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A Boolean topos is an elementary topos in which every subobject has a complement, equivalently its internal truth-value Heyting algebra satisfies excluded middle. Its internal first-order logic is classical. Being Boolean alone does not imply that it has enough set-valued points.

The pullback of the truth point along a local operator $j$ is the subobject $J\hookrightarrow \Omega$. A monomorphism with characteristic map $\chi$ is j-dense exactly when $\chi$ factors through $J$. Its internal order is inherited from the Heyting algebra of truth values. Having a least global element is an internal condition over all parameter objects, not just a condition on global truth values.

In a Heyting algebra, meet with a fixed element is left adjoint to Heyting implication, so it preserves joins:The dual distributive law follows from this one by the lattice absorption laws.

In a Heyting algebra, $a\to b$ is the greatest $x$ with $x\wedge a\le b$. Thus $x\le a\to b$ if and only if $x\wedge a\le b$. Every frame has this operation, obtained by taking the join of all such $x$. The open nucleus attached to $a$ is $x\mapsto a\to x$.

The subobject classifier $\Omega$ of a topos carries internal truth, falsity, meet, join and implication. Its generalized elements over $X$ are subobjects of $X$; implication satisfies $R\le (P\Rightarrow Q)$ exactly when $R\cap P\le Q$. Pullback-compatible operations make this an internal Heyting algebra, not merely a structure on global truth values.

Suppose every regular open element is complemented. In a Heyting algebra, a complemented element is regular: if $a\vee b=1$ and $a\wedge b=0$, then $b=\neg a$ and $a=\neg b$. Finite joins of complemented elements are complemented. Hence $\neg \neg a\vee \neg \neg b$ is regular, and it is the least regular upper bound of $a\vee b$. ThereforeThis proves (ii) implies (iii). The identification of the regular open elements with interiors of closures was established in the preceding part.

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.

Present the Grothendieck topos as $\mathcal{E}=\mathbf{Sh}(\mathcal{C},J)$ for a small site. Let $a:\hat{\mathcal{C}}\to \mathcal{E}$ be sheafification and $i$ the inclusion. The standard sheafification construction supplies $a⊣i$, with $a$ preserving finite limits. Concretely, matching-family constructions commute with finite limits, and their directed refinement over covering sieves does so as well. Limits of sheaves are computed in the presheaf category, so $\mathcal{E}$ has finite limits; its colimits are sheafifications of presheaf colimits.

For sheaves $A,B$, form the presheaf exponentialIt is a sheaf. Indeed, for a covering sieve $S\hookrightarrow yc$, one has $aS\cong a(yc)$. Exponential adjunction, sheafification adjunction and left exactness giveThese are the restriction comparisons, proving the sheaf condition. Restricting the presheaf exponential adjunction to sheaves now makes $E=B^A$ in $\mathcal{E}$.

The subobject classifier is the sheaf ${\Omega }_J$ of J-closed sieves. A sieve $S$ on $c$ is J-closed when, for every $f:d\to c$, $f^{\ast }S\in J(d)$ implies $f\in S$. Pullback of sieves defines its restrictions, and the maximal sieve defines truth. Local sieve data glue by taking the J-closure of their compatible generated sieve; uniqueness follows because membership in a J-closed sieve is local. This proves that ${\Omega }_J$ is a sheaf.

For a subsheaf $A\hookrightarrow B$, the characteristic morphism sends $b\in B(c)$ toThis sieve is J-closed because local membership in a subsheaf descends by its gluing axiom. Pulling truth back recovers exactly $A$. Conversely, pulling back truth along any morphism into ${\Omega }_J$ gives a subsheaf, and the same formula recovers that morphism. Finite limits, exponentials and this classifier make every Grothendieck topos an elementary topos.

Here are the explicit Heyting operations on subsheaves of an object $B$. Intersections give meets:The top is $B$. Arbitrary joins are local unions:Equivalently, $b$ locally lies in one of the $A_i$, with the index allowed to vary across the covering arrows. This is the J-closure of the pointwise union, or its sheafification. It is the smallest subsheaf containing every $A_i$, since any such subsheaf must contain sections locally in it.

The bottom is the empty join, namely the initial subobject. Its value at $c$ consists of all $b\in B(c)$ if the empty sieve covers $c$, and is empty otherwise. This detail matters on sites with empty covers; it is not safe to use an always-empty presheaf as the initial sheaf.

Implication is described without any pointwise-complement assumption:Restrictions preserve this condition. It is also local: pull a covering sieve back along an arbitrary $f$, use membership in $A$ on those restrictions, then descend their membership in $D$. Thus it is a subsheaf. It satisfies the defining Heyting algebra adjunctionFor the forward direction take the identity restriction. For the reverse direction every restriction of a section of $C$ remains in $C$, so membership in $A$ forces membership in $D$. Negation is $\neg A=A\Rightarrow 0$. These formulas give the complete Heyting algebra structure on ${\mathrm{Sub}}_{\mathcal{E}}(B)$; in general negation is a pseudocomplement rather than a set-theoretic complement.

For each object $X$, characteristic maps identify $\mathcal{E}(X,\Omega )$ with $\mathrm{Sub}(X)$. These subobject posets carry a natural Heyting algebra structure. Truth and falsity classify $X$ and the empty subobject. Intersections define $\wedge$, unions define $\vee$, and implication is characterized byThese operations commute with pullback, giving arrows on $\Omega$ and the internal Heyting algebra of truth values. Concretely, the order relation on $\Omega$ is the subobject where $p\wedge q=p$; its characteristic map is implication. Negation is $p\Rightarrow \perp$.

If $U\hookrightarrow 1$ has truth value $u:1\to \Omega$, its quasi-closed local operator isThis is double negation relative to the bottom value $u$. In the Heyting algebra interval $[u,1]$, relative implication is inherited and relative negation is $N_u(p)=p\Rightarrow u$. Also $N_u(p)=N_u(p\vee u)$. Therefore the displayed map is the composite of adjoining $u$ and relative double negation. It is inflationary, idempotent, fixes truth and preserves binary meets, so it is a Lawvere-Tierney topology. Its value at bottom is $u$.

The truth values of its sheaf topos are the fixed elements $p=q(U)(p)$. They have bottom $u$, inherited meet and join $q(U)(p\vee r)$. Their relative negation is $N_u(p)$, which is again fixed because $N_u^3=N_u$. The relative double-negation law holds on every fixed element, andThus every subobject in the sheaf topos has a complement: every quasi-closed subtopos is Boolean. This identifies its internal logic, not merely the global truth-value lattice.

Now work in $\mathcal{E}/\Omega$. The arrow $\top :1\hookrightarrow \Omega$ is a subterminal object there; internally it supplies a freely varying generic truth value $u$. Let $a_q$ be sheafification for a local operator for its quasi-closed topology, and let $\pi :\mathcal{E}/\Omega \to \mathcal{E}$ be the slice geometric morphism. The composite $h$ has inverse image $h^{\ast }=a_q{\pi }^{\ast }$.

To prove surjectivity, take any mono $S\hookrightarrow X$ in $\mathcal{E}$, classified by $p:X\to \Omega$. Its pullback along $\pi$ has predicate $p(x)$, independent of the generic $u$. If $h^{\ast }$ sends the mono to an isomorphism, it is $q$-dense in the slice, sofor all $(x,u)$. Pull back this identity along the graph $x\mapsto (x,p(x))$. Substituting $u=p(x)$ gives $q_{p(x)}(p(x))=p(x)$, and therefore $p(x)=1$. Thus the original mono was already invertible.

For parallel arrows $r,s:X\rightrightarrows Y$, equality of their inverse images makes the inverse image of their equalizer invertible. The preceding mono argument makes the equalizer invertible and hence $r=s$. Therefore $h^{\ast }$ is faithful, and the composite is a surjective geometric morphism. A fixed double-negation subtopos can erase information; allowing the generic relative bottom is what detects every original predicate.

For the completeness consequence, let $\mathbb{T}$ be coherent and use its classifying topos with the generic model. Its conservative syntactic interpretation makes an underivable coherent sequent fail as a subobject inclusion. Pulling this model into the Boolean cover constructed above preserves coherent formulas and still refutes that inclusion, because the inverse image is faithful and reflects containment. In the Boolean topos, the difference between antecedent and consequent is a nonzero complemented subobject. Slicing over that difference gives a nondegenerate Boolean topos with a global tuple satisfying the antecedent and the negation of the consequent.

Classical first-order deduction is sound in a Boolean topos. Hence $\mathbb{T}$ together with constants for that tuple, the antecedent and the negated consequent is classically consistent: a proof of contradiction would hold in the nondegenerate sliced model. The ordinary Henkin construction supplies a set model of this consistent theory. Briefly, extend it by witness constants, complete it to a maximal consistent theory, form the term structure modulo provable equality, and prove the truth lemma by induction on formulas. For possibly empty sorts use the standard encoding by sort predicates and functional graph relations, without asserting sort inhabitance; the constants for the chosen tuple assert only the needed witnesses. The resulting set model is a countermodel to the original sequent.

Consequently a coherent sequent valid in every set-valued model of a coherent theory is derivable in coherent logic. The Boolean cover supplies the crucial passage from the generic intuitionistic countermodel to consistency with classical negation; the final set-model step uses ordinary first-order completeness via its Henkin proof, rather than an assumption that every Boolean topos has points.

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.

A Heyting algebra is a bounded lattice $H$ equipped with an operation $a\Rightarrow b$ satisfyingThus, for fixed $a$, the map $x\mapsto x\wedge a$ is left adjoint to $y\mapsto a\Rightarrow y$. A left adjoint preserves joins, soThis is one distributive law for lattices; the other follows from it and the absorption laws. Hence every Heyting algebra is a distributive lattice. This argument is the distributivity of a Heyting algebra.

A Heyting algebra is a bounded distributive lattice $L$ in which, for every $a,b\in L$, there is an element $a\Rightarrow b$ satisfyingThus $a\Rightarrow b$ is the greatest element whose meet with $a$ lies below $b$.

For a finite distributive lattice, defineDistributivity and finiteness giveEvery $x$ with $x\wedge a\le b$ occurs in the join, so $x\le a\Rightarrow b$. This proves the defining adjunction and makes $L$ a Heyting algebra.