Contemplate the class of summary $sigma$-algebras ${mathcal B} = (0, 1, vee, wedge, bigvee_{n=1}^infty, bigwedge_{n=1}^infty, overline{cdot})$ (Boolean algebras wherein all countable joins and meets exist), with the morphisms being the $sigma$-complete Boolean homomorphisms (homomorphisms of Boolean algebras which protect countable joins and meets). If a morphism $phi: {mathcal A} to {mathcal B}$ between two $sigma$-algebras is surjective, then it’s actually an epimorphism: if $psi_1, psi_2: {mathcal B} to {mathcal C}$ are such that $psi_1 circ phi = psi_2 circ phi$, then $phi_1 = phi_2$. However is the converse true: is each epimorphism $phi: {mathcal A} to {mathcal B}$ surjective?

Setting ${mathcal B}_0 := phi({mathcal A})$, the query might be phrased as follows. If ${mathcal B}_0$ is a correct sub-$sigma$-algebra of ${mathcal B}$, does there exist two $sigma$-algebra homomorphisms $phi_1, phi_2: {mathcal B} to {mathcal C}$ into one other $sigma$-algebra ${mathcal C}$ that agree on ${mathcal B}_0$ however usually are not identically equal on ${mathcal B}$?

Within the case that ${mathcal B}$ is generated from ${mathcal B}_0$ and one further component $E in {mathcal B} backslash {mathcal B}_0$, then all components of ${mathcal B}$ are of the shape $(A wedge E) vee (B wedge overline{E})$ for $A, B in {mathcal B}_0$, and I can assemble such homomorphisms by hand, by setting ${mathcal C} := {mathcal B}_0/{mathcal I}$ the place ${mathcal I}$ is the right splendid

$$ {mathcal I} := { A in {mathcal B}_0: A wedge E, A wedgeoverline{E} in {mathcal B}_0 }$$

and $phi_1, phi_2: {mathcal B} to {mathcal C}$ are outlined by setting

$$ phi_1( (A wedge E) vee (B wedge overline{E}) ) := [A]$$

and

$$ phi_2( (A wedge E) vee (B wedge overline{E}) ) := [B]$$

for $A,B in {mathcal B}_0$, the place $[A]$ denotes the equivalence class of $A$ in ${mathcal C}$, noting that $phi_1(E) = 1 neq 0 = phi_2(E)$. Nevertheless I used to be not capable of then receive the overall case; the standard Zorn’s lemma kind arguments aren’t out there within the $sigma$-algebra setting. I additionally performed round with utilizing the Loomis-Sikorski theorem however was not capable of get sufficient management on the assorted null beliefs to settle the query. (Nevertheless, Stone duality appears to settle the corresponding query for Boolean algebras.)