# Stone Duality: What are $\sigma$-Algebras Dual To?

Stone duality, one of many dualities between certain lattices and certain topological spaces, asserts that there is a contravariant categorical equivalence between the category $$\text{Bool}$$ of boolean algebras and the category $$\text{Stone}$$ of stone spaces. For those who are not familiar with this, here is a brief statement of what this says:

Definition: A boolean algebra is a partially ordered set $$A$$ such that

1) $$A$$ has finite meets and finite joins, including the empty join.

2) Meets distribute over joins in $$A$$

3) For each $$a \in A$$, there is $$b$$ such that $$a \vee b$$ is the greatest element, and $$a \wedge b$$ is the least element.

(1) says that $$A$$ is a bounded lattice, (1) and (2) say that $$A$$ is a bounded distributive lattice, and (3) says that $$A$$ is complemented.

Definition: A stone space $$X$$ is a space occuring as the cofiltered limit $$\text{limit}\ X_i$$ of discrete spaces in the category of topological spaces.

One functor $$\text{Spec} : \text{Bool} \rightarrow \text{Stone}$$ sends a boolean algebra $$A$$ to $$\text{Bool}(A, \{ 0, 1 \})$$ of maps of boolean algebras from $$A$$ to the boolean algebra $$\{ 0, 1\}$$. The other sends a stone space $$X$$ to $$\text{Stone}(X, \{ 0, 1\})$$, the set of maps of stone spaces from $$X$$ to $$\{ 0, 1\}$$.

I am interested in how this might work for $$\sigma$$-algebras. A $$\sigma$$-algebra is a subalgebra of the boolean algebra $$P(X)$$ (power set of a set) closed under countable meets (and therefore countable joins). By the categorical equivalence, $$\sigma$$-algebras $$A$$ on $$X$$ correspond to certain quotient objects of $$\hat{X}$$, the profinite completion of $$X$$ (inverse limit over all quotients onto a finite set).

Question: Let $$X$$ be a set. Can we characterize the quotient stone spaces of $$\hat{X}$$ corresponding to $$\sigma$$-algebras in $$P(X)$$?

Note: it was mentioned in the comments that $$\sigma$$-algebra can also refer to an ambient boolean algebra which has countable meets (and therefore countable joins). Here I mean to specifically fix an embedding into $$P(X)$$, and for joins and meets in the $$\sigma$$-algebr to match joins and meets in $$P(X)$$.

• Note that the term $\sigma$-algebra can also refer to an abstract Boolean algebra which has all countable meets. Such a Boolean algebra need not be isomorphic to a $\sigma$-algebra of sets (in which countable meets are intersections). – Eric Wofsey Apr 27 at 18:52
• I don't understand why quotient objects of the profinite completion though ? Shouldn't it be quotient objects of the Stone-Cech compactification ? (unless they're the same thing ?) – Max Apr 27 at 19:47
• Maybe you're right. What I want is the universal stone space $\text{Stonify}(X)$ with a map $X \rightarrow \text{Stonify}(X)$. That is, for every other stone space $Z$ with a map $X \rightarrow Z$, there is a unique continuous map $\text{Stonify}(X) \rightarrow Z$ which makes a commutative triangle. I thought that is the same as $\text{limit} X_i$, the limit taken over finite quotients of $X$. – Dean Young Apr 27 at 19:58
• @Max: They are the same thing in this case. – Eric Wofsey Apr 27 at 20:59
• @EricWofsey : thank you for confirming that, I'll have to think about it ! – Max Apr 27 at 21:05