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I posted this question on MSE a few days ago, but got no response (despite a bounty). I hope it will get more answers here, but I'm afraid it might not be appropriate as I'm not sure it's actually research-level. Please do tell me if it's not appropriate and if possible tell me how to modify the question so that it may become appropriate (if such modifications exist)

I recently saw a video of a presentation of Andrej Bauer here about constructive mathematics; and there are two examples of locales he mentions that strike me : he explains quickly what the space of random reals might be, by saying that it's the locale of reals that are in every measure $1$ subset of $[0,1]$ (for instance): as he says, of course there are no such reals, but that should not stop us from considering the space of these reals, which may have interesting topological properties even if it has no points.

Similarly in a constructive setting (or classical setting without AC) some rings may have no prime/maximal ideals, and so their spectrum as usually conceived is uninteresting. But that should not stop us from studying the space of prime/maximal ideals with the Zariski topology, even if it has no points.

My questions are related to these examples specifically and to generalizations:

Is the first example of random reals in any way connected to the random reals one mentions in forcing ? e.g. is forcing to add some random reals in any way connected to considering the topos of sheaves on the locale of random reals ?

Has the second example been extensively studied ? What sort of properties can we get from the study of this "Zariski locale" ?

Is there some form of general theory of locales as spaces of imaginary points ? For instance is this how one usually sees locales intuitively; or better is there some actual theory (more than a heuristic) of constructing pointless (or with few points) spaces of objects that we'd like to exist but don't actually exist ? This is very vague so I'll give a further example of what one might envision: if two first-order structures $A$ and $B$ aren't isomorphic but $A\cong_p B$, we might want to study the space of isomorphisms of $A$ and $B$, which would ideally be a pointless locale. One could say something similar about generic filters of a poset when one is trying to do some forcing : from the point of view of the small model, these generic filters don't exist: we could envision a space of generic filters. In these four cases we have some objects that don't exist (random reals, maximal ideals, isomorphisms) but that we can define and that in some very vague sense ought to exist, and so we construct the space of these objects; but it turns out that this space can have no points at all: is there a general theory of this sort of thing ?

These questions are very vague so I hope they're appropriate. I'll appreciate answers with references, but I'd also very much like answers that themselves provide some intuition (though a bit more technical than what I've expressed in the question), and some thoughts.

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  • $\begingroup$ What does $A\cong_p B$ mean? Does it denote elementary equivalence of structures? $\endgroup$ – Qfwfq Jul 20 '18 at 20:30
  • $\begingroup$ I wasn't sure whether it was standard notation. It means they're finitely partially isomorphic, i.e. there are many finite partial isomorphisms between them (many in the sense that you can always enlarge a finite domain or a finite codomain of a finite partial isomorphism to any larger finite domain or codomain). Some authors describe it as "$A$ is isomorphic to $B$, except if they're not; but then it's for stupid reasons, such as cardinality". In particular it's easy to see that this is the same as "there exists a forcing extension where they're isomorphic" @Qfwfq $\endgroup$ – Max Jul 20 '18 at 21:24
  • $\begingroup$ I see. I had never heard of this notion before (not a logician speaking...). I know I'll be asking a slightly off topic thing but could you make a concrete example of such a pair of structures being "not isomorphic just for stupid reasons"? $\endgroup$ – Qfwfq Jul 20 '18 at 22:32
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    $\begingroup$ @Qfwfq : $\mathbb{Q}$ and any dense, linear order without endpoints. It's known that if this second order is countable, then it is in fact isomorphic to $\mathbb{Q}$. If it's not countable though, then by collapsing its cardinal to $\aleph_0$ in a forcing extension makes it isomorphic to $\mathbb{Q}$ $\endgroup$ – Max Jul 21 '18 at 8:29
  • $\begingroup$ Even in classical mathematics, one of the main approach to algebraic geometry is not about spaces, but about toposes, which are already defined in this "point free" manner. $\endgroup$ – Hurkyl Jul 22 '18 at 6:27
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I can only answer some of your questions.

Yes, the Zariski locale is extensively studied. It's one of the ways of setting up scheme theory in a constructive context: Don't define schemes as locally ringed spaces, but as locally ringed locales. The locally ringed locale $\mathrm{Spec}(A)$ always enjoys the universal property we expect of it, namely that morphisms $X \to \mathrm{Spec}(A)$ of locally ringed locales are in one-to-one correspondence with ring homomorphisms $A \to \mathcal{O}_X(X)$. (Only) if the Boolean Prime Ideal Theorem is available (a slightly weaker form of the axiom of choice), one can show that the Zariski locale has enough points. In this case it's isomorphic to the locale induced from the classical topological space of prime ideals (equipped with the Zariski topology).

(Note that the preceding paragraph assumes that you define the Zariski locale of a ring $A$ to be the locale of prime filters of $A$, not the locale of prime ideals. (A prime filter is a direct axiomatization of what's classically simply the complement of a prime ideal.) The locale of prime ideals also exists, but does not coincide with the true Zariski locale; classically, it is isomorphic to the topological space of prime ideals equipped with the constructible topology.)

Yes, there is a general theory of locales as spaces of imaginary points. Briefly, to any propositional geometric theory $T$ (roughly speaking, a collection of axioms of a certain form, such as the axioms of a prime ideal or of a prime filter), there is a classifing locale $\mathrm{Set}[T]$. The points of this locale are exactly the $\mathrm{Set}$-based models of $T$ (that is, the actual prime ideals or the actual prime filters). It can happen that there are no such models, yet still the theory $T$ is consistent. In this case the classifying local doesn't have any points, yet still is not the trivial locale.

Any locale $X$ is the classifying locale of some propositional geometric theory, a theory which deserves the name "theory of points of $X$".

The theory of classifying locales indeed allows you to construct spaces (locales) of things which aren't expected to exist, just by writing down the axioms of the hypothetical objects. A particularly tantalizing example is the locale of surjections from $\mathbb{N}$ to $\mathbb{R}$. There are no such surjections, of course, therefore this locale doesn't have any points. But it's still nontrivial. The topos of sheaves over this locale contains an epimorphism from $\underline{\mathbb{N}}$, the constant sheaf with stalks $\mathbb{N}$, to $\underline{\mathbb{R}}$; this epimorphism could be named the "walking surjection from $\mathbb{N}$ to $\mathbb{R}$", as any such surjection in any topos is a pullback of this one.

(The reals starred in the preceding paragraph only for dramatic purposes. The previous paragraph stays correct if $\mathbb{R}$ is substituted by any inhabited set. The described construction has been used, as one of a series of reduction steps, in Joyal and Tierney's celebrated monograph An Extension of the Galois Theory of Grothendieck.)

An excellent entry point to the business of locales as spaces of ideal points is a very accessible expository note by Steve Vickers. (When you've finished with this one, be sure to check out his further surveys, all available on his webpage, for instance this one.)

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  • $\begingroup$ Thank you very much for a great answer. Could you add a bit of clarification concerning the classifying locale ? $\endgroup$ – Max Jul 21 '18 at 17:00
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    $\begingroup$ Nice answer. So, it looks like you belive that there is no surjection from $\mathbb{N}$ to $\mathbb{R}$. $\endgroup$ – Andrej Bauer Jul 21 '18 at 21:39
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Here is a very brief sketches of the connection between this and forcing. I'll describe you how I understand forcing, this is quite different from how it is generally described by logician, but this how peoples in topos theory/categorical logic understand it. And it makes the connection with those "locales of imaginary points" very clear.

It should be equivalent to the classical description..

Let say I want to construct some forcing extension that add one "thing" where "thing" can be for example "a random real number", "a generic real", "a surjection $\mathbb{N} \twoheadrightarrow X$" for some fixed set $X$, "a non-principal ultrafilter", a "generic filter"...

The first step is to look at the "space of all thing", i.e. the classifying locales of the theory of "thing". So "thing" has to be a nice (geometric) notion so that such a classyfing space exists.

Depending on whether your ground model of ZFC already have "things" this locale can have points or not. (of course the most interesting case is when it does not)

Then I need to check that this classifying locale of 'things' is non trivial. There are a lots of way to do that, and it really depends on the type of 'thing' you consider, though a very common technics to do that is the "Localic Bair category theorem" which says that an arbitrary intersection of dense sublocales is a dense sublocales. Of course this can fail. This happen for example if "things" are surjection $F \twoheadrightarrow \mathbb{N}$ with $F$ a fixed finite set.

If this locale is non-empty, I can look at the category of sheaves over it. It is a topos, and hence it admits something called "internal logic" which makes into a new "set theoretical universe" in which you have a canonical "thing". I refer you to classical books on topos theory for that notion (Moerdijk & MacLane "sheaves in geometry and logics, The volume 3 of Borceux's "Handbook of categorical algebra" are both very classical and very good. Chapter three of Collin McLarty "Elementary categories, elementary toposes" is also very focused on categorical logic so give a shorter introduction to the topic).

This is not quite the end of the story because this new "mathematical universes" is not quite a model of ZF for two reasons:

  • It corresponds to a "structural set theory" whereas ZF is a "material set theory" (to use the terminology of Mike Shulman's excellent paper that I recommend). This means that it is not based on the $\in$ relation, but rather on functions between sets. Fortunately the paper I just mention present some constructions (the "Cole-Mitchell-Osius" construction) that allow to go from a structural set theory to a material set theory, basically by looking at the class of all trees.

  • It might not satisfies the axiome of chocie or the law of excluded middle. But fortunately there is a nice topos theoretic construction which given a locale $L$ (or a more general Grothendieck topos) produces a covering of $L$ by a boolean locale $B$. The internal logic of that boolean locale also has a "thing" and this times satisfies the law of excluded middle and, if your ground model satisfies choice, the axiom of choice.

So to sum up, the forcing extension adding a "thing" is obtained hes the Cole-Mitchell-Osius construction applied to the category of sheaves of a boolean cover of the classifying locale of things. Of course, if you are already doing topos theory or cartegorical logic, you don't really care about getting a model of ZFC exactly, so you tend to get ride of the Cole-Mitchell-Osius construction, and even often of the boolean cover, and so you remeber the slogan that "forcing = sheaves over classyfing spaces".

For example for some structure $A$ and $B$, "being isomorphic in a forcing extention" is exactly the same as saying that the locale of isormorphisms between $A$ and $B$ is non-trivial.

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Since your question is kind of vague, I'll answer the question that I think you're asking:

The theory of generalized spaces usually goes a bit beyond locales, namely to Grothendieck topoi. Locales are a reflective subcategory of Grothendieck topoi, so you can think of Grothendieck topoi as a true generalization of locales. There are many examples of topoi that don't arise from topological spaces that occur frequently in geometry, for example, the topoi of sheaves of sets on the Étale and Nisnevich sites.

I am not a logician, but I think that the examples you're looking for are all usually stated in this language of topoi. Proving that these topoi are the sheaves on a localic site is usually how people think of such things.

Reference: https://ncatlab.org/nlab/show/locale#RelationToToposes

This is covered in great detail in Johnstone's Sketches of an elephant

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