Locales as spaces of ideal/imaginary points

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 flat 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.)

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.

Here are some comments on intuition. You can think of locales as being analogous to affine schemes and frames as being analogous to commutative rings; from this point of view the existence of locales with few or no points is no more surprising than the existence of affine schemes with few or no global points (morphisms from $\text{Spec } \mathbb{Z}$). In fact we have the following table of analogies:

• Frame : commutative ring
• Locale : affine scheme
• Coproducts / joins : addition
• Products / meets : multiplication
• Open of a locale : function on an affine scheme
• Point of a locale : global point of an affine scheme
• Sierpinski space : affine line

etc.

Now, consider the following construction: there is an affine scheme which deserves to be called the "classifying scheme of square roots of $-1$," in that morphisms from $\text{Spec } R$ into this affine scheme correspond to square roots of $-1$ in $R$. Of course this scheme is just $\text{Spec } \mathbb{Z}[x]/(x^2 + 1)$. Note that it has no points, in the sense of no morphisms from $\text{Spec } \mathbb{Z}$, because $-1 \in \mathbb{Z}$ has no square roots. However, the "theory of a square root of $-1$" is nevertheless "consistent," in the sense that $\mathbb{Z}[x]/(x^2 + 1)$ is not the zero ring, and this "theory" does have "models," just in more complicated rings than $\mathbb{Z}$.

The quotation marks are meant to emphasize the following analogies between presenting a commutative ring by generators and relations and presenting a locale as the classifying locale of a propositional geometric theory:

• Propositional geometric theory : collection of variables and polynomial identities between them
• Model of a theory : collection of elements of a commutative ring satisfying some polynomial identities
• Classifying locale of a theory : Spec of a commutative ring presented by generators and relations

Somewhat more explicitly, when we present a commutative ring by generators and relations as $\mathbb{Z}[x_1, \dots x_n]/(f_1, \dots f_m)$, the resulting affine scheme has the universal property that maps from $\text{Spec } R$ into it correspond exactly to solutions of the system of equations $f_1 = \dots = f_m$ in $R$. This is exactly analogous to the universal property of the classifying locale of a theory.