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 $\mathrm{Stone}({\mathcal B})$, and let $M$ be the $\sigma$-ideal of $\Sigma$ consisting of Baire-measurable sets that are meager. Then there is a natural inclusion $\iota \colon {\mathcal B} \to \Sigma/M$ defined by sending each $E \in {\mathcal B}$ to the equivalence class of the clopen set $\{ \alpha \in \mathrm{Stone}({\mathcal B}): \alpha(E) = 1 \}$. This is a Boolean homomorphism (injective by the Baire category theorem) 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 $\tilde f: \Sigma/M \to {\mathcal X}$ 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 $\Sigma/M$ 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 and meager sets to meager sets, thus maps $\Sigma/M$ to the equivalence classes of Baire sets in $\mathrm{Stone}({\mathcal X})$, which is isomorphic to ${\mathcal X}$ 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 $\tilde f: \Sigma/M \to {\mathcal X}$ with the required properties.

In this language, the Loomis-Sikorski theorem can be interpreted as the assertion that the natural inclusion $\iota \colon {\mathcal B}\to \Sigma/M$ is an isomorphism if and only if ${\mathcal B}$ is $\sigma$-complete.

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.]

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 $\mathrm{Stone}({\mathcal B})$, and let $M$ be the $\sigma$-ideal of $\Sigma$ consisting of Baire-measurable sets that are meager. Then there is a natural inclusion $\iota \colon {\mathcal B} \to \Sigma/M$ defined by sending each $E \in {\mathcal B}$ to the equivalence class of the clopen set $\{ \alpha \in \mathrm{Stone}({\mathcal B}): \alpha(E) = 1 \}$. This is a Boolean homomorphism (injective by the Baire category theorem) which is universal in the sense that for every Boolean homomorphism $f \colon {\mathcal B} \to {\mathcal X}$ into a $\sigma$-algebra ${\mathcal X}$ there is a unique lift $\tilde f: \Sigma/M \to {\mathcal X}$ such that $f = \tilde f \circ \iota$.

• Proof of uniqueness: the topology of the Stone space $\mathrm{Stone}({\mathcal B})$ is generated by the clopen sets, hence by Stone-Weierstrass the Baire algebra is also, hence $\Sigma/M$ 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 and meager sets to meager sets, thus maps $\Sigma/M$ to the equivalence classes of Baire sets in $\mathrm{Stone}({\mathcal X})$, which is isomorphic to ${\mathcal X}$ 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 $\tilde f: \Sigma/M \to {\mathcal X}$ with the required properties.

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 $\mathrm{Stone}({\mathcal B})$, and let $M$ be the $\sigma$-ideal of $\Sigma$ consisting of Baire-measurable sets that are meager. Then there is a natural inclusion $\iota \colon {\mathcal B} \to \Sigma/M$ defined by sending each $E \in {\mathcal B}$ to the equivalence class of the clopen set $\{ \alpha \in \mathrm{Stone}({\mathcal B}): \alpha(E) = 1 \}$. This is a Boolean homomorphism (injective by the Baire category theorem) 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 $\tilde f: \Sigma/M \to {\mathcal X}$ 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 $\Sigma/M$ 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 and meager sets to meager sets, thus maps $\Sigma/M$ to the equivalence classes of Baire sets in $\mathrm{Stone}({\mathcal X})$, which is isomorphic to ${\mathcal X}$ 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 $\tilde f: \Sigma/M \to {\mathcal X}$ with the required properties.

In this language, the Loomis-Sikorski theorem can be interpreted as the assertion that the natural inclusion $\iota \colon {\mathcal B}\to \Sigma/M$ is an isomorphism if and only if ${\mathcal B}$ is $\sigma$-complete.