Stone's Representation Theorem and The Compactness Theorem

Solution 1:

Let $\textbf{Bool}$ be the category of boolean algebras, and let $\textbf{Stone}$ be the category of Stone spaces (however you define it). Suppose $F : \textbf{Bool}^\textrm{op} \to \textbf{Stone}$ is a weak equivalence of categories (i.e. fully faithful and essentially surjective on objects); we will deduce the boolean prime ideal theorem.

First, note that $\textbf{Bool}$ has an initial object, namely the boolean algebra $2 = \{ 0, 1 \} = \mathscr{P}(1)$, and it has a terminal object, namely the boolean algebra $1 = \{ 0 \} = \mathscr{P}(\emptyset)$. Since $F$ is a weak equivalence, $F(2)$ must be a terminal object (so a one-point space) and $F(1)$ must be an initial object (the empty space). [It is very easy to check these preservation properties even in the absence of a quasi-inverse for $F$.]

Let $B$ be a boolean algebra. Clearly, prime ideals of $B$ correspond to boolean algebra homomorphisms $B \to 2$, and hence, to continuous maps $F(2) \to F(B)$, which are the same thing as points of $F (B)$. But if $F (B)$ is empty, then the canonical map $F(1) = \emptyset \to F(B)$ is a homeomorphism, and so the canonical homomorphism $B \to 1$ is an isomorphism. [Again, this is easy to check even in the absence of a quasi-inverse for $F$.] Thus, $B$ has a prime ideal if and only if it is non-trivial.