Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 24 / 4 / iii

For a Boolean algebra $B$, let $U(B)$ be its set of ultrafilters, equivalently the Boolean homomorphisms $B\to 2$. Regard $U(B)$ as an object of ${\mathbf{Set}}^{\mathrm{op}}$. A homomorphism $k:B\to B^{\prime }$ acts by inverse image of ultrafilters, a set map $U(B^{\prime })\to U(B)$, hence an arrow $U(B)\to U(B^{\prime })$ in the opposite category.

A Boolean homomorphism $h:B\to \mathcal{P}(A)$ gives the functionConversely, a function $v:A\to U(B)$ gives $h(b)=\{a:b\in v(a)\}$. The ultrafilter laws are exactly what makes this map preserve the Boolean operations, including complements and $0,1$. These constructions are inverse and natural, proving the Boolean ultrafilter-power-set adjunctionIts unit is ${\eta }_B(b)=\{u\in U(B):b\in u\}$.

Take $B=\mathcal{P}(\mathbb{N})$, an object in the image of the right adjoint. There is a nonprincipal ultrafilter on $\mathbb{N}$, by extending the cofinite filter in the usual choice setting. The subset $P\subset U(B)$ consisting of all principal ultrafilters is not ${\eta }_B(b)$ for any $b\subseteq \mathbb{N}$: containing the principal ultrafilter at every integer forces $b=\mathbb{N}$, but ${\eta }_B(\mathbb{N})=U(B)$ also contains the nonprincipal ultrafilters. Thus ${\eta }_B$ is not surjective and is not an isomorphism. By the necessary condition proved in the root solution, this adjunction is not idempotent.