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The category of rectangular bands is Cartesian closed. For nonempty products $A=B\times C$, $D=E\times F$, the exponential is $E^B\times F^C$. Also $D^0=1$ and $0^A=0$ for nonempty $A$. It has no subobject classifier: the classifier would have two elements, hence would be left-zero or right-zero, while its characteristic maps on a two-by-two product could not single out a one-point subalgebra.

An elementary topos is a category with finite limits, a cartesian-closed structure, and a subobject classifier. These axioms support an internal intuitionistic higher-order logic.

The subobject classifier supplies intuitionistic truth values. Subobjects interpret predicates, finite limits interpret finite conjunctions and equality, and suitable image and adjoint constructions interpret quantifiers. One must distinguish internal intuitionistic arguments from classical reasoning in the external category of sets.

A sieve is J-closed when membership is local with respect to the Grothendieck topology $J$. Pullback preserves J-closedness. Their presheaf is the subobject classifier of the sheaf topos: a section of a sheaf is sent to the sieve of arrows on which it belongs to the chosen subsheaf.

A logical functor between elementary toposes preserves finite limits, exponential objects and the subobject classifier, with their canonical comparison maps. It therefore preserves power objects and the double-power-object monad. If it has a left adjoint, power-object monadicity and the adjoint lifting theorem for monad algebra functors give it a right adjoint as well.

In a category with finite limits and a subobject classifier, every monic endomorphism $f$ of $\Omega$ is an involution. The subobject $g:U\hookrightarrow \Omega$ classified by $f$ is subterminal, since $fg=\top {!}_U$ is monic. If $V\hookrightarrow U$ is classified by $g$, pullback uniqueness gives $f\top {!}_U=g$, hence $f^{2}g=g$. If $u$ classifies $U\hookrightarrow 1$, the pullback of $g$ along $f$ is $\top {!}_U$, so $f^2(p)=u\wedge p$. Monicity implies $u=\top$, because $u\wedge u=u\wedge \top$, proving the displayed identity.

For a small category $C$, write $\hat{C}=[C^{\mathrm{op}},\mathbf{Set}]$. Its objects are categorical presheaves and its arrows natural transformations. The terminal presheaf is the constant singleton, and products and equalizers are computed objectwise. Their restriction maps are induced by the corresponding set operations, so the objectwise universal properties are natural and give finite limits in $\hat{C}$.

For presheaves $F,G$, define the exponential object byFor $u:B\to A$, restriction precomposes with $yu\times 1_F$. Evaluation sends $(\alpha ,x)$ at $A$ to ${\alpha }_A(1_A,x)$. Given $h:H\times F\to G$, its transpose sends $z\in H(A)$ to the transformation whose component at $B$ isNaturality of $h$ proves this is a natural transformation and that the assignment is natural in $A$. Conversely evaluation reconstructs $h$. These constructions are inverse, proving the exponential universal property explicitly.

For the subobject classifier, let $\Omega (A)$ be the set of sieves on a category on $A$ and let $\Omega (u)$ take inverse image of sieves on a category. The truth map $1\to \Omega$ selects the maximal sieve on a category at every stage. A monomorphism of presheaves is objectwise injective, so identify it with $U\subseteq F$. DefineThis is a sieve on a category, and restriction gives precisely inverse image, so ${\chi }_U$ is natural. Its value is maximal exactly when $x\in U(A)$, by testing the identity arrow. Thus $U$ is its pullback of truth. Uniqueness follows since membership of $v$ is detected by whether the restriction to $v$ is true. Smallness of $C$ makes all the displayed components and natural-transformation sets genuine sets. Hence $\hat{C}$ is an elementary topos.

The same constructions work componentwise under the weaker slice hypothesis. Indeed,A component on $v:B\to A$ is simply a map $F(B)\to G(B)$, with the naturality conditions for triangles in $C/A$. Restricting to a small skeleton of an essentially small slice therefore makes this a set; the result determines the maps on all isomorphic slice objects. Likewise a sieve on a category on $A$ is a full downward-closed collection of objects of $C/A$, determined by its intersection with that small skeleton, so the collection of sieves on a category is a set. The restriction, evaluation and classifier proofs above are unchanged. This establishes the slice-small presheaf construction.

For an example, take a discrete category with one object for every ordinal and no arrows other than identities. Each slice is the terminal category, but there is a proper class of pairwise nonisomorphic objects, so it is not equivalent to a small category.

The large-category assertion needs a universe convention. The constructions give all elementary-topos axioms in a larger universe in which those diagrams and their transformations are admitted. They do not imply local smallness in the original universe: for the discrete example, transformations from the constant singleton to the constant two-element presheaf are arbitrary ordinal-indexed binary families and do not form a set there. If “topos” is required to be locally small relative to that original universe, the unrestricted large-category assertion is false. With the usual larger-universe interpretation, the explicit component constructions prove the intended result; no assumption that the original category itself is essentially small is needed.

For an idempotent monad, the unit laws give $T\eta =\eta T={\mu }^{-1}$. If $a:TA\to A$ is an algebra, $a{\eta }_A=1$ and naturality giveThus algebras are exactly the objects on which ${\eta }_A$ is invertible, with the unique algebra structure ${\eta }_A^{-1}$. They form a full reflective subcategory with reflector $T$. The assumed preservation of finite limits makes this a left-exact reflective subcategory.

For a subobject $S\hookrightarrow A$, define its closure byusing the unit and the monomorphism $TS\hookrightarrow TA$. This closure operation induced by a left-exact reflector is extensive by unit naturality, stable under pullback and preserves intersections because $T$ preserves finite limits. Applying $T$ to its defining square and using $T\eta$ invertible identifies $T\overline{S}$ with $TS$ over $TA$. Closing a second time therefore gives the same subobject. It is a universal idempotent closure operation, hence defines a local operator $j$ by classifying the closure of $t:1\hookrightarrow \Omega$; closure of any mono is then classified by composing its classifier with $j$.

A mono $s:S\hookrightarrow A$ is $j$-dense exactly when $Ts$ is invertible. One implication follows immediately from the closure square. Conversely if its closure is all of $A$, the unit ${\eta }_A$ factors through $Ts$. Applying $T$ and using the invertible $T{\eta }_A$ makes $Ts$ split epic; since it is also monic, it is invertible. Every fixed object $B$ is consequently a $j$-sheaf, since the adjunction givesand a dense restriction becomes an isomorphism.

Conversely let $B$ be a $j$-sheaf. Its diagonal is closed: the two projections from the closure of the diagonal agree on the dense diagonal, so uniqueness of extension into $B$ makes them agree everywhere. The closure formula identifies this closed diagonal with the kernel pair of ${\eta }_B$, since $T$ takes the diagonal of $B$ to the diagonal of $TB$. Thus ${\eta }_B$ is monic. Also $T{\eta }_B$ is invertible, so this mono is dense. The sheaf property extends $1_B$ across it to a retraction $TB\to B$. Since $TB$ is fixed and hence itself a sheaf, uniqueness across that dense mono also makes the other composite the identity. Therefore ${\eta }_B$ is invertible. We have proved the algebra category is exactly the category of $j$-sheaves.

Its topos structure is concrete. Finite limits are computed in $\mathcal{E}$. For sheaves $A,B$, the ambient exponential $B^A$ is a sheaf: extension across a dense mono reduces, by transposition, to extension into $B$ across its product with $A$, again a dense mono. The classifier is ${\Omega }_j=\mathrm{Eq}(j,1_{\Omega })$ with the induced truth point; it classifies closed subobjects. Colimits are obtained by taking the ambient colimit and applying $T$. These supply an elementary topos, as required.

For the second part, use the square reader monadThe two unit laws select $(a,b)$ again. Each side of associativity selects the entries indexed $(0,0,0)$ and $(1,1,1)$ in an eight-entry array, so the monad laws hold. An algebra is an operation $\ast :A\times A\to A$ satisfyingThese imply associativity, since $(x\ast y)\ast z=(x\ast y)\ast (z\ast z)=x\ast z$ and $x\ast (y\ast z)=(x\ast x)\ast (y\ast z)=x\ast z$. They are the rectangular band laws.

For nonempty $A$, choose $e\in A$, and put $B=\{a\ast e:a\in A\}$, $C=\{e\ast a:a\in A\}$. Define $\varphi (a)=(a\ast e,e\ast a)$ and $\psi (b,c)=b\ast c$. Then $(a\ast e)\ast (e\ast a)=a\ast a=a$, and for $b=x\ast e$, $c=e\ast y$ the laws give $(b\ast c)\ast e=b$ and $e\ast (b\ast c)=c$. Thus $\varphi ,\psi$ are inverse bijections. In these coordinates the operation isConversely any nonempty product with this operation satisfies the algebra laws, proving the requested correspondence with product decompositions. A homomorphism between nonempty products has its first coordinate depending only on the first factor and its second only on the second, as follows by applying it to the displayed operation. Hence it is exactly a pair of factor maps. There is in addition the unique empty algebra, a strict initial object.

This algebra category is Cartesian closed. For nonempty $A=B\times C$ and $D=E\times F$, its exponential is $E^B\times F^C$ with rectangular multiplication: the usual set adjunction in each factor gives the universal property. For the empty cases, $D^0=1$ and $0^A=0$ when $A$ is nonempty; these formulas also give $0^0=1$. They verify the adjunction against empty test objects as well. Thus the algebra category is Cartesian closed.

It is not a topos. If a classifier existed, its global elements would correspond to the two subobjects of the terminal singleton algebra, so its underlying set would have two elements. A two-element rectangular band is either left-zero or right-zero: its two nonempty factors have sizes $(2,1)$ or $(1,2)$. A homomorphism from the four-element band $\{0,1\}\times \{0,1\}$ to a left-zero classifier depends only on the first coordinate, so the preimage of truth cannot be a single point; for a right-zero classifier the same failure occurs in the second coordinate. Yet a single point is a subalgebra and its inclusion is monic. Pullbacks of algebra maps are computed on underlying sets, so this subobject could not be classified. Therefore there is no subobject classifier and the algebra category is not a topos, despite its cartesian closure.

Use the elementary, or weak, topos axioms: finite limits and a power object $PA$ for every object $A$, naturally representing $\mathrm{Sub}(A\times X)$. No arbitrary products or coproducts are assumed. In particular $\Omega =P1$ is the subobject classifier. Inverse image of the membership relation defines $Pf:PB\to PA$ for $f:A\to B$.

Interchanging the factors in a relation gives natural bijectionsThus the contravariant power-object functor $P:{\mathcal{E}}^{\mathrm{op}}\to \mathcal{E}$ has left adjoint $P^{\mathrm{op}}$. Its adjunction unit ${\eta }_A:A\to PPA$ sends a generalized element $a$ to the predicate on $PA$ given by membership of $a$.

This unit is monic. If $x,y:T\to A$ have the same image under ${\eta }_A$, membership of $x$ and $y$ agrees in every $T$-parameterized subobject of $A$. Apply this to the graph of $x$ in $A\times T$: membership of $x$ is true, so membership of $y$ is true, which means $y=x$. This argument uses parameterized relations, not an assumption that global elements detect morphisms.

The functor $P$ reflects isomorphisms. Suppose $Pf$ is invertible. Then $PPf$ is invertible, and naturality $PPf{\eta }_A={\eta }_{B}f$ together with monicity of ${\eta }_A$ shows that $f$ is monic. Regard $f$ as a subobject of $B$. Its inverse image along itself is the whole of $A$, just as for the whole subobject of $B$. Those two subobjects correspond to maps $1\to PB$; since $Pf$ is monic, they coincide. Therefore the subobject $f$ is all of $B$, and $f$ is an isomorphism.

We now prove the preservation condition for the crude monadicity theorem. A reflexive pair in ${\mathcal{E}}^{\mathrm{op}}$ is a coreflexive pair $f,g:B\rightrightarrows A$ in $\mathcal{E}$, with $rf=rg=1_B$ for some $r:A\to B$. Let $e:E\hookrightarrow B$ be its equalizer. For any monomorphism $m$, define ${\exists }_m$ on power objects by composing a parameterized subobject with $m$; no general image construction is needed because $m$ is monic. Inverse image followed by this extension obeys $Pm{\exists }_m=1$.

Here both $f,g$ are monic. For a parameterized subobject $S$ of $B$, an element $b$ belongs to $g^{-1}(f(S))$ exactly when $gb=f{b}^{\prime }$ for some $b^{\prime }\in S$. Applying $r$ gives $b=b^{\prime }$, so this condition is precisely $b\in S$ and $fb=gb$. Consequently, as morphisms of power objects,If $h:PB\to Z$ equalizes $Pf,Pg$, these identities giveIt therefore factors uniquely through $Pe$, which is a split epimorphism with section ${\exists }_e$. Since $fe=ge$, $Pe$ also equalizes $Pf,Pg$. Hence $Pe$ is their coequalizer. This proves explicitly that power objects turn coreflexive equalizers into coequalizers.

Finite equalizers in $\mathcal{E}$ give all coequalizers of reflexive pairs in ${\mathcal{E}}^{\mathrm{op}}$, and we have shown that $P$ preserves them and reflects isomorphisms. The crude monadicity theorem now yields

Fix a set $I$ of cardinality $\kappa$. If $\mathcal{E}$ has $I$-indexed products in a category, its Eilenberg-Moore category for the double-power monad has them, since the forgetful functor creates existing limits. The displayed equivalence then gives $I$-products in ${\mathcal{E}}^{\mathrm{op}}$, hence $I$-coproducts in a category in $\mathcal{E}$.

For the converse, suppose those coproducts exist. The diagonal $\Delta :\mathcal{E}\to {\mathcal{E}}^I$ is a logical functor, because all the elementary topos structure in ${\mathcal{E}}^I$ is pointwise, and it has left adjoint ${\coprod }_I$. Apply power-object monadicity to both toposes. Under their monadic identifications, ${\Delta }^{\mathrm{op}}$ is the lift of $\Delta$ to the double-power Eilenberg-Moore categories. The base functor $\Delta$ is a right adjoint to ${\coprod }_I$, so the adjoint lifting theorem for monad algebra functors gives a left adjoint to its algebra lift. Its required reflexive coequalizers exist in ${\mathcal{E}}^{\mathrm{op}}$, since they are opposites of finite equalizers in $\mathcal{E}$. Taking opposites gives a right adjoint to $\Delta$, which is exactly the $I$-product functor. Thus the products and coproducts of fixed cardinality in a topos satisfy

Let $(G,\epsilon ,\delta )$ be a Cartesian comonad, so its endofunctor preserves finite limits. A coalgebra for a comonad is $(A,a)$ with $a:A\to GA$, ${\epsilon }_{A}a=1_A$ and ${\delta }_{A}a=Gaa$. Its morphisms satisfy $bf=Gfa$. Write $U:{\mathcal{E}}^G\to \mathcal{E}$ for the forgetful functor and $R(X)=(GX,{\delta }_X)$ for the cofree coalgebra. Transposition isThe forgetful functor creates finite limits, since $G$ preserves them. We construct both exponential objects and the subobject classifier.

For coalgebras $(A,a),(B,b)$, set $D=B^A$ and $W=(GB{)}^A$ in the ambient topos. Define $p,q:GD\to W$ by transposingwhere $GD\times GA\cong G(D\times A)$. Their cofree transposes are coalgebra morphismsLet $H$ be their coalgebra equalizer. For a coalgebra $(C,c)$, an arrow $C\to R(D)$ corresponds to a map $h:C\times A\to B$. Its evaluation respects coalgebra structure exactly whenUnder the displayed transpositions this is precisely the condition that it equalize $\hat{p},\hat{q}$. Thus ${\mathcal{E}}^G(C,H)\cong {\mathcal{E}}^G(C\times A,B)$ naturally, and $H$ is the required exponential in a coalgebra topos.

Let $\kappa :G\Omega \to \Omega$ classify $G\top :G1\cong 1\hookrightarrow G\Omega$. In $R(\Omega )$ take the equalizerThe truth map into $R(\Omega )$ is $G\top$ and factors through this equalizer. To verify its universal property, take an underlying subobject $m:S\hookrightarrow A$ of a coalgebra. It carries a compatible coalgebra structure exactly when $a$ maps $S$ into $GS$. Such a structure, if it exists, is unique because $Gm$ is monic; its laws follow by cancellation of $m,Gm,G^{2}m$. Moreover $a^{-1}(GS)\subseteq S$ always: applying ${\epsilon }_A$ to a point of $GS$ lands in $S$. Thus stability is equivalent to $S=a^{-1}(GS)$.

If $\chi :A\to \Omega$ classifies $S$, this last equality is exactlyThe cofree transpose $\hat{\chi }=G\chi a:A\to R(\Omega )$ satisfies this equation precisely when it factors through ${\Omega }_G$. Indeed transposing the equality $\hat{\chi }=G\kappa {\delta }_{\Omega }\hat{\chi }$ gives the equation above. Pulling back its truth map recovers $S$, because $\kappa$ classifies $G\top$. Hence ${\Omega }_G$ is the subobject classifier of a coalgebra topos. We have obtained finite limits, exponentials and a classifier, proving ${\mathcal{E}}^G$ is a topos.

Now let $f:\mathcal{H}\to \mathcal{E}$ be a geometric morphism, and put $L=f^{\ast }$, $V=f_{\ast }$, with unit $\eta$. Both functors preserve finite limits, so $G=LV$ is a Cartesian comonad on $\mathcal{H}$. Put $\mathcal{D}={\mathcal{H}}^G$. Its comparison functoris left exact and has right adjointTo check the adjunction, transpose an underlying map $LX\to A$ to $X\to VA$. The coalgebra-morphism equation becomes exactly the equality of the two composites into $VLVA$, so the transposed arrow factors uniquely through the displayed equalizer.

The counit $KJ(A,a)\to (A,a)$ is invertible. Applying $L$ to the defining equalizer gives the equalizer of $Ga,{\delta }_A:GA\rightrightarrows G^{2}A$. The structure map $a:A\to GA$ is exactly that equalizer: it equalizes by coassociativity; if $t:T\to GA$ equalizes, apply ${\epsilon }_{GA}$ to obtain $t=a{\epsilon }_{A}t$, giving its unique factorization through $a$. Thus the underlying comparison counit is an isomorphism, and the forgetful functor reflects isomorphisms. Therefore $J$ is full and faithful.

The adjunctions $U⊣R$ and $K⊣J$ defineHere $p^{\ast }=U$ is faithful, so $p$ is a surjective geometric morphism; $i_{\ast }=J$ is full and faithful, so $i$ is a geometric embedding. Their inverse-image composite is $UK=L$, establishing the surjection-embedding factorization of a geometric morphism.

For uniqueness, let $S$ be the class of arrows in $\mathcal{E}$ inverted by $f^{\ast }$. In any surjection-embedding factorization $f=i^{\prime }{p}^{\prime }$, conservativity of $p^{\prime \ast }$ shows that $S$ is also the class inverted by $i^{\prime \ast }$. The essential image of the fully faithful $i_{\ast }^{\prime }$ consists exactly of the $S$-local objects $Y$, namely those for which $\mathcal{E}(B,Y)\to \mathcal{E}(A,Y)$ is bijective for every $A\to B$ in $S$. One direction follows from the adjunction. For the converse, the reflector unit $Y\to i_{\ast }^{\prime }{i}^{\prime \ast }Y$ belongs to $S$; locality gives a retraction, and locality of the reflected object makes that retraction also a right inverse. Thus the unit is invertible. The image subcategory is consequently determined by $f$, independently of the factorization. The two intermediate toposes are equivalent over $\mathcal{E}$, and their inverse-image composites with the embeddings determine the surjection legs. This proves uniqueness up to equivalence.

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.

Use the elementary topos definition: a category with finite limits, exponential objects, and a subobject classifier. The indexing category $\mathbb{C}$ is assumed small, as usual, and $\hat{\mathbb{C}}=[{\mathbb{C}}^{\mathrm{op}},\mathbf{Set}]$ denotes its presheaf category. A finite limit of categorical presheaves is formed objectwise: at $c$ take the corresponding limit of sets, and apply the universal property to induce restriction maps. The terminal categorical presheaf is the constant singleton, so this supplies all required finite limits.

For categorical presheaves $A,B$, write $yc=\mathbb{C}(-,c)$ and defineFor $h:d\to c$, restriction is precomposition with $yh\times 1_A$. Evaluation sends $(\theta ,a)$ at $c$ to ${\theta }_c(1_c,a)$. Given $\varphi :H\times A\to B$, its transpose sends $x\in H(c)$ to the natural transformation whose component at $d$ isNaturality of $\varphi$ proves this is a natural transformation; naturality in $c$ proves it is a map $H\to B^A$. Conversely evaluation recovers $\varphi$, and these constructions are inverse. Thereforeso the presheaf category is Cartesian closed.

Define $\Omega (c)$ to be the set of sieves on a category with codomain $c$. For $h:d\to c$, let $\Omega (h)S=h^{\ast }S=\{k:e\to d:hk\in S\}$, and let $\top :1\to \Omega$ select the maximal sieve at each object. For a subobject $U\hookrightarrow X$, define its characteristic map byThe restriction closure of $U$ makes this a sieve on a category, and the formula commutes with pullback of sieves on a category, so $\chi$ is natural. It is maximal exactly when $1_c$ belongs to it, equivalently when $x\in U(c)$; hence the pullback of $\top$ along $\chi$ is $U$. Uniqueness is also explicit: naturality forces $h\in {\chi }_c(x)$ exactly when the restriction $X(h)x$ lies in the truth pullback at $d$. Thus the sieve presheaf is a subobject classifier, and every small presheaf category is an elementary topos.

The defining square for the subobject classifier is a pullback in a category:The inverse image functor of a geometric morphism preserves this square and the terminal object. By definition, the inverse-image truth arrow $f^{\ast }{\top }_{\mathcal{E}}$ is itself the pullback of ${\top }_{\mathcal{F}}$ along $\bar{\lambda }$. Composing these two pullback squares shows that $f^{\ast }A\hookrightarrow f^{\ast }X$ is the truth pullback along $\bar{\lambda }\circ f^{\ast }a$. Uniqueness of the characteristic map gives

Let ${\Omega }_j\hookrightarrow \Omega$ be the equalizer of $j$ and the identity. By part (iv), maps $X\to {\Omega }_j$ correspond exactly to closed subobjects of $X$. Closed subobjects of $X$ in turn correspond to subobjects of $LX$ in $\mathcal{L}$: send $S$ to $LS\hookrightarrow LX$, and in the reverse direction pull a fixed subobject $N\hookrightarrow LX$ back along ${\eta }_X$. Left exactness shows that these operations are inverse; the pulled-back object has reflection $N$, making it closed by the defining formula.

ConsequentlyReplacing $X$ by $LX$ and using $L{\eta }_X$ invertible shows that maps into ${\Omega }_j$ extend uniquely across every unit. The criterion of part (i) therefore proves ${\Omega }_j$ itself is fixed, so belongs to $\mathcal{L}$. Truth factors through it because $j\top =\top$. For fixed $X$, the displayed bijection becomesWith the factored truth arrow this is precisely the subobject classifier universal property. Thus the closed-subobject classifier supplies the classifier in the reflective subcategory.

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.

Take the definition of an elementary topos as a category with finite limits, exponential objects and a subobject classifier $\top :1\hookrightarrow \Omega$. Its power object is $PX={\Omega }^X$, and $Pf:PY\to PX$ is precomposition with $f:X\to Y$. Transposing a predicate on $X\times Y$ in either variable givesThus $P^{\mathrm{op}}:\mathcal{E}\to {\mathcal{E}}^{\mathrm{op}}$ is left adjoint to the contravariant power-object functor $P:{\mathcal{E}}^{\mathrm{op}}\to \mathcal{E}$.

Here are the remaining hypotheses for the crude monadicity theorem, obtained from those topos axioms. The unit of this adjunction is ${\eta }_X:X\to PPX$, internally ${\eta }_X(x)(S)=(x\in S)$. It is monic: if two such evaluations agree, evaluate on the singleton predicate $\{x\}$ classified by the diagonal to obtain $x=y$. If $Pf$ is invertible, naturality $PPf{\eta }_X={\eta }_{Y}f$ therefore makes $f$ monic. The characteristic map of this mono, regarded as a global element of $PY$, pulls back along $f$ to the everywhere-true predicate on $X$. Since $Pf$ is monic, that characteristic map was already everywhere true on $Y$. The classifier pullback then says that $f$ is invertible. Hence $P$ is a conservative functor.

It remains to check preservation of reflexive coequalizers in the opposite category. Equivalently, let $f,g:B\rightrightarrows A$ be a coreflexive pair, with $r:A\to B$ satisfying $rf=rg=1_B$, and let $e:E\hookrightarrow B$ be their equalizer. Both $f$ and $g$ are monic. For any mono $i$, there is a direct-image map ${\exists }_i$ between power objects: a subobject is sent to its composite with $i$. This uses only classification of monos, not the prior existence of general images or colimits. We have $Pi{\exists }_i=1$.

For a predicate $W\hookrightarrow B$, consider its direct image along $f$. Its pullbacks along the two sections areTo verify the second equality, $g(b)=f(w)$ with $w\in W$ implies $b=w$ after applying $r$, and therefore $f(b)=g(b)$; conversely $b\in W\cap E$ supplies that witness. This argument works for parameterized subobjects as well, so it is an equality of arrows between power objects.

Now if $h:PB\to Z$ satisfies $hPf=hPg$, then $h=h{\exists }_{e}Pe$. Hence $h{\exists }_e:PE\to Z$ is its unique factorization through $Pe$, uniqueness following from the section ${\exists }_e$. Thereforeis a coequalizer. Finite limits supply every required coreflexive equalizer. The right adjoint $P$ reflects isomorphisms and preserves the corresponding reflexive coequalizers, so $P$ is monadic. In particular ${\mathcal{E}}^{\mathrm{op}}$ is equivalent to the Eilenberg-Moore category of the double-power-object monad on $\mathcal{E}$.

Let $F:\mathcal{E}\to \mathcal{F}$ be a logical functor, and suppose $L⊣F$. Preservation of exponentials and the classifier gives $F{P}_{\mathcal{E}}\cong P_{\mathcal{F}}{F}^{\mathrm{op}}$, compatibly with the units, counits and resulting monads. Thus $F^{\mathrm{op}}$ is the algebra functor lifting the base functor $F$ through the two monadic power-object functors. The adjoint lifting theorem for monad algebra functors applies to the base adjunction $L⊣F$: its required coequalizers exist in ${\mathcal{E}}^{\mathrm{op}}$, because $\mathcal{E}$ has finite equalizers. It supplies a left adjoint $K⊣F^{\mathrm{op}}$. Taking opposites gives $F⊣K^{\mathrm{op}}$, the required right adjoint to $F$.

Preserving either of the two logical structures by itself is insufficient. For the exponential example take $F:\mathbf{Set}\to \mathbf{Set}$ constant at $1$. The constant-empty functor is its left adjoint, since both relevant hom-sets are singletons. Its canonical exponential comparison is $1\cong 1^1$, so it preserves exponentials. It has no right adjoint: a functor with a right adjoint would preserve the initial object, whereas $F(\varnothing )=1$.

For the classifier example take $G:(\mathbb{Z}/2)\text{-}\mathbf{Set}\to \mathbf{Set}$ to be fixed points. The trivial-action functor is left adjoint to $G$. In a group-action topos the classifier is the trivial-action two-element set, since invariant subsets have ordinary equivariant characteristic maps. Therefore $G$ preserves the classifier, its true arrow and the terminal object. But $G$ does not preserve the coequalizer of the identity and the nontrivial translation on the regular two-element group set: that coequalizer is $1$, while the fixed-point sets of the domain and codomain of the parallel pair are empty. Their set-theoretic coequalizer is empty, not $G(1)=1$. Thus $G$ has no right adjoint.

Write the Cartesian comonad as $(G,\epsilon ,\delta )$, with $G$ preserving finite limits. A coalgebra for a comonad is $(A,a)$ with $a:A\to GA$, ${\epsilon }_{A}a=1_A$ and ${\delta }_{A}a=Gaa$. Its forgetful functor $U:{\mathcal{E}}^G\to \mathcal{E}$ is faithful, creates finite limits, and has the cofree coalgebra right adjoint $R(X)=(GX,{\delta }_X)$. We construct the two remaining topos structures explicitly.

For exponential objects, take coalgebras $(A,a),(B,b)$ and start with $R(B^A)$. Put $X=G(B^A)$, with structure $\xi ={\delta }_{B^A}$, and letTranspose the following two maps $X\times A\to GB$ into maps $u,v:X\to (GB{)}^A$:using $G(X\times A)\cong GX\times GA$. The cofree adjunction transposes $u,v$ once more into coalgebra morphisms $R(B^A)\rightrightarrows R((GB{)}^A)$. Let $Q$ be their equalizer. A coalgebra map $(C,c)\to R(B^A)$ corresponds to an arbitrary ambient map $h:C\times A\to B$. It factors through $Q$ precisely whenwhich is exactly the condition that $h$ be a coalgebra morphism. Therefore $Q$ represents ${\mathcal{E}}^G(C\times A,B)$ and is the required exponential in a coalgebra topos.

For the subobject classifier, let $\kappa :G\Omega \to \Omega$ classify the mono $G\top :G1\cong 1\hookrightarrow G\Omega$. In the cofree coalgebra $R\Omega$, formBoth arrows are coalgebra morphisms. The cofree transpose of $\top :1\to \Omega$ factors through this equalizer and gives its true arrow. To verify classification, let $S\hookrightarrow X$ have ambient characteristic map $\chi :X\to \Omega$. It supports a subcoalgebra of $(X,x)$ exactly when it is invariant under $x$, equivalentlyThe counit proves the reverse containment in the first equation; the forward containment supplies the restricted structure map, whose coalgebra laws follow through the mono. Under the cofree adjunction, the second equation says exactly that $G\chi x:X\to R\Omega$ factors through ${\Omega }_G$. Pulling back its true arrow recovers $S$, since $\kappa G\chi x=\chi$. This proves the universal property of the subobject classifier of a coalgebra topos. Hence ${\mathcal{E}}^G$ is a topos.

Now let $f:\mathcal{E}\to \mathcal{F}$ be a geometric morphism, with $L=f^{\ast }⊣H=f_{\ast }$. The comonad $G=LH$ is Cartesian: $L$ preserves finite limits and the right adjoint $H$ preserves limits. Put $\mathcal{D}={\mathcal{E}}^G$, already a topos. The forgetful adjunction $U⊣R$ defines $p:\mathcal{E}\to \mathcal{D}$ with $p^{\ast }=U$, so $p^{\ast }$ is faithful.

The comparison functorpreserves finite limits. It has a right adjoint $J$, given on a coalgebra $(A,a)$ byIndeed, the transposed arrow $Y\to HA$ corresponds to a coalgebra map $KY\to (A,a)$ exactly when it equalizes these two maps. Applying the finite-limit-preserving $L$ shows that $LJ(A,a)$ is the equalizer ofThat equalizer is $a:A\to GA$. If $t:Z\to GA$ equalizes the pair, then $a{\epsilon }_{A}t={\epsilon }_{GA}Gat={\epsilon }_{GA}{\delta }_{A}t=t$, proving the claimed universal property. Consequently the counit $KJ(A,a)\to (A,a)$ is invertible. The fully faithful adjoint criterion makes $J$ full and faithful.

Thus $K⊣J$ defines a geometric embedding $i:\mathcal{D}\to \mathcal{F}$, and $UK=L$ identifies the composite with $f$. The requested factorization iswhere $p$ is a surjective geometric morphism and $i_{\ast }=J$ is full and faithful.

Write the comonad as $(G,ϵ,\delta )$ and its category of coalgebras for a comonad as ${\mathcal{E}}^G$. A coalgebra for a comonad is a map $a:A\to GA$ with ${ϵ}_{A}a=1_A$ and ${\delta }_{A}a=Gaa$. A morphism $f:(A,a)\to (B,b)$ satisfies $bf=Gfa$. Let $U:{\mathcal{E}}^G\to \mathcal{E}$ be the forgetful functor and let $R(X)=(GX,{\delta }_X)$ be the cofree coalgebra. The adjunction $U⊣R$ has the explicit correspondenceWe construct the three pieces of the elementary topos structure.

Because $G$ preserves finite limits, each underlying finite limiting cone has a unique coalgebra structure induced by the structures on its vertices. The counit and coassociativity equations can be checked after its jointly monic projections. Thus $U$ creates finite limits and reflects isomorphisms. A morphism of coalgebras is monic exactly when its underlying morphism is monic, by the diagonal criterion using the created pullback.

For exponentials in a coalgebra topos, fix coalgebras $(A,a)$ and $(B,b)$ and put $E=B^A$ in $\mathcal{E}$. On the cofree coalgebra $R(E)$ there is an underlying evaluationThe two mapsuse $G(GE\times A)\cong GGE\times GA$ in the second expression. Transpose them in $\mathcal{E}$ to maps $r,s:GE\rightrightarrows (GB{)}^A$, and then transpose across $U⊣R$ to coalgebra morphisms $\tilde{r},\tilde{s}:R(E)\rightrightarrows R((GB{)}^A)$. Take their equalizer $C$ in ${\mathcal{E}}^G$.

An underlying map $UX\times A\to B$ corresponds to a coalgebra map $h:X\to R(E)$. The equation saying that the original map is a coalgebra morphism is precisely $rh=sh$, since ${\delta }_{E}h=Ghx$ for the structure $x:X\to GX$. By the cofree adjunction, this is equivalent to $\tilde{r}h=\tilde{s}h$, hence to unique factorization through $C$. Thereforenaturally in $X$. This constructs the required exponential object.

For the subobject classifier of a coalgebra topos, let $\top :1\hookrightarrow \Omega$ be the underlying subobject classifier, and let $\kappa :G\Omega \to \Omega$ classify the mono $G\top :G1\cong 1\hookrightarrow G\Omega$. Its cofree transpose is the coalgebra endomorphismDefine ${\Omega }_G$ as the equalizer of $k$ and $1_{R\Omega }$. The transpose of $\top$ factors through this equalizer and gives ${\top }_G:1\to {\Omega }_G$.

Indeed, for a subobject $S\hookrightarrow UX$ classified by $\chi :UX\to \Omega$, the pullback $x^{-1}(GS)$ has characteristic map $\kappa G\chi x$. It is always contained in $S$, by naturality of the counit. Equality holds exactly when $x$ restricts to a coalgebra structure on $S$; its axioms then follow by composing with the monomorphisms $GS\hookrightarrow GX$ and $GGS\hookrightarrow GGX$. Under $U⊣R$, the classifying map becomes $h=G\chi x:X\to R\Omega$. The equality of subobjects is ${ϵ}_{\Omega }h=\kappa h$. Transposing this equality gives $h=kh$, so exactly the coalgebra subobjects correspond to maps $X\to {\Omega }_G$. Their pullback of ${\top }_G$ is the desired subcoalgebra, and uniqueness follows from uniqueness of $\chi$.

Thus we have finite limits, exponentials and a subobject classifier:The construction does not assume that $G$ preserves the underlying exponentials or underlying subobject classifier.