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Stream: theory: category theory

Topic: are Stone spaces free relational beta-modules?

Matteo Capucci (he/him) (Sep 12 2023 at 15:16):

A relational beta-module is a way to presenting topological spaces by a convergence relation. A topology on $S$, in fact, is equivalent to a relation $\beta S \to S$ (where $\beta SS$ is the set of [[ultrafilters]] on $S$) which also satisfy unitality and associativity (suitably weakened, you can read about them on the nLab: [[relational beta-module]]) saying that the principal ultrafilter generated by $s \in S$ has to converge to $s$ and that 'convergence is transitive'.
Now the question is, are [[Stone spaces]] (compact Hausdorff totally disconnected spaces) free such modules?

I'd say yes, because, first of all, the multiplication $\beta \beta S \to \beta S$ is a functional relation, hence the free $\beta$-module on $S$ is compact and Hausdorff. As for the total disconnected... well, we have strict unitality of this module since $m(\eta(F))) = \{A \mid [A] \in \eta(F)\}$ where $[A]$ denotes the set of ultrafilters containing $A$. But $[A] \in \eta(F)$ means $A \in F$, so that $m(\eta(F))=F$. I think this corresponds to being totally disconnected because it says that not only the principal ultrafilter associated to $F$ converges to it, but also that no other ultrafilter does, or in positive terms, that if an ultrafilter $\cal G$ converges to $F$ then $\cal G$ must have been principal at $F$ to begin with. But then this is saying $\{F\}$ is open, since such a subset is open iff, for every ultrafilter $\cal G$ converging to $F$, $\{F\} \in \cal G$, and that's the case by what we have discussed.

So if I didn't make any mistakes, free relational beta-modules spaces are Stone spaces, while viceversa kind of eludes me at the moment.

Matteo Capucci (he/him) (Sep 12 2023 at 15:20):

I guess my hunch is that Stone duality allows us to conclude that a given Stone space $X$ is actually of the form $\beta X'$, where $X'$ is the Boolean algebra of clopens of $X'$. But somehow I find Stone duality very slippery, so I don't trust my instinct here.

Kevin Arlin (Sep 12 2023 at 15:51):

I think this probably isn’t quite right. You seemed to claim to show that $\{F\}$ was open for any ultrafilter $F\in\beta S$, no? But that would imply $\beta S$ is discrete, which definitely isn’t true.

One relevant thing that’s known is that extremally disconnected spaces are the retracts of free algebras for the ordinary, i.e. non-relational, ultrafilter monad. So in that situation you can’t get all the way to totally disconnected even from retracts of frees. I don’t understand anything in general about free relational algebras though.

Todd Trimble (Sep 13 2023 at 00:34):

I understand a "free object" $FX$ as a representing object for a functor of the form $\hom(X, U-)$ for some "forgetful functor" $U$. Could you clarify what is supposed to be playing the role of $U$ here?

It's true that a space $\beta S$ is compact Hausdorff totally disconnected, because that is the Stone space corresponding to the power set Boolean algebra $P(S)$ (in other language, the spectrum of the Boolean ring $P(S)$ as commutative ring). But most Stone spaces will not be of the form $\beta S$, for the same reason that most Boolean algebras are not power sets (using the fact that Stone duality guarantees that a Stone space uniquely determines the Boolean algebra that it comes from).

dusko (Sep 13 2023 at 00:42):

Matteo Capucci (he/him) said:

A relational beta-module is a way to presenting topological spaces by a convergence relation. A topology on $S$, in fact, is equivalent to a relation $\beta S \to S$ (where $\beta SS$ is the set of [[ultrafilters]] on $S$) which also satisfy unitality and associativity (suitably weakened, you can read about them on the nLab: [[relational beta-module]]) saying that the principal ultrafilter generated by $s \in S$ has to converge to $s$ and that 'convergence is transitive'.
Now the question is, are [[Stone spaces]] (compact Hausdorff totally disconnected spaces) free such modules?

I'd say yes, because, first of all, the multiplication $\beta \beta S \to \beta S$ is a functional relation, hence the free $\beta$-module on $S$ is compact and Hausdorff. As for the total disconnected... well, we have strict unitality of this module since $m(\eta(F))) = \{A \mid [A] \in \eta(F)\}$ where $[A]$ denotes the set of ultrafilters containing $A$. But $[A] \in \eta(F)$ means $A \in F$, so that $m(\eta(F))=F$. I think this corresponds to being totally disconnected because it says that not only the principal ultrafilter associated to $F$ converges to it, but also that no other ultrafilter does, or in positive terms, that if an ultrafilter $\cal G$ converges to $F$ then $\cal G$ must have been principal at $F$ to begin with. But then this is saying $\{F\}$ is open, since such a subset is open iff, for every ultrafilter converging to $F$, $\{F\} \in F$, and that's the case by what we have discussed.

So if I didn't make any mistakes, free relational beta-modules spaces are Stone spaces, while viceversa kind of eludes me at the moment.

is it well-known that, over $Set$, the stone-gelfand duality (of CH-spaces = $\beta$-algebras and commutative $C^\ast$-alegebras) restricts to the stone duality (of stone spaces and boolean algebras)? i guess i should know this if i read johnstone more carefully.

but even assuming that that is true, the pons asinorum might be the assumption that free $\beta$-modules exist. i may be missing something essential, since i just looked them up on the n-catlab page, but lifting a monad from $Set$ to $Rel$ and assuming that the free algebras in $Set$ will still be free in $Rel$ for the same algebraic theory is a lot. are the $\beta$-modules equivalent to binary relations over compact hausdorff spaces? do they then correspond to congruences between commutative $C^\ast$-algebras?

i think people wanted to lift grothendieck's slogan "a sheaf of rings is a ring of sheaves" to "a relation on algebras is an algebra on relations" and maclane wrote a paper about it, but people (barr and grillet) worked out the calculus of relations in regular categories to avoid the side conditions...

Kevin Arlin (Sep 13 2023 at 00:49):

No, I don't think relational $\beta$-modules have anything to do with binary relations on compact Hausdorff spaces. The relation is between the underlying set of a space and its set of ultrafilters.

Todd Trimble (Sep 13 2023 at 01:15):

Kevin Arlin said:

No, I don't think relational $\beta$-modules have anything to do with binary relations on compact Hausdorff spaces. The relation is between the underlying set of a space and its set of ultrafilters.

Arguably off-topic and gossipy, but this reminds me of a cat that some of you may have heard of named Victor Porton. He's earned some infamy as someone who, for instance, nominated himself for the Abel prize, for his "revolutionary" work called Algebraic General Topology, and therefore as someone widely regarded as a crackpot over at, e.g., MathOverflow. Anyway, I actually spent some time looking at his stuff (on his website), and discovered that his main concept in life, which he calls a "funcoid" between sets $X, Y$, is tantamount to an internal relation from $\beta X$ to $\beta Y$ in the pretopos of compact Hausdorff spaces. This is connected with a side topic that Mike Shulman was looking at for a while, that of syntopogenous space. So maybe Porton isn't quite all he's crackpotted up to be.

Kevin Arlin (Sep 13 2023 at 01:52):

Interesting! That's pretty nice that you took the time to parse through his stuff.

Mike Shulman (Sep 13 2023 at 02:56):

Or maybe I'm more of a crackpot than I appear.

dusko (Sep 13 2023 at 06:47):

Kevin Arlin said:

No, I don't think relational $\beta$-modules have anything to do with binary relations on compact Hausdorff spaces. The relation is between the underlying set of a space and its set of ultrafilters.

oh i see so the limit is a relation. which it of course is if the space is not hausdorff or not compact. thanks!

dusko (Sep 13 2023 at 07:45):

so stone spaces are definitely projective as $\beta$-relational algebras but i don't understand why this category works and don't see that they are free. does everyone except me understand the explanations on ncatlab page? i get lost in Rel as double category and framed bicategory. what i do understand is that we can lift the functor $\beta$ to Rel, but not the monad because the distributivity with the semilattice monad is wrong way. but in general, without distributivity, we can take algebras for a monad in the kleisli category of another monad (here Rel) and require that the morphisms are strict. the resulting category is not monadic either over the kleisli category or over the original one, so i don't see how do we know that free compact hausdorff spaces are free in this new bigger category, which is matteo's starting point. sorry, maybe that is on ncatlab page and i am just blind.

but what i am dying to know is what happens with stone-gelfand duality when we expand from $\beta$-algebras to $\beta$-relational modules. (i guess we don't call them algebras because the category is not monadic.) sure, it doesn't go through. but the step from compact hausdorff spaces to arbitrary topological spaces by wiggling the limit operation, which barr pulled off, must correspond to something on the $C^\ast$-algebra side. any ideas? or maybe it is even known? the link to walter tholen et al book on ncatlab is dead...

Matteo Capucci (he/him) (Sep 13 2023 at 08:32):

Kevin Arlin said:

I think this probably isn’t quite right. You seemed to claim to show that $\{F\}$ was open for any ultrafilter $F\in\beta S$, no? But that would imply $\beta S$ is discrete, which definitely isn’t true.

Ah uhm that's a bummer indeed. I see that I misunderstood what strict unitality ($m(\eta(F)) = F$) implies... usually, one has $m \circ \eta \geq 1_{\beta S}$ meaning that $\forall F \in \beta S, \eta(F) \rightsquigarrow F$, where in our case $\rightsquigarrow$ is given by

$$\cal G \rightsquigarrow F \iff m(\cal G) = F \iff \{A \mid [A] \in \cal G\} = F$$

Asking for $m \circ \eta \leq 1_{\beta S}$ would mean then

$$\eta(F) \rightsquigarrow G \implies F=G$$

which is true but doesn't imply what I claim above (that the only ultrafilter convergin at $F$ is $\eta(F)$).

Graham Manuell (Sep 13 2023 at 10:10):

Since a relational beta-module is the same thing as a topological space (and the underlying sets are the same), surely the free ones are the same as the 'free' topological spaces --- that is, the discrete spaces.

Graham Manuell (Sep 13 2023 at 10:24):

dusko said:

is it well-known that, over $Set$, the stone-gelfand duality (of CH-spaces = $\beta$-algebras and commutative $C^\ast$-alegebras) restricts to the stone duality (of stone spaces and boolean algebras)? i guess i should know this if i read johnstone more carefully.

What do you mean by this? It's clearly not literally true, since $C^*$-algebras aren't Boolean algebras, but it's also clearly true that you can find a subcategory of $C^*$-algebras that correspond to Stone spaces.

dusko said:

but what i am dying to know is what happens with stone-gelfand duality when we expand from $\beta$-algebras to $\beta$-relational modules. (i guess we don't call them algebras because the category is not monadic.) sure, it doesn't go through. but the step from compact hausdorff spaces to arbitrary topological spaces by wiggling the limit operation, which barr pulled off, must correspond to something on the $C^\ast$-algebra side. any ideas? or maybe it is even known? the link to walter tholen et al book on ncatlab is dead...

Do you know of an approach that proves the usual Gelfand duality directly in terms of beta-algebras? Cause that would be the first step if this is the approach you want to take. Though I'm pretty skeptical that such an extension is possible. The idea behind C*-algebras is to model the complex-valued functions on a space, but complex-valued functions cannot distinguish between spaces with the same Tychonoff reflection.

Todd Trimble (Sep 13 2023 at 12:42):

dusko said:

so stone spaces are definitely projective as $\beta$-relational algebras but i don't understand why this category works and don't see that they are free. does everyone except me understand the explanations on ncatlab page? i get lost in Rel as double category and framed bicategory. what i do understand is that we can lift the functor $\beta$ to Rel, but not the monad because the distributivity with the semilattice monad is wrong way. but in general, without distributivity, we can take algebras for a monad in the kleisli category of another monad (here Rel) and require that the morphisms are strict. the resulting category is not monadic either over the kleisli category or over the original one, so i don't see how do we know that free compact hausdorff spaces are free in this new bigger category, which is matteo's starting point. sorry, maybe that is on ncatlab page and i am just blind.

but what i am dying to know is what happens with stone-gelfand duality when we expand from $\beta$-algebras to $\beta$-relational modules. (i guess we don't call them algebras because the category is not monadic.) sure, it doesn't go through. but the step from compact hausdorff spaces to arbitrary topological spaces by wiggling the limit operation, which barr pulled off, must correspond to something on the $C^\ast$-algebra side. any ideas? or maybe it is even known? the link to walter tholen et al book on ncatlab is dead...

I'm not quite following all of this, starting with "stone spaces are definitely projective as $\beta$-relational algebras", but for the usual ultrafilter monad $\beta$ on sets, being a retract of a free algebra $\beta S$ is a stronger condition than being a Stone space, according to Gleason's theorem, as touched upon in Wikipedia.

The simple reason why the monad (as opposed to just the functor) $\beta$ fails to lift from $Set$ to $Rel$ is that the multiplication $m: \beta\beta \to \beta$ and unit $u: 1 \to \beta$ fail to be natural transformations when considered for $Rel$. I wasn't really following the thing about distributivity going the wrong way (not that I'm doubting it) -- is the semilattice monad here supposed to be the sup-lattice monad (for which $Rel$ is the Kleisli category)?

I've got nothing to say about the deal with laxifying Stone-Gelfand duality (sounds intriguing though!). The nLab link to the article "One Setting for All: Metric, Topology, Uniformity, Approach Structure" by Clementino, Hofmann, Tholen -- indeed it wasn't working: I've noticed this also with other links to stuff hosted by York University, or maybe just some stuff uploaded by Walter Tholen. Anyway, I've replaced that pdf link with a Springer link, and put in the doi, and hopefully that doi is enough to exploit the usual Russian underground connections (it is: I checked).

Todd Trimble (Sep 13 2023 at 12:55):

Graham Manuell said:

Since a relational beta-module is the same thing as a topological space (and the underlying sets are the same), surely the free ones are the same as the 'free' topological spaces --- that is, the discrete spaces.

Yeah, that's why I wanted clarification above on which forgetful functor $U$ Matteo meant, or even which categories are meant. Continuous maps $f: X \to Y$ from the relational $\beta$-module point of view -- well, those are "lax" in the sense of noninvertible 2-cells $f \circ \xi_X \leq \xi_Y \circ \beta(f)$ (where $\xi_X, \xi_Y$ denote the convergence relations), but they are "strict" in the sense that these are functional (well- and totally defined) relations, so $Top$ from this point of view is still a plain old category: no 2-cells $f \leq g$ between such beta-module maps.

Kevin Arlin (Sep 14 2023 at 03:24):

Well, I guess you also have a double category of relational $\beta$-modules over the double category of sets, functions, and relations, and that $\beta$ is a legitimate monad in double cat. But then the $\beta$ algebras are constructed as monoids in the horizontal Kleisli category, not as the objects of an Eilenberg-Moore object, and I think Cruttwell-Shulman say they don't know a universal property for the horizontal Kleisli category, so perhaps this is a bad place to think about free things?

dusko (Sep 14 2023 at 08:01):

Graham Manuell said:

dusko said:

is it well-known that, over $Set$, the stone-gelfand duality (of CH-spaces = $\beta$-algebras and commutative $C^\ast$-alegebras) restricts to the stone duality (of stone spaces and boolean algebras)? i guess i should know this if i read johnstone more carefully.

What do you mean by this? It's clearly not literally true, since $C^*$-algebras aren't Boolean algebras, but it's also clearly true that you can find a subcategory of $C^*$-algebras that correspond to Stone spaces.

stone duality is obtained by homming into 2 in the category of boolean algebras on one hand and in the category of stone spaces on the other. geland duality is homming into R in the category of commutative rings on one hand and of compact hausdorff spaces on the other. boolean algebras live as boolean rings in the category of rings. stone spaces live in the category of compact hausdorff spaces. 2 lives as a subspace and... maybe a quotient ring of real numbers? at the first sight, the mismatch is that the boolean ring homomorphisms are not boolean algebra homomorphisms. but the relationship is subtle...

logically, both dualities are completeness results: the points of stone spaces are models of propositional theories, and the points of compact hausdorff spaces are the maximal ideals of C*-algebras... relating them might be an instance of the perennial question: we compute using numbers and reason using propositions. rings and modules and abelian categories tell us about numbers; frames and sheaves and toposes tell us about propositions. how are they related? when rings become noncommutative, abelian categories need to be filerted through toposes. if SGA ever gets completed, they might.

dusko (Sep 14 2023 at 08:09):

Graham Manuell said:

Since a relational beta-module is the same thing as a topological space (and the underlying sets are the same), surely the free ones are the same as the 'free' topological spaces --- that is, the discrete spaces.

discrete spaces are coalgebras for an idempotent comonad on topological spaces. topological spaces are not monadic over sets. the question barr pursued in his paper was: how much (or how little) do we need to tweak a monad (into a non-monad for sure) to get topological spaces as whatever our tweak outputs as its tweak-algebras.

dusko (Sep 14 2023 at 08:29):

@Todd Trimble thank you :) the reason why $\beta$ does not lift to Rel as a monad is clear enough. if it did, we would have the category of relational $\beta$-algebras (which wiggle the limits) and limit congruences as morphisms. but barr said "i'll let the limits present non-hausdorff and non-compact spaces by taking $\beta$-algebras in the category of free $\wp$-algebras, i.e. relations, but keep the homomorphisms strict, i.e. functions". so the "$\beta$-algebras" aren't really algebras and the category is not a category of algebras for any monad... it is a very elegant reconstruction, and i got all excited thinking what do we need from two monads to play this game. but i didn't get anywhere, and if there was anything there, barr would have prolly worked it out :) ("rabbi, there is $100 on the road." "nonsense, if there was $100 someone would have found it already.")

Reid Barton (Sep 14 2023 at 19:10):

FWIW I actually found a $100 bill on the ground a couple years ago, thereby disproving the efficient market hypothesis.

John Baez (Sep 14 2023 at 19:53):

You should have published a paper about it.

Todd Trimble (Sep 15 2023 at 00:19):

Ross Street and I, walking to Carlisle Hall one day for the Sydney Category Seminar, saw an Australian 20-dollar bill on the ground at the same time. He got to it first.