Stone duality

A Boolean homomorphism $\phi :A\to B$ induces ${\phi }^{\ast }:\mathrm{Ult}(B)\to \mathrm{Ult}(A)$, ${\phi }^{\ast }(U):={\phi }^{-1}(U)$, which is well defined by [L3] and continuous because $({\phi }^{\ast }{)}^{-1}([a])=[\phi (a)]$ for $a\in A$; moreover $({\mathrm{i}\mathrm{d}}_A{)}^{\ast }=\mathrm{i}\mathrm{d}$ and $(\psi \circ \phi {)}^{\ast }={\phi }^{\ast }\circ {\psi }^{\ast }$ for composable homomorphisms $\phi :A\to B$, $\psi :B\to C$.

A continuous map $f:X\to Y$ between Stone spaces induces $f^{\ast }:\mathrm{Clop}(Y)\to \mathrm{Clop}(X)$, $f^{\ast }(C):=f^{-1}(C)$, which is a Boolean homomorphism by [L4]; identity and composition are preserved in the reversed order, since $(g\circ f{)}^{-1}=f^{-1}\circ g^{-1}$.

For a Stone space $X$ the map ${\epsilon }_X:X\to \mathrm{Ult}(\mathrm{Clop}(X))$ is a bijection: ${\epsilon }_X(x)$ is a proper filter of clopens (it contains $X$, omits $\mathrm{\varnothing }$, is closed under intersections and upward closed) which decides every clopen $C$ because exactly one of $x\in C$, $x\in X\setminus C$ holds, so it is an ultrafilter by the dichotomy; it is injective because distinct points are separated by a clopen set by [L4]; and it is surjective, since for an ultrafilter $U$ of clopens the family $U$ has the finite intersection property, so ${\bigcap }_{C\in U}C\mathrm{\ne }\mathrm{\varnothing }$ by compactness, and if $x\mathrm{\ne }y$ both lie in the intersection then a clopen $C$ containing exactly one of them belongs to $U$ by the dichotomy and excludes the other, a contradiction, so the intersection is a single point $x$ and then $C\in U$ for every clopen $C$ containing $x$ (because $\{C^{\mathrm{\prime }}\setminus C:C^{\mathrm{\prime }}\in U\}$ has finite subfamily with empty intersection, giving $C^{\mathrm{\prime }}\subseteq C$ for some $C^{\mathrm{\prime }}\in U$).

For every Stone space $X$ the map ${\epsilon }_X$ is continuous, since ${\epsilon }_X^{-1}([C])=C$ is open, and it is a homeomorphism because it is a continuous bijection from the compact space $X$ to the Hausdorff space $\mathrm{Ult}(\mathrm{Clop}(X))$.

Naturality: for a Boolean homomorphism $\phi :A\to B$ and $a\in A$ one has ${\eta }_B(\phi (a))=[\phi (a)]=({\phi }^{\ast }{)}^{-1}([a])={\phi }^{\ast \ast }({\eta }_A(a))$, that is, ${\eta }_B\circ \phi ={\phi }^{\ast \ast }\circ {\eta }_A$; for a continuous $f:X\to Y$ and $x\in X$ one has ${\epsilon }_Y(f(x))=\{C:f(x)\in C\}=\{C:x\in f^{-1}(C)\}=f^{\ast \ast }({\epsilon }_X(x))$, that is, ${\epsilon }_Y\circ f=f^{\ast \ast }\circ {\epsilon }_X$.

The two functors are mutually inverse on hom-sets: given $\phi :A\to B$ the naturality of [step 2.2] and invertibility of ${\eta }_A,{\eta }_B$ (from [L1]) give $\phi ={\eta }_B^{-1}\circ {\phi }^{\ast \ast }\circ {\eta }_A$, so $\phi$ is determined by ${\phi }^{\ast }$, and symmetrically for continuous maps using [step 2.1]; since [step 1.1] and [step 1.2] show that the assignments preserve identities and composition, they define a contravariant equivalence of categories.

The statement is proved: $\eta$ is a natural isomorphism by [L1] and [step 2.2], $\epsilon$ is a natural isomorphism by [step 2.1] and [step 2.2], and the arrow assignments are mutually inverse by [step 3.1].