TITLE: Localic locales?  Towards very pointless spaces by iterated internalization.
QUESTION [14 upvotes]: One can think of locales as (generalizations of) topological spaces which don't necessary have (enough) points.  Of course when one studies locales, one "actually" studies frames,
certain sorts of lattices, but reverses the arrows so that story has a less algebraic and more geometrical/topological flavor.
So while "localic spaces" make lack for a "sufficient" set of point, by their very nature they do have a sufficient set of "open sets" (scare quotes because these open sets arise as a primitive notion and not as actual sets of anything).

Curiosity drive us on: could one go a step further and make the open sets of locales as ghostly as locales make the points of spaces ghostly?

Let me put the question more technically (but correct me if you think now that I'm asking the wrong question):  One often interprets algebraic objects such as "group objects" in sufficiently nice categories.  So locales with ghostly open sets might show up as localic locales, or what amounts to the same, "coframic frames" -- diagrams in the opposite category to the category of frames, specifically diagrams that define internal frames.  Though such things must have at least two points (the empty set and the total space) corresponding to the nullary operations in the algebraic theory (just as a localic group must have at least one point, the identity, a priori it seems that they might have no more.
In this spirit

How would one construct interesting examples of coframic frames = localic locales?

Even if such things have few classical points, and few classical open sets, they will still have plenty of classical (open) sets of open sets.  

By what formalism might one keep on going, iterating the internalization?

The previous question implicitly suggests levels of internalization indexed by natural numbers?

Could someone propose a definition for a level of internalization indexed by a limit ordinal?

REPLY [2 votes]: Let me give an example of a category whose objects should be thought of as being “localic locales.”
Consider the category $\mathcal{F}_{U}$ consisting of all pairs $(X,M)$ where $X$ is a set and $M$ is an ultrafilter on $X$. If $(X,M),(Y,N)$ are objects in $\mathcal{F}_{U}$, then a function $f:X\rightarrow Y$ is said to be a morphism if $f^{-1}[R]\in M$ whenever $R\in N$. Let $\mathcal{G}_{U}$ be the quotient category $\mathcal{F}_{U}/\simeq$ where if $f,g:(X,M)\rightarrow(Y,N)$ are morphisms, then $f\simeq g$ if and only if $\{x\in X\mid f(x)=g(x)\}\in U$. This category was originally formulated in Andreas Blass's dissertation.
One should consider the objects in $\mathcal{G}_{U}$ as being open set free locales. If $(X,M)$ is an object in $\mathcal{G}_{U}$, then the “points” in this “space” are the elements in the intersection $\bigcap M$. If $M$ is non-principal, then $\bigcap M=\emptyset$, so the “space” $M$ is completely point-free. The “open sets” in the “space” $(X,M)$ are the equivalence classes in $P(X)/M$. In this case, $(X,M)$ only has two points, namely $X/M$ and $\emptyset/M$.
The category $\mathcal{G}_{U}$ even has its own version of Stone-duality since the dual of $(X,M)$ will be an ultrapower $\mathcal{A}^{X}/M$ for a suitable structure $\mathcal{A}$.