the category of Boolean algebras with monomorphisms

Question

Is there a nice characterization of $$\mathbf{Bool}_m^{op},$$

the opposite category of Boolean algebras with monomorphisms ?

I've just learnt that $$\mathbf{Set}^{op}$$ is CABA, which looks inetersting for me.

Answer 1

Yes, through the Stone duality (see also Stone's representation theorem). That is, if we write $\mathbf{Stone}$ for the category of Stone spaces with continuous maps then there is a duality $$ \mathbf{Bool} \simeq \mathbf{Stone}^\text{op}. $$ I am not entirely certain if by $\mathbf{Bool}_m^\text{op}$ you mean $(\mathbf{Bool}_m)^\text{op}$ or $(\mathbf{Bool}^\text{op})_m$, but it is easy to characterise what happens in both cases.

Through the Stone duality $(\mathbf{Bool}_m)^\text{op}$ corresponds to $\mathbf{Stone}_e$, where the subscript $e$ stands for epimorphisms. In other words, $(\mathbf{Bool}_m)^\text{op}$ is equivalent to the category of Stone spaces with epimorphisms.

Similarly, but even a bit easier, we get that $(\mathbf{Bool}^\text{op})_m$ is equivalent to $\mathbf{Stone}_m$, the category of Stone spaces with monomorphisms.