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An object is j-separated when two maps into it which agree on a j-dense monomorphism are equal. Equivalently its diagonal is j-closed. Quotienting an arbitrary object by the j-closure of its diagonal gives its separated reflection, used in constructing the sheaf reflector for a local operator.

A local operator, also called a Lawvere-Tierney topology, is a map $j:\Omega \to \Omega$ which internally satisfiesThe closure operation of a local operator sends a mono with characteristic map $\chi$ to the subobject classified by $j\chi$. It is inflationary, idempotent and pullback-stable. A mono is j-dense if its closure is its whole codomain, and j-closed if it equals its closure. A j-sheaf $S$ is an object for which restrictionis a bijection for every j-dense mono $A\hookrightarrow B$; requiring only injectivity defines a j-separated object.

Here is a construction underlying the sheaf reflector for a local operator. The closed-subobject classifier ${\Omega }_j=\{p:jp=p\}$ is a j-sheaf: closed subobjects on a dense subobject extend uniquely by taking their closure in the larger object. Powers ${\Omega }_j^X$ are also sheaves, because products of a dense mono with $X$ remain dense. A j-closed subobject of a sheaf is a sheaf: first extend a map into the ambient sheaf, then use density to force its image into the closed subobject.

Close the diagonal of $X$. Its j-closure is an equivalence relation, using preservation of finite meets and pullback-stability to verify transitivity. The effective quotient $X\twoheadrightarrow X_s$ is the separated reflection: every map from $X$ into a separated object identifies that closed diagonal and factors uniquely. For separated $X_s$, the closed-singleton mapis monic. Its j-closed image closure $a_{j}X$ is a sheaf, and $X_s\hookrightarrow a_{j}X$ is dense. Unique extension across that mono, following the separated quotient factorization, provesfor every sheaf $S$. This proves reflectivity. The closure construction is pullback-stable; equivalently, separated quotients and the subsequent dense embeddings commute with the finite limiting comparisons, giving the usual left-exact sheaf reflector.

Finite limits of sheaves are computed in $\mathcal{E}$, because unique extensions can be taken componentwise. If $S$ is a sheaf, $S^T$ is a sheaf for any $T$, by the same product-with-dense-mono argument. Thus sheaf exponentials are the ambient exponentials. Monos between sheaves have j-closed images: their closure is a sheaf, and the dense inclusion into it splits by the extension property, hence is an isomorphism. Therefore ${\Omega }_j$ classifies precisely their subobjects. These observations establish ${\mathbf{sh}}_j(\mathcal{E})$ is a reflective topos.

Now let $u:1\to \Omega$ classify the given subterminal object. Its open local operator and closed local operator areThe Heyting algebra identities verify all local-operator axioms: implication by fixed $u$ preserves meets and is idempotent, while adjoining $u$ preserves meets by distributivity and is idempotent.

For a mono in $B$ with characteristic predicate $p$, closedness for $c(U)$ means $u\vee p=p$, or $u\le p$. Density for $o(U)$ means $(u\Rightarrow p)=\top$, again $u\le p$. Thus the c(U)-closed monos are exactly the o(U)-dense monos.

Both densities together force $u\le p$ and $u\vee p=\top$, hence $p=\top$: the only jointly dense monos are isomorphisms. More explicitly, the meet of these operators is pointwise andso $o(U)\wedge c(U)={\mathrm{id}}_{\Omega }$.

For their join, every mono $A\hookrightarrow B$ factors through the union with the pullback $U_B=U\times B$:The first mono is c(U)-dense, because adjoining $U$ fills its codomain; the second is o(U)-dense, because its image contains $U_B$. Any local operator above both must therefore make every mono dense, since its dense monos are closed under composition. It is the largest operator $p\mapsto \top$. ConsequentlyThese are the complementary open and closed local operators in the ordered lattice of local operators, with order given by pointwise implication.

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.