4. Boolean Duality

This is the case where ${\cal C}$ is the category of Boolean Algebras with Boolean homomorphisms and ${\cal D}$ is the category of Boolean spaces with continuous functions. For a Boolean algebra $A$, $F(A)$ is $\Pi(A)$. For a Boolean space $X$, $G(X)$ is Clopen$(X)$. For a homomorphism $\phi: A _ 1 \rightarrow A _ 2$ of Boolean algebras, $F(\phi)$ is the function $\phi ^ \star : \Pi(A _ 2) \rightarrow \Pi(A _ 1)$ given by $\phi ^ \star ({\cal P}) = \phi ^ {-1}({\cal P})$. For a continuous function $f: X\rightarrow Y$ between Boolean spaces, $G(f)$ is the map $f ^ \dagger:$   Clopen$(Y) \rightarrow$   Clopen$(X)$ given by $f ^ \dagger (C) = f ^ {-1} (C)$.