TITLE: Locales as geometric objects
QUESTION [17 upvotes]: There is the following analogy:
$$\begin{array}{cc} \text{frames} & - &  \text{commutative rings} \\  | && | \\\text{locales} &  - & \text{affines schemes}\end{array}$$
Here, $\bigvee$ is analagous to $\sum$ and $\bigwedge$ is analogous to $\prod$. The category of locales is defined as the dual of the category of frames. We may also define the category of affines schemes as the dual of the category of commutative rings, this is done for example by Toen and Vaquié in relative algebraic geometry. But usually we define affine schemes geometrically as locally ringed spaces. Also,  affine schemes are the local building blocks for schemes. 
This leads to the following two questions:


Is there a geometric definition of locales?
Are locales the local building blocks for some geometric objects?


So, the second question is about completing the following analogy:
$$\begin{array}{cc} \text{frames} & - &  \text{commutative rings} \\  | && | \\\text{locales} &  - & \text{affines schemes} \\  | && | \\ \text{?} & - & \text{schemes}\end{array}$$
After skimming through the nlab article on locales, I would guess that locales may be defined "geometrically" as localic toposes and that perhaps the glued geometric objects are toposes, too (which ones?). This is motivated by the fact that we have a fully faithful left adjoint functor
$$\mathrm{Sh} : \mathsf{Loc} \to \mathsf{Topos}$$
from locales to toposes given by taking sheaves, and that toposes are sometimes regarded as geometric objects. (It is noted at the nlab that locales coincide with Grothendieck $(0,1)$-toposes. But this is just a reformulation of the definitions.)
But this doesn't really look similar to the construction of the spectrum of a commutative ring. It rather looks like the fully faithful embedding from (affine) schemes into the category of cocomplete symmetric monoidal categories, given by taking quasi-coherent sheaves. And this is not how we usually define affine schemes geometrically.
Perhaps there is an embedding from the category of locales into the category of locally ringed spaces? Of course, we cannot take the associated topological space of points, since that may turn out to be empty. Perhaps there is a different construction, though.

REPLY [9 votes]: While Simon's answer is very good, I think one can also say something a little more along the lines of what you may be thinking.  I haven't seen this written out in this way before, so I may have made some errors, but I think the general idea is valid.
Note that the "space of points of a locale", when regarded as a functor on frames, is very similar to the maximal or prime spectrum of a ring.  For a ring $R$, the points of $\mathrm{Spec}(R)$ are the prime ideals in $R$, and the open sets are generated by the sets $O_f$ of prime ideals not containing $f$ for $f\in R$.  Similarly, for a frame $A$, the points of "$\mathrm{Spec}(A)$" are the completely prime filters in $A$ (filters and ideals being complementary in a lattice), and its open sets are generated by sets $O_u$ of filters containing $u$ for $u\in A$.
Now, as we know, a ring $R$ induces a sheaf of rings on $\mathrm{Spec}(R)$, whose ring of sections over the open set $O_f$ generated by some $f\in R$ is the localization $R_f$. Similarly, from a frame $A$ we can try to induce a sheaf of frames on $\mathrm{Spec}(A)$, whose frame of sections over $O_u$ is the set $A_u$ of elements of $A$ that are $\le u$; this is the "localization" at $u$ in frames, since the only "unit" in a frame is the top element.
The structure sheaf of a ring $R$ is a sheaf of local rings, in the sense that its stalks are local rings.  A ring is local if $\sum_{i=1}^n a_i = 1$ implies there exists an $a_i$ that is invertible.  Similarly, a frame is local if $\bigvee_i a_i = \top$ implies there exists an $a_i=\top$, i.e. the set $\{\top\}$ is a completely prime filter, or equivalently (in localic language) there is a point whose only neighborhood is the whole space (a "focal point").  The above sheaf of frames would be a local one; indeed, its stalk over a point $x\in \mathrm{Spec}(A)$ is the "localization of $A$ at $x$" in an appropriate sense (localically, it is the "germ at $x$", the intersection of all the open sublocales containing $x$).
The problem, of course, is that unlike rings, frames may not have enough points.  One manifestation of this is that a localization of a ring $R_f$ depends only on the radical of the principal ideal $(f)$, and hence only on the set of prime ideals containing (or not containing) $f$; thus the definition $O_{\mathrm{Spec(R)}}(O_f) = R_f$ is well-defined in that if $O_f = O_g$ then $R_f \cong R_g$.  However, this is not true for frames: we can have $O_u=O_v$ but $A_u\ncong A_v$.  The most drastic example is that there are nontrivial frames without any points at all, so that $O_u=O_v$ for any $u,v$.  This means that our above attempted definition of a "structure sheaf of frames" on $\mathrm{Spec}(A)$ doesn't work.
We can, I believe, fix it by taking colimits of $A_u$ over all appropriate $u$, and the stalks will still be as I claimed above.  But the problem remains that the space $\mathrm{Spec}(A)$ itself could be completely empty even if $A$ is highly nontrivial, and so frames will not embed contravariantly into "locally framed spaces" in the same way that rings embed into locally ringed spaces.
The solution, of course, is that we need to consider locally framed locales rather than spaces.  Since this is obvious to locale theorists, they don't generally go through the above detour.  But now the analogous construction looks very tautological: the locale version of the spectrum $\mathrm{Spec}_\ell(A)$ is just $A$ itself regarded as a locale.  The "structure sheaf" is the subobject classifier of the sheaf topos $\mathrm{Sh}(A)$, which as is well-known is an internal frame, and the colimit-over-$u$ fix I described above is the pullback of this internal frame to the topos of sheaves over the topological space of points $\mathrm{Spec}(A)$.