# 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:

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?