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For a subterminal object classified by $u$, the closed local operator is $p\mapsto u\vee p$. Its closed monos contain the pullback of that subterminal, hence are exactly the monos dense for the corresponding open local operator.

For a subterminal object classified by $u$, the open local operator is $p\mapsto (u\Rightarrow p)$. Its dense monos are exactly those whose image contains the pullback of that subterminal. It is complementary to the corresponding closed local operator.

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.

Work internally with the Heyting algebra $\Omega$ and the truth value $u$ of $U\hookrightarrow 1$. For $o(u)(p)=u\Rightarrow p$, the identitiesprove that it is an open local operator.

For a monomorphism $m:S\hookrightarrow X$ with characteristic map $\chi$, density means $u\Rightarrow \chi =\top$, equivalently $u\le \chi$. This says exactly that $U\times X$ is contained in $S$, or that the pullback mono $U\times S\to U\times X$ is invertible. Consequently

An object $X$ is an $o(U)$-sheaf exactly when the canonical map ${\eta }_X:X\to X^U$, taking a value to its constant function on $U$, is invertible. Indeed $T\times U\hookrightarrow T$ is dense for every $T$. The sheaf condition makes $\mathcal{E}(T,X)\to \mathcal{E}(T\times U,X)\cong \mathcal{E}(T,X^U)$ bijective for every $T$, which proves ${\eta }_X$ invertible by the Yoneda lemma. Conversely, if ${\eta }_X$ is invertible and $m:S\hookrightarrow T$ is dense, the isomorphism $S\times U\cong T\times U$ givesso $X$ is a sheaf.

The pullback functor $L:\mathcal{E}\to \mathcal{E}/U$ has right adjoint ${\Pi }_U$, sending $A\to U$ to $A^U$. Every map $X\times U\to A$ is automatically over $U$, since $U$ is subterminal object, proving this adjunction directly. Its counit $U\times A^U\to A$ is evaluation and is invertible. The inverse sends $a\in A$ to its image in $U$ together with the constant function $U\to A$ of value $a$; all elements of $U$ agree, so the composites are identities. The right adjoint is therefore full and faithful, and its image consists of objects fixed by $X\to X^U$. The sheaf characterization above proves

For a general local operator $j$, its dense monomorphism classifier is the ordered subobjectA mono is j-dense exactly when its characteristic map factors through $J$. If $j=o(u)$, then $J=\{p:u\le p\}$, which has least element $u$. This is least internally over every parameter object, not just among global points.

Conversely suppose a global $u:1\to J$ is internally least. Upward closure of $J$ gives $jp=\top$ exactly when $u\le p$, in every context. For $S\hookrightarrow X$ let $\overline{S}$ be its $j$-closure, with characteristic predicate $jp$. The mono $S\hookrightarrow \overline{S}$ is dense: restricting $jp$ to $\overline{S}$ gives truth. Its characteristic map therefore lands in $J$, so leastness of $u$ gives $U\cap \overline{S}\subseteq S$. Internally this isOn the other hand $ju=\top$, and applying $j$ to $u\wedge (u\Rightarrow p)\le p$ gives $j(u\Rightarrow p)\le jp$. Inflationarity then gives $u\Rightarrow p\le jp$. Both inequalities yieldThis proves the least dense truth characterizes an open local operator criterion without replacing its internal order condition by the weaker condition on global truth values.

A regular category has finite limits, coequalizers of kernel pairs, and regular epimorphisms stable under pullback. Equivalently, every morphism factors as a regular epimorphism followed by a monomorphism, and these factorizations are stable under pullback. A regular functor preserves finite limits and regular epimorphisms, hence also preserves these image factorizations.

For this question, the capital regular category convention is the one in which every well-supported object is a well-pointed object in a category. Here an object $A$ is well-supported if $A\to 1$ is a regular epimorphism. It is well-pointed if a subobject $m:B\hookrightarrow A$ through which every point $1\to A$ factors must be the whole object. This convention is documented in Definitions 1.1–1.3. It imposes no such point-detection requirement on objects whose support is a proper subobject of $1$. In particular, it should not be replaced here by the stronger assertion that $\mathcal{A}(1,-)$ reflects every isomorphism.

First prove two facts about a capital regular category $\mathcal{A}$. Every well-supported object $A$ has a point. Otherwise $A\times A$ is well-supported and has no points, so its diagonal $A\hookrightarrow A\times A$ contains all points and is invertible. That makes $A$ a subterminal object; the regular epimorphism $A\to 1$ is then also monic and hence invertible, contradicting the absence of a point. If $e:X\to Y$ is a regular epimorphism and $y:1\to Y$ is a point, its pullback in a category is a well-supported object over $1$, so it has a point lifting $y$. Thus the representable functorpreserves regular epimorphisms; it already preserves all existing limits. Hence $\Gamma$ is a regular functor. Second, if $\Gamma X$ is a singleton, its unique point $x:1\to X$ is a monomorphism, $X$ is well-supported, and $x$ contains all points of $X$. Well-pointedness makes $x$ invertible. Thus $\Gamma$ reflects terminal objects, even though it need not reflect arbitrary isomorphisms.

Now let $\mathcal{C}$ be a small category that is a regular category. For each object $B$, its slice category $\mathcal{C}/B$ is small and regular: finite limits, images and the stability of regular epimorphisms are computed using the corresponding constructions in $\mathcal{C}$. By the permitted capitalization result choose a conservative functor that is regular,with ${\mathcal{A}}_B$ a small capital regular category. The base-change functor $B^{\ast }:\mathcal{C}\to \mathcal{C}/B$, given by $X\mapsto (X\times B\to B)$, preserves finite limits and regular epimorphisms. So doesTaking these as the coordinates defines a regular functorThe exponent is a set of indices, so this is a power of the Category of sets, with finite limits and regular epimorphisms computed coordinatewise.

To prove reflection of isomorphisms, suppose $R(f)$ is invertible for $f:A\to B$. In $\mathcal{C}/B$, the object $(A,f)$ is the pullback of $B^{\ast }f:(A\times B\to B)\to (B\times B\to B)$ along the point given by the diagonal $B\to B\times B$. Applying ${\Gamma }_B{J}_B$ shows that ${\Gamma }_B{J}_B(A,f)$ is the fibre of the bijection $R_B(f)$ over this diagonal point, and is a singleton. The preceding lemma gives $J_B(A,f)\cong 1$ through its canonical arrow to $1$. Since $J_B$ reflects isomorphisms, $(A,f)\to (B,1_B)$ is invertible. Its underlying morphism is $f$. Therefore

In the small-category setting of this representation result, the elementary condition for replacing the power by one copy of $\mathbf{Set}$ is:This is almost total support for a regular category. “Strict initial” means initial and every arrow into it is invertible; the condition allows the case in which all objects are well-supported and no initial object exists.

For necessity, let $V:\mathcal{C}\to \mathbf{Set}$ be regular and conservative. If $VA\ne \varnothing$, factor $A\to 1$ through its support of an object in a regular category $S\hookrightarrow 1$. Preservation of images gives $VS=1$. Conservativity makes $S\to 1$ invertible, so $A$ is well-supported. If $VA=\varnothing$, then for every $X$ the projection $A\times X\to A$ is sent to the bijection $\varnothing \to \varnothing$, hence is invertible. Its inverse followed by the other projection supplies an arrow $A\to X$. For any two such arrows, their equalizer is sent to a bijection onto $VA$, hence is invertible, proving uniqueness. Thus $A$ is initial. If $X\to A$ exists, then $VX\to \varnothing$ exists, so $VX=\varnothing$ and that arrow is sent to a bijection. Conservativity makes it invertible. Hence $A$ is strict initial.

For sufficiency, choose the permitted conservative regular functor $J:\mathcal{C}\to \mathcal{A}$ into a small capital regular category, and put $V=\Gamma J$. This is regular. A well-supported object of $\mathcal{C}$ is sent to a well-supported object of $\mathcal{A}$, which has a point. A non-well-supported object is strict initial by the condition, and is consequently a proper subterminal object. The functor $J$ preserves subterminality and reflects the noninvertibility of its arrow to $1$; its image is a proper subterminal object and has no points. Thus $VA$ is empty exactly on the non-well-supported objects.

Suppose $Vf$ is a bijection for $f:A\to B$. If both sets are empty, $A$ and $B$ are strict initial objects and $f$ is invertible. Otherwise $A,B$ are well-supported. Factor $Jf$ as a regular epimorphism followed by a monomorphism $m:I\hookrightarrow JB$. Surjectivity of $Vf$ makes every point of $JB$ factor through $m$; capitality of $\mathcal{A}$ makes $m$ invertible. Hence $Jf$ is a regular epimorphism. Its kernel pair $P=JA{\times }_{JB}JA$ is well-supported, because $P\to JA$ and $JA\to 1$ are regular epimorphisms. Injectivity of $Vf$ implies that every point of $P$ has equal coordinates, and therefore factors through the diagonal $JA\hookrightarrow P$. Since $P$ is well-pointed, this diagonal is invertible. Thus $Jf$ is also monic, hence an isomorphism, and conservativity of $J$ makes $f$ invertible. This proves sufficiency with actual point and image arguments, rather than asserting that global sections of every capital regular category is conservative.

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 $B$ be a subterminal object in a Cartesian closed category. For every $X$ and $A$, the exponential object adjunction givesThe set on the right has at most one element because $B$ is subterminal. Hence $B^A$ is subterminal, proving that $\mathrm{Sub}(1)$ is an exponential ideal.

Now let $\mathcal{D}\subseteq \mathcal{C}$ be reflective, with reflector $L$ and unit ${\eta }_A:A\to LA$. Recall the useful form of its universal property: an object $D$ lies in $\mathcal{D}$ precisely when every map $f:A\to D$ factors uniquely through ${\eta }_A$.

Suppose first that $\mathcal{D}$ is an exponential ideal. Reflective subcategories are closed under ambient limits, so $LA\times LB$ lies in $\mathcal{D}$. Given $f:A\times B\to D$ with $D\in \mathcal{D}$, curry in the first variable to obtain $A\to D^B$. Since $D^B\in \mathcal{D}$, this factors uniquely through ${\eta }_A$ and uncurries to $LA\times B\to D$. Curry once more, now in $B$; since $D^{LA}\in \mathcal{D}$, the result factors uniquely through ${\eta }_B$. Thus every $f$ factors uniquely throughThis makes $LA\times LB$ a reflection of $A\times B$, so uniqueness of reflections gives

Conversely, suppose $L$ preserves binary products, and take $D\in \mathcal{D}$. To prove $D^A\in \mathcal{D}$, start with $f:X\to D^A$ and let $\tilde{f}:X\times A\to D$ be its transpose. Since $D$ is reflective, $\tilde{f}$ factors uniquely throughComposing the resulting map $LX\times LA\to D$ with $1_{LX}\times {\eta }_A$ and currying produces an extension $LX\to D^A$ of $f$. Product preservation identifies the unit on $LX\times A$ with $1_{LX}\times {\eta }_A$, so the same universal property proves uniqueness. Therefore $D^A$ is reflective, and $\mathcal{D}$ is an exponential ideal. This proves the reflector product criterion for an exponential ideal.

Finally consider the arrow category $[2,\mathbf{Set}]$. For arrows $a:A_0\to A_1$ and $b:B_0\to B_1$, letThen the exponential $b^a$ is the arrowIndeed, a commutative square from $c:C_0\to C_1$ to this arrow is, after currying, exactly a commutative square $a\times c\to b$. This establishes the required exponential adjunction and proves that $[2,\mathbf{Set}]$ is cartesian closed.

If $b$ is injective, two elements $(u_0,u_1)$ and $(u_0^{\prime },u_1)$ of $F$ haveso injectivity gives $u_0=u_0^{\prime }$. Hence $F\to B_1^{A_1}$ is injective. The category of injective functions is therefore an exponential ideal in the arrow category. Its terminal object and binary products are inherited pointwise, so these same exponential objects make $\mathrm{Mono}(\mathbf{Set})$ cartesian closed.

The support of an object $A$ in a regular category is the subterminal object appearing as the image of a morphism in a regular category of $A\to 1$. A regular functor preserves this construction. In the Category of sets, the support is the empty set if $A$ is empty and a singleton otherwise.