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Let $\mathcal{E}$ be an elementary topos with subobject classifier $\top :1\to \Omega$. A local operator is a map $j:\Omega \to \Omega$ which is inflationary, idempotent, preserves truth, and preserves binary meets:If $m:A^{\prime }\hookrightarrow A$ is classified by ${\chi }_m:A\to \Omega$, its closure operation of a local operator is classified by $j{\chi }_m$. The mono is j-dense monomorphism when this closure is all of $A$, and $j$-closed when it equals its closure. An object $X$ is a j-sheaf when every map $A^{\prime }\to X$ along a $j$-dense mono $A^{\prime }\hookrightarrow A$ extends uniquely to $A\to X$.

The closed-subobject classifier is the equalizerThus maps to ${\Omega }_j$ classify precisely the $j$-closed subobjects. Idempotence factors $j$ as

To prove that ${\Omega }_j$ is a $j$-sheaf, let $m:B\hookrightarrow A$ be dense and let $f:B\to {\Omega }_j$ classify a closed subobject $C\hookrightarrow B$. Take the closure in $A$ of the composite $C\hookrightarrow A$. Pullback stability of closure givesso the classifier $A\to {\Omega }_j$ of ${\overline{C}}^{,A}$ extends $f$. If two closed subobjects of $A$ restrict to the same subobject of dense $B$, the equalizer of their classifiers is a closed subobject containing $B$; it is both closed and dense and hence equals $A$. The extension is therefore unique.

Letbe the sheaf reflector for a local operator. The subobject classifier in the sheaf topos is ${\Omega }_j$. We prove the four assertions through the cycle

The canonical map comparing the reflected ambient classifier with the sheaf classifier isConsequently $L$ preserves the subobject classifier exactly when $L(q)$ is an isomorphism. This proves $(i)⟺(ii)$.

Since $qi=1_{{\Omega }_j}$, one hasIf $L(q)$ is an isomorphism then $L(i)$ is its inverse. Conversely, if $L(i)$ is an isomorphism, the same equation makes $L(q)$ its inverse. A monomorphism is sent to an isomorphism by sheafification exactly when it is $j$-dense, so $(ii)⟺(iii)$.

Assume $(iii)$ and let $m:A^{\prime }\hookrightarrow A$ have characteristic map $\chi :A\to \Omega$. Form the pullbackBecause dense monos are pullback-stable, $d:A^{\prime \prime }\hookrightarrow A$ is $j$-dense. The original $A^{\prime }$ factors through $A^{\prime \prime }$, and its characteristic map inside $A^{\prime \prime }$ is the top horizontal map followed by $i$, which is fixed by $j$. Hence $A^{\prime }\hookrightarrow A^{\prime \prime }$ is $j$-closed. This proves $(iv)$.

Finally assume $(iv)$ and apply it to $i:{\Omega }_j\hookrightarrow \Omega$:where $c$ is closed and $d$ is dense. Since $d$ is monic and $i=dc$, the map $c$ is the pullback of $d$ along $i$. It is therefore dense as well as closed, and hence is an isomorphism. Thus $i$ is, up to an isomorphism, the dense mono $d$, proving $(iii)$. All four conditions are equivalent, as summarized by the subobject-classifier preservation criterion for a sheaf reflector.