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Here is one way of saying what a pyknotic set is. Fix an inaccessible cardinal $\kappa$, and let $Proj_\kappa$ be the category of $\kappa$-small, extremally disconnected compact Hausdorff spaces. Recall that Stone duality restricts to an equivalence between the opposite category $Proj_\kappa^{op}$ and the category $CBool_\kappa$ of $\kappa$-small, complete Boolean algebras, and all Boolean algebra homomorphisms. Barwick and Haine define a pyknotic set $X$ to be a sheaf on the category $Proj_\kappa$ with respect to the canonical Grothendieck topology. It turns out to be very easy to say what this means explicitly. That is, we have:

Definition: A pyknotic set comprises

  1. A functor $X: CBool \to Set$;

(i.e. for each complete boolean algebra $B$ we have a set $X(B)$, and for each boolean algebra homomorphism $B \to B'$ we have a function $X(B) \to X(B')$ respecting identities and composition)

  1. such that $X$ respects finite products.

(i.e. $X(1)$ is a singleton, where $1$ is your favorite 1-element Boolean algebra, and the canonical map $X(B \times B') \to X(B) \times X(B')$ is a bijection for any pair of complete Boolean algebras $B,B'$.)


I've chosen to state the definition of a pyknotic set this way; the condensed sets of Clausen and Scholze are similar, but avoid the requirement of choosing an inaccessible cardinal. This is done by defining $Proj_\kappa$ as above where $\kappa$ is merely a strong limit cardinal, and then taking a direct limit over all $\kappa$'s to arrive at the final notion.


Now, the thing about complete Boolean algebras is that it seems they're more commonly studied by set theorists than by anybody else. My understanding is that in the "Boolean-valued models" approach to forcing, the forcing extension is more-or-less identified with sheaves on the forcing poset, which is a complete Boolean algebra. From this perspective, it sounds like a pyknotic set is somehow a set which "lives in all forcing extensions at once".

Question:

  1. Do set theorists have a notion of a "set which lives in all forcing extensions at once"?

  2. If so, how similar is such a thing to the data of a pyknotic / condensed set?

  3. In any event, do there exist theoretical frameworks in set theory for talking about objects like pyknotic / condensed sets?

My initial guess is that in multiverse-type frameworks, one probably considers morphisms of forcing posets which preserve a little more structure than just an arbitrary boolean algebra homomorphism, so that perhaps the connection is not so tight. But I really have no idea!

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  • $\begingroup$ So I'm not a category theorist, but at a glance I don't think that pyknotic sets should be thought of as sets living in every forcing extension. A pyknotic set $X$ assigns to each c.b.a. an already-existing set in the ground model. So I don't really see that there's any forcing happening here. That said, the idea of looking at things existing in all forcing extensions at once is certainly an interesting one in my opinion, and I have Thoughts on the matter; should I post them, even if it's not really related to the original motivation? $\endgroup$ – Noah Schweber Feb 28 at 3:09
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    $\begingroup$ The question doesn't parse for me, since there is surely a massive difference between looking at sheaves on a single complete boolean algebra, and cosheaves on the category of all complete Boolean algebras. Regarding 1. sure, that's something like set-theoretic geology, but you'd want to add "in all nontrivial forcing extensions", otherwise you would (I think) just get things in the base model. $\endgroup$ – David Roberts Feb 28 at 3:19
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    $\begingroup$ @DavidRoberts Re: the second half of your comment, I think actually the right picture is a mixed-quantifier ones: what are the objects for which there is some forcing notion such that every generic for that forcing yields a universe with the object (up to whatever appropriate notion of equivalence) in it? $\endgroup$ – Noah Schweber Feb 28 at 3:47
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    $\begingroup$ Tim, what i suspect were thinking of is the fibred topos over the category of complete boolean algebras, where the fibre over $B$ is $Sh(B)$. Modulo size issues, one could then take the total topos of this à la SGA4. But this would be something like the iterated forcing over all possible forcing notions (the link here was pointed out by Scedrov), but where we are not doing something sequential, so it would be a complete mess. $\endgroup$ – David Roberts Feb 28 at 10:52
  • $\begingroup$ I agree I've overlooked the fact that a pyknotic set $X$ doesn't really have much to do with sheaves over any given complete boolean algebra $B$. One thing to say is that $X$ could still be viewed as assigning a constant sheaf to each boolean algebra $B$. A constant sheaf can still see some of the structure of the site its defined on (since it's not a constant presheaf but rather the sheafification thereof) so it's conceivable that viewing a pyknotic set this way could still be fruitful. @NoahSchweber I'd be interested to hear your thoughts. $\endgroup$ – Tim Campion Feb 28 at 14:14
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Great question — for some reason this tight relation between extremally disconnected profinite sets and forcing had elapsed me!

I've just been trying to read a bit about it. From what I understand, the sheaf-theoretic approach to forcing, as in MacLane-Moerdijk "Sheaves in geometry and logic" Chapter VI, consists of three steps.

Start with any extremally disconnected profinite set $S$. Consider the category of open and closed subsets of $S$, with the following notion of cover: $\{U_i\subset U\}_i$ is a cover of $U$ if $\bigcup_i U_i\subset U$ is dense in $U$. (This is what, I believe, the "double negation topology" amounts to.) In this topology, the subsheaves of $\ast$ are exactly given by the open and closed subsets $U\subset S$, so one has a boolean topos. The first step is thus completed: The construction of the boolean topos of sheaves $\mathrm{Sh}(S)$ on $S$.

The second step is to pick any point $s\in S$, and take the colimit $\varinjlim_{U\ni s} \mathrm{Sh}(U)$. Here, the subsheaves of $\ast$ are just $\emptyset$ and $\ast$, so it's a boolean topos with only two truth values.

The third step is to start with the topos $\varinjlim_{U\ni s} \mathrm{Sh}(U)$ and somehow make it into a model of ZFC. When I've previously tried myself to contemplate forcing from the sheaf-theoretic point of view, this is the point that got me very confused: In a model of ZFC, elements of sets should be sets, but there's no meaningful way to talk about elements of objects in this topos, and certainly they won't be objects of this topos. I have to read more about this step; MacLane-Moerdijk cite work of Fourman. Apparently the idea is to redo the iterative construction of $V_\alpha$'s by iteratively taking the powerset, but now internally in this topos. [Edit: This third step deals with the problem of turning a structural set theory into a material set theory. A very nice discussion of this is in the paper Comparing material and structural set theories of Shulman. In particular, he explains how to very cleanly go back and forth between models of ECTS + a structural form of replacement, which $\varinjlim_{U\ni s} \mathrm{Sh}(U)$ satisfies, and models of ZFC, see Corollary 9.5.] (I might at this point be sold on structural set theory — forcing seems to have an extremely clean formulation in terms of structural set theory, namely just $\varinjlim_{U\ni s} \mathrm{Sh}(U)$. Please correct me if I'm misunderstanding something!)

In any case, the third step seems to be orthogonal to the question at hand. More salient is that the category of sheaves on $S$ is actually incompactible with the category of sheaves on $S$ that we would consider, where covers are just open covers. This is critical! If $S$ is the Stone-Cech compactification of a discrete set $S_0$, then sheaves in our sense are equivalent to functors on subsets of $S_0$, taking finite disjoint unions to products. But in the forcing-sense, they are equivalent to such functors taking all disjoint unions to products; equivalently, they are just sheaves on the discrete $S_0$. Most condensed sets of interest (like $\mathbb Z$ or $\mathbb R$) do not have this property; actually, the condition singles out the compact Hausdorff condensed sets if I'm not mistaken.

So even before analyzing the question of how to put this together for varying $S$, I think there are slightly different things happening even for individual $S$. But I agree that it's definitely worth finding out if there's something more to this!

Addendum in response to Mike Shulman's question in the comments below: I think $\varinjlim_{U\ni s} \mathrm{Sh}(U)$ is well-pointed. The key seems to be the following: If $f: B\to A$ is a map in $\mathrm{Sh}(S)$ that is surjective in the stalk at $s$, then $f|_U$ is surjective for some $U$ containing $s$. (This property does seem surprising to me. Is it a formal consequence of being a boolean topos?) To prove this, we prove that if $f|_U$ is not surjective for all such $U$, then also $f$ is not surjective in the stalk at $s$. Look at pairs $(V\subset S,a\in A(V))$ of an open and closed subset $V\subset S$ and a section $a$ of $A$ over $V$, such that $a\times_{A|_V} B|_V=\emptyset$. There is an obvious partial order on such; by Zorn's lemma and as any section over a union $\bigcup_i V_i$ extends to the closure by the notion of covering, there is some maximal such $(V,a)$. If $s\in V$, then in particular $f$ is not surjective on the stalk at $s$, as desired. Otherwise, let $U=S\setminus V$, which contains $s$. By assumption, $f|_U$ is not surjective, so there is some $U'\subset U$ and $a'\in A(U')$ such that $a'\times_{A|_{U'}} B|_{U'}$ (a subsheaf of $\ast|_{U'}$) is not all of $\ast|_{U'}$, and thus given by $\ast_{U''}$ for some $U''\subsetneq U'$; replacing $U'$ by $U'\setminus U''$, we can assume that $a'\times_{A|_{U'}} B|_{U'}=\emptyset$. But then $(V\sqcup U',a\sqcup a')$ extends $(V,a)$, contradicting maximality of $(V,a)$.

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  • $\begingroup$ Oh that's right ! The last comment of my post is wrong : The site I considered has less morphisms, but it also has more cover than the one used to define Pyknotic/condensed sets. So a Pyknotic/condensed set will indeed define a presheaves on the the category of all boolean algebra, but It won't be a sheaves in general. Thank you ! $\endgroup$ – Simon Henry Mar 1 at 14:30
  • $\begingroup$ "the third step seems to be orthogonal to the question at hand" absolutely. For set theorists it's important, but it's where all the tricky business in the material definition of forcing comes from ("names" and so on), and is safe to ignore for most people. $\endgroup$ – David Roberts Mar 1 at 23:37
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    $\begingroup$ I'm glad you like my paper. I guess $\varinjlim_{U\ni s} \mathrm{Sh}(U)$ is the filterquotient, right? This is the step that's always been a bit unclear to me. It makes sense intuitively that one passes to the filterquotient to get from an arbitrary topos back to a model of structural set theory, but although the filterquotient is two-valued and Boolean, I don't know how to show that it's actually well-pointed (i.e. that 1 is a generator). Is there something obvious that I'm missing? $\endgroup$ – Mike Shulman Mar 2 at 15:32
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    $\begingroup$ See the beginning of my answer: I look at the Grothendieck topology on the category of open and closed subsets $U\subset S$ where $\{U_i\subset U\}_i$ is a cover if $\bigcup_i U_i\subset U$ is dense. Then $\mathrm{Sh}(S)$ denotes the category of sheaves for this Grothendieck topology. $\endgroup$ – Peter Scholze Mar 3 at 9:43
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    $\begingroup$ Regarding well-pointedness, thanks for the reference (on the other question) to ML+M. I added an abstract proof at ncatlab.org/nlab/show/well-pointed+topos#boolean_properties. $\endgroup$ – Mike Shulman Mar 4 at 16:51
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So, I wanted to say something, I don't think this answer the questions, but that was definitely too long for a comment. In short, this is just the result of me trying to make sense of this idea:

Coming from topos theory, a "forcing extension" is for me a category of sheaves on a boolean locale, (that is a complete boolean algebras) $\mathcal{B}$. These are functorial on morphisms of boolean locales:

That is given a morphisms of boolean locale $f:\mathcal{B'} \to \mathcal{B}$, I can think of $Sh(\mathcal{B'})$ as a further forcing extention of $Sh(\mathcal{B})$, with the fullback $f^*: Sh(\mathcal{B}) \to Sh(\mathcal{B'})$ embedding the "$\mathcal{B}$-sets" in the " $\mathcal{B}'$-set". Thisespecially make sense as in this case one can think of $Sh(\mathcal{B}')$ as a forcing model internally in $Sh(\mathcal{B})$ in the sense that there will be an (internal) complete boolean algebra in $Sh(\mathcal{B})$ whose category of $Sh(\mathcal{B})$-valued sheaves identify with $Sh(\mathcal{B'})$.

From this perspective, I can give a notion of "set which lives in all forcing extensions at once" which somehow fit in the picture you describe:

If I ignore size issue for now (because they are present and can be handled in the same way for what I'm going to talk about and for Pyknotic sets), I can take a kind of (co)lax (co)limits (the "co" depending on how you fix the variance of the functoriality) of all these forcing extensions:

To be precise, I'm looking at the category of collections of sheaves $X_{\mathcal{B}} \in Sh(\mathcal{B})$ for each Boolean locale $\mathcal{B}$ with comparison maps $f^* X_\mathcal{B} \to X_\mathcal{B'}$ for each morphism of boolean algebra (asking for isomorphism would collaps the notion)

An example of such an object is if you look at the "set" of functions from $X$ to $Y$. Indeed, in each new forcing model you get potentially more functions from $X$ to $Y$, so you have comparison map $f^*[X,Y]_\mathcal{B} \to [X,Y]_{\mathcal{B}'}$ which in general are not isomorphismes).

Now, this category of collection is equivalent to the category of sheaves on the categoy of all boolean locales, with the topology of open covering.

Small side remark: Note that as a morphism of boolean locales is automatically an open maps one can also consider the topology where epimorphism (hence open surjection) and coproducts give the covering. This topology is a bit stronger, but also makes a lot of sense in this picture, in fact I tend to find it more natural. For example the object [X,Y] described above is a sheaf for this stronger topology. This however doesn't really affect the next point.

So, this looks a lot like Pyknotic/condensed sets, but there is one big difference:

If we stop here, The category of boolean algebras under consideration are not the same !

Indeed, here I look at the category of boolean locales, so up to the variance, the morphisms are map of boolean algebras that preserve supremums.

In Condensed/Pyknotic mathematics one considers the category of complete boolean algebra with arbitrary maps of boolean algebras. So this category has more maps. (The variance seems to match though)

Now, that is probably not be the end of the story: these additional maps actually also make sense from the point of view of forcing models: They corresponds to a generalized ultraproduct (or should I say filter-quotient?) construction. For example, if $\mathcal{B}$ is a complete boolean algebra, a map of boolean algebra $\mathcal{B} \to \{0,1\}$ is justs an ultrafilter on $\mathcal{B}$ and I have a functoriality $Sh(\mathcal{B}) \to Set$ given by taking germs along this ultrafilter.

So, there might a way to think of Pyknotic sets in these terms, but that is still not completely clear to me.

(Edit: the next sentence is wrong. see Peter Scholze's answer and my comment below it.) In any case, this suggests that we do have at least a forgetful functor from Pyknotic sets to the category of sheaves I described.

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  • $\begingroup$ Not sure I understand your viewpoint but I believe extremally disconnected spaces mentioned in the OP must enter your picture one way or another. In other words, along with the Boolean locales you have the Stonean locales, whose frames of opens are the frames of all ideals of some complete Boolean algebra. I believe they are some sort of compactifications of Boolean locales and must play some rôle, no? $\endgroup$ – მამუკა ჯიბლაძე Feb 28 at 19:42
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    $\begingroup$ Absolutely. In the way I phrased the above, Stonean locales enter the picture beacause the category of Stonean locale is equivalent to the opposite of the category of complete boolean algebra with arbitrary boolean algebra map between them. But they are indeed also exactly the Stone-Czech compactification of Boolean locales. That's probably relevant as well, but so fat that fact doesn't play a role in the way I phrased things above (at least not in a clear way). $\endgroup$ – Simon Henry Feb 28 at 19:46
  • $\begingroup$ The (co)lax (co)limit is, I think, the same as the total topos I mentioned in a comment on the question. But you make a very good point about what sort of morphisms are allowed between Boolean algebras. $\endgroup$ – David Roberts Feb 28 at 23:19
  • $\begingroup$ I wonder if there's some kind of gros topos / petit topos distinction to be made. Maybe the site of Stonean spaces and all maps could be our big topos and the site of Stonean spaces and open maps could be our little topos... $\endgroup$ – Tim Campion Mar 1 at 13:55
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    $\begingroup$ Related to this, it might be relevant to note that to specify a condensed set one need only specify its values on a very special class of complete boolean algebras, namely ones that are the boolean algebra of subsets of some set S (corresponding to the extr. disc. space given by the Stone-Cech compatification of S). The values on more exotic complete boolean algebras are determined from these, because they are retracts... but only in the category of boolean algebras, not in the category of complete boolean algebras and sup-preserving maps (cf the distinction Simon draws). $\endgroup$ – Dustin Clausen Mar 3 at 9:22

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