# Point-free topology, but with $\sigma$-algebras instead of spaces

I have a question about $$\sigma$$-algebras in relation to point-free topology. The question was inspired by a comment on a similar question I had:

If abstract $$\sigma$$-algebras (i.e. certain boolean algebras) are to $$\sigma$$-algebras on a set as the lattice of open sets is to a topological space, then we may wonder whether there is a correspondence between abstract and embedded $$\sigma$$-algebras, just like the correspondence between frames and sober spaces in topology. That is, do we have a correspondence between abstract $$\sigma$$-algebras and certain embedded $$\sigma$$-algebras? Those certain embedded $$\sigma$$-algebras would be analogous to sober spaces in topology.

Specifically, I want to think about how to modify this theorem:

Theorem: Let $$\text{Frame}$$ be the category of frames and let $$\text{Top}$$ be the category of frames. There is a functor $$\text{Top} \rightarrow \text{Frame}$$ sending a topological space $$X$$ to the hom-set $$\text{Top}(X, \{ 0, 1\})$$ endowed with a certain lattice structure. There is a functor $$\text{Frame} \rightarrow \text{Top}$$ sending a frame $$F$$ to the hom-set $$\text{Frame}(X, \{0, 1\})$$ endowed with a certain topology. These functors form an idempotent adjunction $$\text{Frame} \leftrightarrow \text{Top}$$, which then factors as two adjunctions $$\text{Frame} \leftrightarrow \text{Sober} \leftrightarrow \text{Top}$$, where $$\text{Sober}$$ is the full-subcategory of $$\text{Top}$$ consisting of sober spaces.

To get this theorem (which may require tweaking):

Theorem: Let $$C$$ be the category of ambient $$\sigma$$-algebras and let $$D$$ be the category of embedded $$\sigma$$-algebras. There is a functor $$D \rightarrow C$$ sending an embedded $$\sigma$$-algebra $$(X, A)$$ to the hom-set $$\text{D}((X, A), (\{ 0, 1\}, P(\{ 0, 1\}))$$ endowed with a certain boolean algebra structure with countable meets and joins. There is a functor $$C \rightarrow D$$ sending a $$\sigma$$-algebra $$A$$ to the hom-set $$\text{C}(A, \{0, 1\})$$ endowed with a certain topology. These functors form an idempotent adjunction $$\text{C} \leftrightarrow \text{D}$$, which then factors as two adjunctions $$\text{C} \leftrightarrow \text{?} \leftrightarrow \text{D}$$, where $$\text{?}$$ is some category which exists from the fundamental equivalence of definitions of an idempotent adjunction.

That is, $$?$$ should be a full (either reflective or co-reflective) subcategory of the category of embedded $$\sigma$$-algebras which I suspect would play the role of sober spaces under this analogy. And, if the theorem above holds, then I am interested to know:

Question: can we characterize the category $$?$$ in the theorem above in more concrete terms?