ct.category theory – Sober spaces and $ sigma $ -Algebras

I have a question about $ sigma $-algebras compared to the topology without points. The question was inspired by a comment on a similar question that I had:

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If abstract $ sigma $-algebras (that is, some Boolean algebras) must $ sigma $-algebras on a set as the network of open sets is a topological space, one can then wonder if there is a correspondence between the frames and the sober spaces, as in the topology of points set. That is, do we have a correspondence between summary $ sigma $-algebras and some incorporated $ sigma $-algebras? These certain measurable spaces would be analogous to the sober spaces of the topology.

More specifically, I want to think about how to modify this theorem:

Theorem: Let $ text {Frame} $ to be the category of the frames and let $ text {Top} $ to be the category of executives. There is a functor $ text {Top} rightarrow text {Frame} $ sending a topological space $ X $ to the hom-set $ text {Top} (X, {0, 1 }) $ 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 }) $ with a certain topology. These functors form an idempotent addition $ text {Frame} leftrightarrow text {Top} $, which then counts as two additions $ text {Frame} leftrightarrow text {Sober} leftrightarrow text {Top} $, or $ text {Sober} $ is the complete subcategory of $ text {Top} $ composed of sober spaces.

To obtain this theorem (which may require an adjustment):

Theorem: Let $ C $ to be the category of ambient $ sigma $-algebras and leave $ D $ to be the category of embedded $ sigma $-algebras. There is a functor $ D rightarrow C $ send an embedded $ sigma $-algebra $ (X, A) $ to the hom-set $ text {D} ((X, A), ( {0, 1 }, P ( {0, 1 }) $ with a certain Boolean algebra structure with meetings and accounting joins. There is a functor $ C rightarrow D $ send one $ sigma $-algebra $ A $ to the hom-set $ text {C} (A, {0, 1 }) $ with a certain topology. These functors form an idempotent addition $ text {C} leftrightarrow text {D} $, which then counts as two additions $ text {C} leftrightarrow text {?} leftrightarrow text {D} $, or $ text {?} $ is a category that exists from the fundamental equivalence of the definitions of an idempotent addition.

C & # 39; is, $? $ should be a complete subcategory (reflective or co-reflective) of the category of $ sigma $Algebras I suspect would play the role of sober spaces under this analogy. And, if the above theorem is valid, then I am interested in knowing:

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