ and $\prec$ that manifestly do all the work regarding the equivariant structure: For $X \in \mathbf{H}$ 0-truncated and concrete (hence $\sharp_1$-modal) and equipped with the action of a discrete group $G$, it is $\prec (X \!\sslash\! G)$ which “is” the corresponding cohesive orbispace in the equivariant homotopy theory $G(\mathbf{H})$, the one whose shape $ʃ \prec (X \!\sslash\! G)$ in $G Grpd_\infty$ is the expected underlying proper $G$-equivariant homotopy type.
The right adjoint is underappreciated, but it has a lot of power. It’s the best way to embed “non-cohesive” mathematics into the cohesive world, since the discrete types are not lex-reflective.
Some comments by Scholze from the thread:
Either you cut off profinite sets (or compact Hausdorff spaces) at some (strongly inaccessible) cardinal $\kappa$, to get what Barwick-Haine call pyknotic sets and Clausen and I call $\kappa$-condensed sets, or you take the (large) filtered colimit over all $\kappa$ of these categories, to get condensed sets.
Regarding language: Clausen and I define κ-condensed sets as sheaves on the site of $\kappa$-small profinite sets, for any strong limit cardinal $\kappa$. If $\kappa$ is strongly inaccessible, these things are also called pyknotic sets (and the assumption of strong inaccessibility is often far too strong for what one is doing, strong limit cardinal is usually enough), and let me stress here that pyknotic sets depend on this implicit choice of $\kappa$. Now condensed sets are the union of $\kappa$-condensed sets along canonical fully faithful embeddings. These can also be defined as small sheaves on the category of all profinite sets – i.e. those sheaves that are small colimits of representable sheaves. So from the condensed point of view, one can easily talk about the pyknotic one by saying κ-condensed.
(Barwick and Haine have a different perspective, where they organize their choices of universes in a different way and end up seeing condensed sets as a subcategory of pyknotic sets. I admit that I find that perspective confusing – they first cut off their profinite sets at some strongly inaccessible $\kappa$, and then their condensed sets are the ones of $V_{\kappa}$, so they are certain sheaves on $\kappa$-small profinite sets with values in $\kappa$-small sets. So even what they call condensed sets depends on the choice of $\kappa$; arguably, they misuse the term.)
The advantage of the pyknotic approach is that you really have a topos, which comes with some benefits: You can apply adjoint functor theorems with no worries, your topos even has enough points, you have indiscrete pyknotic sets, …
For Barwick and Haine, there’s a functor $Pyk$ which applies to all finite-product categories, and sends toposes to toposes.
Pyknotic sets are defined relative to an a priori fixed inaccessible cardinal $\kappa$, and are then the resulting (honest!) topos of $\kappa$-condensed sets, sitting inside the pretopos of all condensed sets. Scholze explained this point of view to me, whereas before I was assuming that condensed sets were somehow the subcategory of small sheaves of the topos of pyknotic sets. One could in principle talk about condensed sets taken inside some Grothendieck universe, but this is not the intended definition, and Scholze is very careful to avoid assuming universes.
But what’s the answer to my question: Any established terminology for the (objects of) the sheaf topos over the pro-étale site of the point? And for the corresponding hypercomplete $\infty$-topos?
Assuming you’re happy with a non-locally small category?
I wouldn’t think I am being cryptic: I am asking, straightforwardly, how the condensed community prefers to refer to the base topos/$\infty$-topos that we are talking about.
It looks to me like a good naming convention for the base might be pro-étale sets and pro-étale $\infty$-groupoids and then for the almost-cohesive gros topos over it pro-étale algebraic sets and pro-étale algebraic $\infty$-groupoids. But that’s just me. I am not invested in this and will not further insist.
But if there is – or when there emerges – some agreed-upon terminology for this in the condensed community (or the pro-étale community ;-), somebody might want to adjust the wording in the entry.
After what I cited above
For a scheme $X$, one can consider the small pro-etale topos $X_{proet}$ built from the site of schemes pro-etale over $X$, and pro-etale coverings, and the big pro-etale topos $X_{Proet}$ built from the site of all schemes over $X$, and pro-etale coverings. There is a natural map of topoi $f: X_{Proet} \to X_{proet}$.
Proposition. $f^{\ast}$ has a left adjoint, and in the pyknotic formalization, $f_{\ast}$ has a right adjoint. Moreover, $f^{\ast}$ is fully faithful,
Scholze writes
Taking $X$ to be a geometric point $Spec k$, the target $X_{proet}$ here is the topos of pyknotic sets… all assertions are true both on the level of usual topoi, and on ∞-topoi of hypercomplete sheaves.
Does that settle it? Mind you I see the Stacks project being cautious about saying “a big/small pro-étale site” rather than “the”. Does that matter?
Re #41: One just says ’the’ pro-étale topos (big/gros or small/petit), where ’the’ means either that one has chosen an appropriately sized cardinal to work below, or that one uses universes. See the original article of Bhatt and Scholze for instance (https://arxiv.org/abs/1309.1198). This can be the (big or small) pro-étale topos of the point, or of a scheme. At the level of objects, one just says a pro-étale sheaf on the point, or on the scheme in question.There is really nothing deep going on!
Re #35: Alas, I have no time to think about this properly, but are you saying that one can express equivariance in a purely axiomatic/formal way using the right adjoint? That would be fascinating if so, and I might be converted to its importance:-). If that is correct, is there any way you could summarise in a few lines how that works axiomatically?
Something that I would love to do (or would have loved to have done 10-15 years ago when I was working on this picture) for example would be to work with something like $\mathbb{G}_{m}$-equivariant étale homotopy types, as these I believe would encode arithmetic aspects of étale cohomology theories that are usually only captured by the Tate twists of objects one maps into, i.e. it would give a kind of way to deloop étale homotopy types with respect to Tate twists. If we did have the right adjoint in the pro-étale, or Weil-pro-étale setting, could you provide me with such a theory axiomatically?
Regarding terminology:
Okay, thanks, so I have now edited the entry here to say that the cohesion is “over pyknotic sets”. That’s what I thought in #35, hoping somebody would confirm, as our entry pyknotic set gives no indication that this the way to speak. I am not thrilled about the term “condensed cohesion over pyknotic sets”, but I really leave this to you/others now.
Regarding the singular right adjoint:
I have now started an entry on this at singular cohesion.
The main point is this: With $Singularities$ denoting the “global orbit category”, i.e. the full sub-$(2,1)$-category of groupoids on those that are deloopings $B G$ of finite groups $G$, we have for any (cohesive) $\infty$-topos $\mathbf{H}_{\subset}$ that
$\mathbf{H} \,\coloneqq\, Glo(\mathbf{H}_{\subset}) \,\coloneqq\, PSh_\infty(Singularities, \mathbf{H}_{\subset})$is cohesive over $\mathbf{H}_{\subset}$, simply because $Singularities$ has a terminal object and we are considering just presheaves instead of sheaves.
Here the direct image morphism, which I am going to denote $Smth$, evaluates on that terminal object. In particular, it sends the delooping of $G$ regarded as an object of $Singularities \xhookrightarrow{y}\mathbf{H}$ to the same delooping but regarded as a geometrically discrete $\infty$-groupoid.
Therefore the right adjoint modality, which I now denote “$\prec$”, sends the homotopy quotient $X \sslash G$ of a 0-truncated concrete object $X$, regarded in the slice over $\ast \sslash G$, to the presheaf on $Singularities_{/\prec B G}$ which to any subgroup $H$ of $G$ assigns the the space of morphisms from $\ast \sslash H$ to $X \sslash G$ sliced over $\ast \sslash G$, and that’s just the $H$-fixed points of $X$.
This way, for $X \,\in\, \mathbf{H}_{\subset} \xhookrightarrow{\;} Glo(\mathbf{H}_{\subset})$ 0-truncated and concrete, its “orbi-singularization” $\prec (X \sslash G) \,\in\, Glo(\mathbf{H}_{\subset})$ is essentially the presheaf of fixed loci of $X$, and its shape $ʃ \prec (X \sslash G) \,\in\, Glo(\mathbf{H}_{\subset})$ is $X$ regarded as a $G$-space in equivariant homotopy theory under Elmendorf’s theorem.
If we did have the right adjoint in the pro-étale, or Weil-pro-étale setting, could you provide me with such a theory axiomatically?
The singular cohesion I described works well only for discrete equivariance groups, as otherwise the cohesion implicit in the group is “not known” to the ambient axioms and they don’t respect it (Rem. 3.64 p. 63 here).
In your example, $G = \mathbb{G}_{m}$ is not discrete, so what I said about singular cohesion does not directly apply to this case. Sorry :-)
[ On the other hand, are you sure you need “proper” $\mathbb{G}_m$ equivariance (i.e. Bredon-style, parameterized over an orbit category) instead of Borel equivariance? Just to remark the obvious, for completeness: that Borel equivariant homotopy theory exists for every $\infty$-topos: It’s the homotopy theory of its slice over the delooping of the equivariance group. ]
Of course this is an issue we have been wondering about. While we are at it, I can mention some thoughts about it, but please feel free to ignore the following monologue, as it goes way off tangent with respect to the discussion in this thread here.
On the other hand, singular cohesion works just as well for higher equivariance groups, as long as they are geometrically discrete. This suggests for any non-discrete equivariance group $G$ (typically this means: a non-discrete compact Lie topological group) to regard equivariance under the cohesive 1-group $G$ through its shadow as equivariance under the geometrically discrete $\infty$-group $ʃ G$ which is the group’s cohesive shape.
There is the following canonical comparison morphism (if shape preserves finite products! :-)
$ʃ Maps\big( \mathbf{B}K, \, \mathbf{B}G \big) \xrightarrow{\;\;\;\;} Maps\big( \mathbf{B} ʃ K ,\, \mathbf{B} ʃ G \big)$from, on the left, the hom-spaces in the usual global orbit category for cohesive 1-groups, to, on the right, the hom-spaces between their associated shape $\infty$-groups. By left Kan extension $ʃ_{!}$ along the corresponding $\infty$-functors between global and higher-global orbit categories
$ʃ \;\colon\; Glo_{CptLie} \xrightarrow{\;\;\;} Glo_{\infty Disc}$every object in the ordinary equivariant homotopy theory over cohesive (i.e. in practice: compact Lie topological) 1-groups re-incarnates as an object in a novel singular cohesive $\infty$-topos relative to higher equivariance groups to which the discussion at singular cohesion applies without caveat 3.64.
What somebody should work out is how much $ʃ_{!}$ really forgets. Maybe much of traditional equivariant homotopy theory with non-discrete compact Lie equivariance groups survives under $ʃ_{!}$? Or maybe not. I have no good idea towards the answer at this point, but at least I managed to realize that this is a good question to ask :-)
(Edit: this was in reply to #46, I have not read #47 yet!) Thank you, this sounds promising! My main question at this point is what is probably a rather stupid one: how general are $0$-truncated and concrete objects of $\mathbf{H}$? I suppose that $0$-truncated is no problem: in the pro-étale case, I suppose this would just say that we have a scheme rather than a higher stack? But what would concrete mean in that context?
Re #48:
Concreteness over a site with terminal object means that morphisms between spaces are determined by what they do to all the “global points” given by maps out of this terminal object. You’ll have to help me translate what exactly this means for schemes… Certainly it implies being reduced relative to the base scheme.
But in what I said the concreteness is needed only in order to understand the $H$-fixed loci as spaces of $H$-fixed points in the ordinary sense of fixed points. If The $X$ is not concrete, then $\prec (X \sslash G)$ still encodes its system of $H$-fixed loci if $X$, with whatever structure these fixed-loci may have, point-supported or not.
On the other hand, the 0-truncation condition is really crucial (yes, this means $X$ is a scheme instead of a stack) for what I said above, since if $X$ is already stacky, then $\prec (X \sslash G)$ would pick up stacky automorphisms in $X$ as fixed loci of a spurious group action.
On the other hand, if one does need $X$ to be a non-0-truncated stack with $G$-equivariance, then one can consider the analogous constructions not in the plain singular-cohesion described above, but in it’s slice over the $G$-orbi-singularity.
how the condensed community prefers to refer to the base topos/$\infty$-topos that we are talking about.
That’s the thing: I don’t think they do, because of the size issues.
Re #45: I don’t think this is correct unfortunately.
Okay, thanks, so I have now edited the entry here to say that the cohesion is “over pyknotic sets”.
I’m pretty sure that the right adjoint does not exist either when global sections land in pyknotic sets or condensed sets. What Scholze was saying in the n-category café discussion was that the right adjoint of the global sections functor from the big pyknotic topos of some scheme to sets exists (it does not exist for condensed sets just for trivial set-theoretic reasons, namely that there is no upper bound on size), not that it exists to pyknotic sets. As Scholze also wrote there, there is really no hope of getting a right adjoint to condensed or pyknotic sets, basically because these are quite rich gadgets, and a scheme (for instance) is ultimately just built on sets.
For somewhat related reasons, with regard to the following…
Concreteness over a site with terminal object means that morphisms between spaces are determined by what they do to all the “global points” given by maps out of this terminal object. You’ll have to help me translate what exactly this means for schemes
…I think there are more or less no interesting schemes with this property, just things like coproducts of copies of $Spec(k)$. To take the case that $k$ is a field for the purpose of illustration, the point is really that fields have no interesting prime ideals, whereas any interesting scheme will have (affine patches which have) some, and morphisms of schemes must respect prime ideals. This is the same reason I have always been skeptical that the right adjoint can be useful in algebraic geometry, or any situation which is not closely tied to sets.
Re #47 and #49: thanks! I’ve not fully digested this yet, but if you can make equivariance magically appear like this, this is very cool regardless of whether it applies in algebraic geometry!
In summary: with sets as base, there is no left adjoint in either the condensed or pyknotic worlds. There is a right adjoint in the pyknotic world, but there is no known use for it. With condensed/pyknotic sets as base there is a left adjoint but no right adjoint in both cases. The left adjoint does not preserve finite products in full generality either case, but this is probably repairable by adding in extra structure, perhaps the ’Weil-pro-étale’ topoi of the paper I linked to above, and finite products are preserved in a good degree of generality.
With regard to the question of the naming of the page, if one wishes the avoid the term ’cohesion’, one could entitle the page ’condensed local contractibility’, which I think is a very important breakthrough. Though of course calling it ’condensed cohesion’ does not preclude discussing why we do not have cohesion.
Okay, thanks, Richard. I have reworded that bit here once more, making it say now the lamest “some base topos related to condensed or pyknotic sets”. I should really bow out of this since if what you say is true then I don’t even understand what the words mean in the Scholze verses whose exegesis we are discussing.
I have also removed the bits about stable and singular cohesion, since it seems out of place to go on about cohesion here.
Then I followed your suggestion and renamed the entry.
Richard wrote #52
With condensed/pyknotic sets as base there is a left adjoint but no right adjoint in both cases.
Wait a minute. Scholze writes (my emphasis):
OK, here is another try to get something that looks closer to cohesion.
For a scheme $X$, one can consider the small pro-etale topos $X_{proet}$ built from the site of schemes pro-etale over $X$, and pro-etale coverings, and the big pro-etale topos $X_{Proet}$ built from the site of all schemes over $X$, and pro-etale coverings. There is a natural map of topoi $f: X_{Proet} \to X_{proet}$.
Proposition. $f^{\ast}$ has a left adjoint, and in the pyknotic formalization, $f_{\ast}$ has a right adjoint. Moreover, $f^{\ast}$ is fully faithful.
Taking $X$ to be a geometric point $Spec k$, the target $X_{proet}$ here is the topos of pyknotic sets. (In the condensed formalization, I’m afraid the right adjoint may fail to exist. Again, I’m a little suspicious about its use. Concreteness also does not seem like a fundamental concept to me.)
Again, all assertions are true both on the level of usual topoi, and on ∞-topoi of hypercomplete sheaves.
Yes, that’s why I said I may not even understand what these words mean: “in the pyknotic formalization” or “in the condensed formalization”. If Richard is right, this means something which we don’t understand – it’s related to my continuing question as to what the base topos is actually meant to be.
What we really want is just a definition of a morphism of sites and a proposition that the induced geometric morphism of sheaf toposes has this or that property.
It looks like either somebody has to work that out (Richard seems to have the details at hand but might not have the time) or somebody needs to ask Peter S. for a more precise statement. I feel the approach of exegesis of scattered blog comments has been a little frustrating, and we need to turn to extracting some actual maths here.
Yes, it is frustrating. What Scholze means by “in the pyknotic formalization” or “in the condensed formalization” is what I cited in #37, just a difference about how to deal with set-theoretic size issues, the former having nicer category-theoretic properties.
But what stops us working out the adjoint quadruple that Scholze mentions as being generated by $f: (Spec k)_{Proet} \to (Spec k)_{proet}$? Is it that we don’t know the pyknotic way to form the domain, $(Spec k)_{Proet}$?
what stops us working out the adjoint quadruple
Exactly, that’s what the entry should spell out. That was the idea, originally, to condense sketchy ideas that were floating around the web into an $n$Lab entry that one could turn to for reference.
Nobody may have the leisure and inclination to do this, but before this is done, we should not muddy the waters by stating further sketchy claims.
The next step should be to write a Definition (we need to define two toposes and a geometric morphism between them), then a Proposition (saying that this geometric morphism has some extra properties). Ideally we’d also indicate the proof, but just knowing the actual claim would already be helpful. If you know how to extract that from those various quotes you are referring to, please do!
Maybe I see now a distinction between cases which do and don’t have the right adjoint.
$f: X_{\mathrm{Proet}} \to X_{\mathrm{proet}}$ does have the right adjoint (pyknotically), but $f: X_{\mathrm{proet}}\rightarrow \ast_{\mathrm{proet}}$ does not.
Is the former case sufficiently interesting?
Maybe it’s more like your singular cohesion case, where there’s cohesion for each $G$. Here there’s one for each $X$.
I’d suggest to focus on the case over the point for starters, in the spirit of taking one step at a time.
First step:
To type out (best: inside a definition
-environment) the actual definition of the sites $\ast_{Proet}$ and $\ast_{proet}$.
Or rather, as there seem to be at least two definitions:
To type out the definition of the sites
$\ast^{cond}_{Proet}$ and $\ast^{cond}_{proet}$
$\ast^{pyk}_{Proet}$ and $\ast^{pyk}_{proet}$.
I gather it’s going to be $\ast^{pyk}_{Proext}$ that we want. In which case the next step would be:
Define a functor
$\ast^{pyk}_{Proet} \longleftarrow \ast^{pyk}_{proet}$and verify that it is a morphism of sites.
These definitions are all bound to be straightforward and trivial/easy. Nonetheless, they want to be written down in our entry.
Let’s see. From the Stacks project we see how to define the big and small sites for a scheme $S$, and note that there is an inclusion from the small site to the big site in (2).
Variants on constructing big sites are mentioned here.
Sure, if we disregard the hypnotic fine-print then
$\ast_{Proet}$ should be roughly the category of all schemes over $Spec(k)$
$\ast_{proet}$ should be roughly the category of those that are pro-étale over $Spec(k)$,
the morphism of sites $\ast_{proet} \to \ast_{Proet}$ should be the forgetful functor,
the Grothendieck topology should be the pro-étale topology.
But now to sort out the missing fine-print! What is $\ast^{pyk}_{Proet}$? What is $\ast^{pyk}_{proet}$? What is $\ast^{cond}_{Proet}$? What is $\ast^{cond}_{proet}$?
These definitions must add some fine-print. The suggestion was that this fine-print can be easily deduced from the quote in #37. Can it?!
Let’s try to straighten this out. Firstly, it is true in quite great generality that the inclusion of a small site into a large site induces a ’constant sheaf’ / global sections / ’push forward of the inclusion’ type adjoint triple, such that the large topos is local over the small one. This is proven for the Zariski topology in Lemma 34.3.13 here. If one traces through everything, the only ingredient in the proof which holds for the Zariski topology and which might not hold for other topologies is that the small site needs to have fibred products. This is true for the étale and pro-étale (see Lemma 4.1.8 in Bhatt-Scholze) topologies (I think it might not be true for things like the various flat topologies, though I don’t fully remember).
So, since we also have the left adjoint to the ’constant sheaf’ functor in the pro-étale setting, we do, for big topoi, have the adjoint quadruple that we are looking for when working with universes, or below a certain cardinal, i.e. the pyknotic topoi.
If we take the étale topology and take the small site over $Spec(k)$ for some appropriate $k$, then the small topos will just be sets, and the ’push forward of inclusion’ functor can I think (although I’ve not checked it carefully) explicitly be described as sending a set $X$ to $Y \mapsto \Pi_{\left| Y \right|} X$, where $\left| Y \right|$ denotes the cardinality of the underlying set of the underlying topological space of the scheme $Y$. I think it is essentially the same in the pro-étale case, replacing sets by pyknotic sets.
If we wished to have it for condensed as opposed to pyknotic sets, the adjoint quadruple would have to be preserved under a colimit of base topoi, which is unlikely (although if one replaced the big pro-étale topos by a colimit of topoi in the same way as one defines condensed sets, I wonder if it might then work). Indeed, if my explicit description of the right adjoint in the case of $Spec(k)$ is correct, I think one runs into the problem that I mentioned in #51 if one tries to carry it out for condensed sets (or just the analogue of condensed sets for sets if one does the same kind of thing for the étale topology), namely that there is no upper bound on size for condensed sets; I think this maybe manifests itself in the fact that one would need to be able to replace a colimit of products of $X$ with a product of colimits, which I think will not be possible in general without an upper bound (but don’t take this as gospel, I have not checked this carefully due to lack of time).
In particular, I am very sorry for all the confusion; I was thinking in terms of cohesion for small topoi in #51 and #52, for some unknown reason (no doubt the writing of it in snatched moments as usual), where what I wrote is I believe true.
Let me then change my summary, hopefully for the last time! For the big pro-étale topos over a separably closed $k$, I think we do have cohesion with base pyknotic sets except for the failure of the leftmost adjoint to preserve finite products in full generality, which might well be repaired by using the Weil-pro-étale topos. The rightmost three parts of the adjoint quadruple will work over any arbitrary base scheme, but the base for cohesion is then not pyknotic sets but the small pro-étale topos over that base scheme; I don’t know whether the left adjoint exists in that generality, but I would guess not. The reason I believe in the existence of that left adjoint with base pyknotic/condensed is the work of Bhatt and Scholze, which strongly suggests that it can be done with such a base; I don’t know whether any actual proof is in the literature of the existence of the left adjoint yet, though.
Thanks Richard. Allow me to check in terms of concrete definitions, as I still don’t really know what it means to say “condensed” in any of this:
For $k$ a suitable(?) ground field, write
$\ast_{Proet}$ for the large site of schemes over $Spec(k)$, with Grothendieck topology being the pro-étale topology;
$\ast_{proet}$ for the site of schemes pro-étale over $Spec(k)$, with Grothendieck topology being the pro-étale topology;
$\ast_{Proet} \overset{\; i \;}{\longleftarrow} \ast_{proet}$ be the canonical forgetful functor.
Then what is the claim? (No worries about whether it’s fully correct, or fully proven, just to know what the claim is.)
I gather the claim is that $i$ is a morphism of large sites, that sheaf toposes on these large sites make sense, and that the resulting geometric morphism of toposes
$Sh(\ast_{Proet}) \underoverset {\underset{i_\ast}{\longrightarrow}} {\overset{i^\ast}{\longleftarrow}} {\;\;\; \bot \;\;\;} Sh(\ast_{proet})$has
$i^\ast$ is fully faithful;
$i^\ast$ is a right adjoint;
$i_\ast$ might also be a left adjoint.
Is that the claim you think is true?
Is $Sh(\ast_{proet})$ called the topos of pyknotic sets?
Is $Sh(\ast_{proet})$ called the topos of pyknotic sets?
Where the definition goes via a choice of strongly inaccessible cardinal to limit the size of the schemes in the site. Is the idea then that little depends on such a choice?
Ah, is that then the definition of $\ast_{proet}^{pyk}$, that it is the subcategory of $\ast_{proet}$ on schemes that are small relative to some $\kappa$?
Let’s write it $\ast_{poet}^{pyk, \kappa}$ then. So maybe the claim is that
$\ast^{pyk,\kappa}_{Proet} \hookleftarrow \ast^{pyk,\kappa}_{proet}$is a morphism of sites and that
$Sh\big(\ast^{pyk,\kappa}_{Proet}\big) \underoverset {\underset{i_\ast}{\longrightarrow}} {\overset{i^\ast}{\longleftarrow}} {\;\;\; \bot \;\;\;} Sh\big(\ast^{pyk,\kappa}_{proet}\big)$is locally connected, for all strongly inaccessible $\kappa$?
And saying”condensed” maybe refers to taking the colimit over $\kappa$s of these geometric morphisms, in the arrow category of toposes? Or in the arrow category of (presentable?)categories?
Extracting from #37, the category of condensed sets, $Sh(\ast^{cond}_{proet})$ is formed via
$\kappa$-condensed sets are sheaves on the site of $\kappa$-small profinite sets, for any strong limit cardinal $\kappa$. The (large) filtered colimit over all $\kappa$ of these categories gives the category of condensed sets.
This colimit is formed where?
Right, but that’s not formulated in terms of the category $\ast_{proet}$ which we need.
Or is $\ast_{proet}$ equivalent to profinite sets?
I expect these things are all defined in Scholze’s published articles. To get to the bottom of this, somebody needs to look at these, not just his blog comments.
Myself, I am really busy elsewhere. But I’ll keep asking here for precise definitions now until they emerge :-)
I would guess that the colimit is in the (arrow) category of pretoposes and cocontinuous lex functors (i.e. the inverse image functors). The colimit doesn’t exist in the (arrow) category of toposes, since condensed sets don’t form a topos. And I strongly suggest they don’t form a presentable category, either, since the category of condensed sets satisfies Giraud’s axioms except there’s a proper class separator/generator. If it was presentable, I think you’d get the implication it was a topos.
I think he doesn’t work via $\ast_{proet}$. In his lectures
This definition presents set-theoretic problems: The category of profinite sets is large, so it is not a good idea to consider functors defined on all of it.
In a first step, we will choose a cardinal bound on the profinite sets, i.e. fix a suitable uncountable cardinal $\kappa$ and use only profinite sets S of cardinality less than $\kappa$ to define a site $\ast_{\kappa-proet}$.
The category of sheaves on this site is then $\kappa$-condensed sets
We will then define the category of condensed sets as the (large) colimit of the category of $\kappa$-condensed sets along the filtered poset of all uncountable strong limit cardinals $\kappa$.
This is no longer a topos, I take it.
Re D. Roberts in #69:
Okay, so you are getting closer to answering my question regarding the name of the base topos:
It sounds now you believe that
$PyknoticSets \,=\, Sh\big( \ast^{pyk, \kappa}_{proet} \big)$and that
$CondensedSets \,=\, \underset{\underset{\kappa}{\longrightarrow}}{\lim} \, Sh\big( \ast^{pyk, \kappa}_{proet} \big) \;\;\; \in \; Cat \,.$One would think that with all the publicity and attention ekpyrotic sets are getting, by comment #70 there would have emerged some references to quote and some Definitions/Propositions to look at. I’ll keep asking questions until somebody finds them… ;-)
It is all laid out in those lectures. Since this doesn’t give a topos
Remark 2.12.The category of condensed sets is a (large) category like the category of all sets;the essential difference is that it does not admit a set of generators (but merely a class of generators),
is it that for this page to work we have to stick to the pyknotic approach, and then there will be an inevitable strongly inaccessible cardinal hanging about?
Really in a hurry here, but let me try to reply quickly to #64 and #66. I’ve not read any of the other messages except #67.
Is that then the definition of $*^{pyk}_{proet}$, that it is the subcategory of $*_proet$ on schemes that are small relative to some $\kappa$?
Yes, exactly, where small refers to the cardinality of the underlying topological space.
Let’s write it $*^{pyk, \kappa}_{proet}$ then.
Very good idea!
What David wrote in #67 is correct regarding the definition of condensed sets. In particular, the colimit is not in an arrow category, just in the category of categories. As I remarked in an aside in my previous comment, I think the failure of cohesion for condensed sets may come from trying to use just the big pro-étale topos $*_{Proet}$. If one instead did the same kind of thing as for the base topos. i.e. took a colimit of all the categories $*^{pyk, \kappa}_{Proet}$, maybe one would get cohesion.
suitable(?)
To get pyknotic/condensed sets, one has to let $k$ be separably closed, i.e. it is only in this case that $*^{pyk, \kappa}_{proet}$ gives pyknotic sets.
canonical forgetful functor
Usually called the inclusion functor :-).
morphism of large sites, that sheaf toposes on these large sites make sense
One has to either use universes, or work below $\kappa$ to get a small site. This is exactly the reason for working with $\kappa$, otherwise no-one would bother!
Is that the claim you think is true?
I always get mightly confused about the direction of things when it comes to geometric morphisms, they have to be one of the worst entities notationally in mathematics! Thus I’ll avoid them and just describe things directly.
What is in the literature is that, from $i$, one gets two functors $*^{pyk, \kappa}_{proet} \rightarrow *^{pyk, \kappa}_{Proet}$, which I’ll denote by $i_{!}$ and $const$, and a functor $*^{pyk, \kappa}_{Proet} \rightarrow *^{pyk, \kappa}_{proet}$ which I’ll denote by $\Gamma$. The functor $i_{!}$ is fully faithful and is right adjoint to $\Gamma$. The functor $const$ is left adjoint to $\Gamma$. This is proven at the link I gave in my previous comment for the Zariski topology, and the same proof works for both the étale and pro-étale topologies, and I imagine also the Weil-pro-étale and Nisnevich topologies for example; the only thing one needs as far as I see is that the small site has fibred products (which I don’t think is true for topologies like fppf). There are some brief remarks on this story at big and little toposes; ideally, we should give the proof on that page for all of the afore-mentioned topologies, I think this would be a good service to the community :-)!
So certainly this much is claimed, but I don’t think there’s anything contentious about this.
In addition, Scholze claims that $const$ has a left adjoint. This is what I consider to be the really interesting thing, and I believe it because the work of Bhatt and Scholze strongly suggests that it should exist. But I don’t think it’s in the literature yet, although I don’t follow it on a day-to-day basis, so don’t know for sure.
Moreover, this left adjoint to $const$ preserves finite products in a number of cases, but not in full generality. The Weil-pro-étale topology might repair this.
Thanks. Okay, I have typed up what seems to be the formal definition and the proposition in question, now here.
(I have removed the “pyk”-superscripts, as that seemed to be neither correct nor necessary, after all. )
Regarding “colimit in the arrow category”: Since we are after the adjoint triple/quadruple between them and not just the categories themselves, we do need to look at some kind of arrow category of categories to see that the local connectivity or cohesion passes to there. But maybe best to ignore any colimits for the time being.
Note that the category of condensed sets is not the colimit of the various toposes of pyknotic sets, unless there are a proper class of inaccessible cardinals. In particular, it is consistent that there are no pyknotic sets, but the category of condensed sets always exists, or there is exactly one category of pyknotic sets, because there is exactly one inaccessible.
I think Peter S would be happy to explain more about this comparison functor and adjoint stuff over at the n-café in the threads there, were he asked.
Thanks for re-confirming, Richard. Glad this has materialized now.
The next question I’d have is (not urgent, just saying):
If the image of the terminal sheaf under the extra left adjoint is not terminal: what is it?? It must be some “special” condensed set.
Time to rename the page again? Pyknotic quasi-cohesion (with a view to the Weil-pro-étale topology giving us Pyknotic cohesion).
Re #77: I’m pretty sure it’s terminal. This is basically asking for the étale homotopy type of the point, which would be the point :-). If you’re thinking about the failure to preserve products, a typical example would be a product with the affine line, because ’Artin-Schreier’ phenomena lead to Künneth theorems not holding. This is what the Weil-pro-étale topos aims to rectify.
Re #78: I think my vote would be to keep as it is, because the local contractibility is the main non-trivial ingredient, but I don’t feel strongly about it :-).
Okay, if at least the terminal object has terminal shape, then the term “cohesion” does make sense at least conceptually – as all the axioms of cohesion except the request for preservation of binary products say that the tops “looks like a fat point” in various ways.
How conjectural is that fix of binary products via the “Weil-pro-étale” topology?
And if we do change the sites in the first place, can we maybe even arrange for a reflection of sites where the left adjoint already preserves finite products? That would imply the claim for the toposes and would make the eventual proof more transparent.
As for the naming: If there is cohesion we should say it. For, while I know and value the definition of “right adjoint”, I am unsure what it means to have an “unimportant right adjoined”. :-)
On this point of concrete schemes, coming back to your #51:
Doesn’t the GAGA theorem (the fully faithfulness here) say that at least all complex-analytic spaces are concrete as schemes over $Spec(\mathbb{C})$?
Does it matter that Barwick and Haine chose their cardinal to be strongly inaccessible, rather than strong limit, as we have it now?
Have you figured out what it is meant to mean to consider the geometric morphism “in the pyknotic formalization” as opposed to “in the condensed formalization”?
Each time I feel I understood what people want to mean by these words, somebody says something that suggests the opposite.
Doesn’t the GAGA theorem say that at least all complex-analytic spaces are concrete as schemes over $Spec(\mathbb{C})$?
GAGA is for the Zariski topology only, but yes, in that topology, I think that is true. More generally, in the Zariski topology, one has at least a chance if one is working over an algebraically closed field; if not, then I think there is little hope. Just think about a polynomial with some real roots for instance: if one works over $\mathbb{R}$, looking at the $\mathbb{R}$-valued points alone cannot possibly, if it has a complex root as well, completely describe a scheme morphism with it as its domain. Even over an algebraically closed field one needs further things to hold. In the étale topology there is even less chance, but I should have been a bit more thorough in #51; even in the étale and pro-étale topologies there are probably some schemes which are concrete which are more interesting than those I mentioned. I still don’t think they’re very general, though.
Regarding the naming of the page, I am fine with mentioning cohesion :-).
Regarding the Weil-pro-étale topology, all we have to go on is my intuition based on a quick glance at the paper I linked to. In other words, very little. But it does look promising to me. Since Peter Scholze is the only person we know of who has thought about the existence of the left adjoint, and since I would have thought he will be aware of the Weil-pro-étale paper, probably the best course of action if we would like to obtain a more informed opinion would be to ask him what he thinks.
Have not thought about anything beyond #80 in the thread yet.
Have you figured out what it is meant to mean to consider the geometric morphism “in the pyknotic formalization” as opposed to “in the condensed formalization”?
I think there is really far too much being read into this; I’ve tried to already say this, but will try once more!
One chooses a cardinal $\kappa$ to work below to get a small site: call the result $\kappa$-condensed sets,. One would like some flexibility to change cardinal, in particular so that one has the convenience of all small limits and colimits, etc: take a colimit over categories of $\kappa$-condensed sets for varying $\kappa$, and call the result condensed sets.
Does it matter that Barwick and Haine chose their cardinal to be strongly inaccessible, rather than strong limit, as we have it now?
To the extent that people wish to obsess about getting a topos in the usual sense (personally I think it is irrelevant!), it matters. See footnote 1 in the lectures you linked to of Scholze. $\kappa$-condensed sets are sheaves of sets on a site of $\kappa$-small profinite sets. Barwick and Haine take $\kappa$ to be a strongly inaccessible cardinal, and take the size of their sets to be one universe higher. Thus this is not a topos in the usual universe, but rather one universe higher.
When $\kappa$ is a strong limit cardinal, $\kappa$-condensed sets are a topos in the usual universe. Condensed sets are not a topos in any universe, but are a (large) category in the usual universe.
In other words, the ’pyknotic formalism’ could equally well be called the ’$\kappa$-condensed’ formalism, i.e. the one using $\kappa$-condensed sets for some $\kappa$. That Barwick and Haine happened to choose $\kappa$ to be a strongly inaccessible cardinal makes no difference to anything compared to any other cardinal (except, of course, that if one wishes to have all small limits/colimits in the usual sense of small, then one needs something at least the size of the first strongly inaccessible cardinal). Hence why I have written ’$\kappa$-condensed/$\kappa$-pyknotic’ on the nLab entry. The condensed formalism is the one involving a colimit of categories; taking such a colimit obviously does have some consequences compared to the $\kappa$-condensed setting. Scholze argues that what one loses are inessential things.
Yes, I was going to say that Barwick and Haine’s definition of pyknotic set is defined relative to two inaccessible cardinals, $\delta_0$ and $\delta_1$, the smallest two that exist under their assumption of the axiom of universes. Then pyknotic sets are then the sheaves of the form $CH_{\lt\delta_0}^{op}\op Set_{\lt\delta_1}$. Since $Set_{\lt\delta_1}$ is a topos, the sheaves give a topos. It’s not a matter of sheaves (of sets of unrestricted size) on small sites, but small sheaves on tiny sites (where ’tiny’ and ’small’ refer to the two smallest inaccessibles).
So I was wondering in what sense it is sensible to take the colimit of toposes of pyknotic sets, given the definition is, as given, tied to two adjacent inaccessibles (clearly one doesn’t need to take the smallest two in the definition), but one could also just take any old pair. I guess with B&H’s assumption of the axiom of universes mean that taking the colimit over ordinals $\alpha$ indexing adjacent inaccessibles $(\delta_\alpha,\delta_{\alpha+1}$, we probably recover all condensed sets.
To point out the obvious: given the definition of pyknotic set, even the generalised version depending on two arbitrary inaccessible cardinals $\delta_\alpha, \delta_{\alpha+1}$, it is not true that a $\delta_{\alpha+1}$-condensed set is pyknotic. The cardinal bounds are completely decoupled. I know it sounds like quibbling, but given the communication difficulties we seem to be having, being careful with the definitions and not leaning into technicality-hiding neologisms is worthwhile here, IMHO.
Is it clear then how to construct the big site pyknotic topos, along the lines of the distinction in #64?
Not to me. The details are hidden in the definition of proétale, which is where the cardinality bounds are hiding. Personally, I’d like to see people referring to Bhatt and Scholze’s paper, where the definitions are, rather than offhand comments in a blog discussion. I’m not personally invested enough or have the time right now to dig out the references and repeat them here, I’m afraid.
Is it clear then how to construct the big site pyknotic topos
As far as I know, Barwick and Haine’s formalism does not cover such things. They only treat pyknotic sets and internalisations of these (pyknotic spaces, pyknotic abelian groups, etc).
What Urs meant was just the pro-étale topology where one restricts the cardinality of (the underlying sets of the underlying topological spaces of) the schemes involved to those less than $\kappa$ for some strong limit cardinal $\kappa$. No problem there. As we have discussed earlier in the thread, if one replaces $\kappa$-condensed sets with condensed sets in the base, it is hardly surprising that one cannot get away with just the $\kappa$-big site if one would still like cohesion; one will presumably have to do something akin to what one does to construct condensed sets, namely take a colimit of the $\kappa$-big topoi over different $\kappa$ in the category of categories, or possibly in an arrow category.
If one chooses $\kappa$ to be a strongly inaccessible cardinal in both the definition of $\kappa$-condensed sets and the $\kappa$-big-pro-étale site, everything is fine with regard to cohesion, it is just that both topoi (the base and the big pro-étale one) are not topoi in the usual universe, but in the next one up (one has to map into $\lambda$-small sets for $\lambda$ the next strongly inaccessible cardinal up); or else one has to restrict to small sheaves, whereupon one does not get a topos, but one may well end up with the same thing as if one takes a colimit over different $\kappa$ for $\kappa$ a strong limit cardinal.
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