Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 24 / 3

A local operator is an inflationary, idempotent, finite-meet-preserving map $j:\Omega \to \Omega$ with $j(\top )=\top$, expressed in the internal logic of a topos. For a subterminal object $U$ with truth value $u$, the closed local operator and quasi-closed local operator are respectivelyThe first is a local operator by distributivity. For the second put $n(p)=p\Rightarrow u$. Then $p\le n^{2}p$, $n$ is order-reversing, and $n^{3}p=np$: the first inequality gives $n^{3}p\le np$, while its application to $np$ gives the reverse inequality. Thus $q=n^2$ is inflationary and idempotent, and $q(\top )=\top$.

For meet preservation, monotonicity gives $q(p\wedge r)\le qp\wedge qr$. Conversely put $t=(p\wedge r)\Rightarrow u$. From $t\wedge p\le r\Rightarrow u$ and $qr\wedge (r\Rightarrow u)\le u$, we obtain $t\wedge qr\le p\Rightarrow u$. Intersecting with $qp$ gives $qp\wedge qr\wedge t\le u$, hence $qp\wedge qr\le q(p\wedge r)$. This proves all local-operator axioms. Also $q(u)(0)=u$ and $q(u)(u)=u$. Equivalently, it is the largest local operator leaving $u$ fixed: if $k(u)=u$, thenThis explains the quasi-closed terminology as relative double negation above the closed bottom $u$.

The subobject classifier of ${\mathbf{sh}}_j(\mathcal{E})$ is the closed-subobject classifier ${\Omega }_j=\{p:jp=p\}$. Its bottom is $j0$, its meet is inherited, its join is $j(p\vee r)$, and implication between closed truth values is the ambient implication. To verify the last claim, if $jv=v$, thenIt follows that $j(p\Rightarrow v)\le jp\Rightarrow v\le p\Rightarrow v$; inflationarity gives equality and also $jp\Rightarrow v=p\Rightarrow v$.

Suppose $j=q(u)$. Every closed $p$ lies above $u$, and its complement in ${\Omega }_j$ is $np=p\Rightarrow u$, also closed because $n^3=n$. We have $p\wedge np=u$. Moreover $n(p\vee np)=np\wedge n^{2}p=np\wedge p=u$, so $q(p\vee np)=\top$. Thus the fixed truth values form a Boolean algebra, and the sheaf topos is a Boolean topos.

Conversely suppose the sheaf topos is Boolean and put $u=j0$. Complementation in its classifier is $p\mapsto p\Rightarrow u$. Applying the double-complement identity to $jp$, and using the implication calculation above, givesTherefore

For the cover, work in the slice category $\mathcal{E}/\Omega$. Its generic subterminal object is $T=(\top :1\hookrightarrow \Omega )$. Form its quasi-closed Boolean subtopos $\mathcal{B}={\mathbf{sh}}_{q(T)}(\mathcal{E}/\Omega )$, and compose its embedding with the slice projection to obtain $b:\mathcal{B}\to \mathcal{E}$. The slice projection has inverse image $X\mapsto X\times \Omega$, and sheafification is left exact, so this is a geometric morphism.

To prove surjectivity, take a monomorphism $S\hookrightarrow X$ with characteristic predicate $p(x)$, and suppose $b^{\ast }$ makes it invertible. Pullback to the slice followed by sheafification makes this exactly the assertion that it is $q(T)$-dense. Internally on $X\times \Omega$, therefore,for the generic truth value $u$. Substitute $u=p(x)$ by pulling back along the graph $X\to X\times \Omega$. The left side becomes $(\top \Rightarrow p(x))=p(x)$, so $p(x)=\top$ and the original mono is invertible. Thus $b^{\ast }$ reflects invertible monos. If it identifies two parallel arrows, their equalizer becomes invertible; reflection then shows the arrows were already equal. Hence $b^{\ast }$ is faithful, andThe generic truth value is essential: substituting only a fixed global value such as $0$ would give a Boolean subtopos, but need not give a surjection onto the original topos.