TITLE: Is there such a thing as the sigma-completion of a Boolean algebra?
QUESTION [7 upvotes]: Hi all,
Suppose that $\mathcal{B}$ is a Boolean algebra. It there a way to extend $\mathcal{B}$ to a smallest Boolean algebra $\mathcal{B}'$ that contains an isomorphic copy of $\mathcal{B}$ and is countably complete, i.e. every countable subset of $\mathcal{B}'$ has a least upper bound in $\mathcal{B}'$? By "smallest" I mean that the inclusion $i: \mathcal{B} \hookrightarrow \mathcal{B}'$ has the obvious universal property, i.e. for every homomorphism $f$ from $\mathcal{B}$ to a countably complete Boolean algebra $\mathcal{C}$ there exists a unique homomorphism $g: \mathcal{B}' \to \mathcal{C}$ such that $g \circ i = f$ (it would be nice if $g$ turned out to commute with countable sups too). If no such $\mathcal{B}'$ exists, is there some other useful definition of "smallest" countably complete Boolean algebra containing $\mathcal{B}$?
If it makes any difference, I'm mostly interested in the special case where $\mathcal{B}$ is a direct limit of a sequence of finite Boolean algebras.
Edit: Thanks very much for the replies, it's a shame I can only mark one as the answer. It will take me a while to absorb the various references I've been given, so if I run into difficulty I'll bump the thread with an edit.
Edit 2: Bumping with followup question, please see my answer below.

REPLY [7 votes]: In the course of a forthcoming research project with Asgar Jamneshan (in ergodic theory), we managed to discover a rather explicit answer to this question, which I am recording here if anyone is interested.
Let ${\mathcal B}$ be a Boolean algebra, let $\mathrm{Stone}({\mathcal B})$ be its Stone space (the space of Boolean homomorphisms $\alpha$ from ${\mathcal B}$ to $\{0,1\}$), let $\Sigma$ be the Baire $\sigma$-algebra of $\mathrm{Stone}({\mathcal B})$.  Then there is a natural inclusion $\iota \colon {\mathcal B} \to \Sigma$ defined by sending each $E \in {\mathcal B}$ to the clopen set $\{ \alpha \in \mathrm{Stone}({\mathcal B}): \alpha(E) = 1 \}$.  This is a Boolean homomorphism which is universal in the sense that for every Boolean homomorphism $f \colon {\mathcal B} \to {\mathcal X}$ into a $\sigma$-complete Boolean algebra ${\mathcal X}$ there is a unique lift $\tilde f: \Sigma \to {\mathcal X}$ such that $f = \tilde f \circ \iota$.

Proof of uniqueness: the clopen sets in the Stone space $\mathrm{Stone}({\mathcal B})$ separate points, hence by Stone-Weierstrass the Baire algebra is generated by the clopen sets, hence $\Sigma$ is generated by $\iota({\mathcal B})$, giving uniqueness.
Proof of existence: Stone duality gives a continuous map from $\mathrm{Stone}({\mathcal X})$ to $\mathrm{Stone}({\mathcal B})$, which pulls back Baire sets to Baire sets, thus maps $\Sigma$ to Baire sets in $\mathrm{Stone}({\mathcal X})$, each one of which can be associated to an element of ${\mathcal X}$ by the Loomis-Sikorski theorem (every Baire set in $\mathrm{Stone}({\mathcal X})$ is equivalent up to meager sets to a unique clopen set).  One easily verifies that this gives a map $\tilde f: \Sigma \to {\mathcal X}$ with the required properties.

[EDIT: a previous version erronously assumed that the pullback of a meager set was meager, leading to an incorrect conclusion.  Now corrected.]