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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:

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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?

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