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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 signature $\Sigma$ specifies sorts, finite-arity function symbols and finite-arity relation symbols. A coherent formula is built from atomic relations and equality using $\top ,\perp$, finite conjunction, finite disjunction and existential quantification. A coherent theory consists of coherent sequents $\varphi {⊢}_{\vec{x}}\psi$ in finite typed contexts. The context expresses universal quantification of the free variables; unrestricted universal quantification or negation is not a formula constructor here.

Construct the coherent syntactic category ${\mathcal{C}}_{\mathbb{T}}$ as follows. Objects are formulas in context $\{\vec{x}.\varphi \}$, modulo renaming and provable equivalence. An arrow to $\{\vec{y}.\psi \}$ is a formula $\rho (\vec{x},\vec{y})$, modulo provable equivalence, for which the theory provesThese say that the relation is supported on the specified objects, total and single-valued. The identity is equality on the context, and composition isThe logical rules give associativity and the identity laws. The empty-context true formula is terminal; conjunction in disjoint contexts gives products; conjunction with equality of the two functional relations gives equalizers. Existential quantification gives images, and finite disjunction gives unions of subobjects. Substitution preserves these constructions, so images and unions are stable under pullback. This makes ${\mathcal{C}}_{\mathbb{T}}$ a coherent category.

Let $M$ be a $\mathbb{T}$-model in a topos $\mathcal{E}$. Interpret each context as a finite product of its sort objects and each formula as its corresponding subobject. DefineThe interpretation of an arrow relation $\rho$ is a total single-valued relation between these objects. Its projection to the domain is both an epimorphism, by totality, and a monomorphism, by single-valuedness, hence invertible in a topos. Its other projection therefore determines a unique morphism. Composition agrees with relational composition, so $F_M$ is a functor. Finite limits interpret equality and conjunction; images interpret existential quantification; finite unions interpret disjunction, including the empty union for $\perp$. Consequently $F_M$ is a coherent functor.

Conversely there is a canonical syntactic model $U$ in ${\mathcal{C}}_{\mathbb{T}}$. Interpret a sort $s$ by $\{x:s.\top \}$, a function symbol by its term graph, and a relation symbol by its atomic formula-in-context subobject. Induction on formulas identifies the interpretation of every $\varphi$ in $U$ with its formula-in-context object. Every axiom is then an inclusion of the associated subobjects, because it is provable in $\mathbb{T}$.

For a coherent functor $F:{\mathcal{C}}_{\mathbb{T}}\to \mathcal{E}$, set $M_F=F(U)$. Preservation of finite limits, images and unions proves inductively that $F$ preserves all coherent interpretations, so every axiom holds in $M_F$. The same induction supplies natural isomorphismscompatible with the functional relations defining arrows. Thus $F\cong F_{M_F}$; directly on the sorts, operations and relations, $M_{F_M}\cong M$.

Finally these constructions agree on morphisms. A model homomorphism $h:M\to N$ induces maps on finite contexts. Atomic relations are preserved by definition; finite conjunction and disjunction and existential witnesses then show that each formula object maps to the corresponding formula object. Compatibility with functional relations makes these maps a natural transformation $F_M\to F_N$. Conversely a natural transformation between coherent functors restricts to its components on sort objects; naturality for term graphs preserves operations, and naturality for relation inclusions preserves relations. It is therefore a model homomorphism. The two correspondences are inverse. We have proved the categorical, not merely objectwise, equivalence

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.

Use one ring sort with $0,1,+,-,\cdot$ and the usual unital commutative ring equations. Write $\mathrm{Inv}(x)$ for the coherent formula $\exists y(xy=1)$. Add the coherent sequentsThis presents the coherent theory of local rings. For an ordinary nonzero ring satisfying these axioms, the nonunits form an ideal: a multiple of a nonunit cannot be a unit; and if nonunits $a,b$ had unit sum, multiplying by $(a+b{)}^{-1}$ would make two nonunits sum to $1$, contradicting the second axiom. They are also closed under additive inverses. This proper ideal contains every proper ideal, since a proper ideal contains no unit, and its quotient is a field; it is the unique maximal ideal. Conversely in a local ring, $x$ and $1-x$ cannot both belong to the unique maximal ideal, so one is a unit. The first sequent excludes the zero ring. In a topos, the disjunction asserts that these unit alternatives hold locally.

Let $\mathcal{C}$ be a small skeleton of the opposite of the category of finitely presented unital commutative rings; write $C_A$ for the object corresponding to $A$. The Zariski coverage on finitely presented rings is generated byand the empty cover of $C_0$. Localization is finitely presented, and its compatibility with base change proves the covering axioms.

Here is the classifying argument. For an internal commutative ring $M$ and a presentation $A=\mathbb{Z}[x_1,\dots ,x_n]/(p_1,\dots ,p_r)$, define $F_M(C_A)$ as the subobject of $M^n$ satisfying the polynomial equations $p_i=0$. Ring maps induce maps of these solution objects in the site direction. This is independent of presentation, and it preserves finite limits: finite colimits of finitely presented rings correspond to joining finite generators and equations, whose solution objects are finite limits. Conversely a left-exact functor $F:\mathcal{C}\to \mathcal{E}$ reconstructs a commutative ring $F(C_{\mathbb{Z}[t]})$ using the polynomial maps for addition, multiplication, constants and additive inverse. Presentations by generators and equations then show $F\cong F_M$. Thus flat functors on $\mathcal{C}$ correspond to internal commutative rings.

Under $F_M$, the localization arrow above identifies its domain with the locus where $f_i$ evaluates to a unit; the inverse witness is unique. The family is jointly epimorphic exactly when these unit loci cover the solution object for $A$. The locality axiom implies this for every unit-ideal family: from a finite sum $\sum f_i{g}_i=1$, repeatedly apply the two-summand unit alternative, rescaling a unit partial sum when necessary; some $f_i{g}_i$, hence some $f_i$, is a unit locally. Conversely the cover of $\mathbb{Z}[t]$ by its localizations at $t$ and $1-t$ gives precisely the locality axiom, while the empty cover of the zero ring gives $0=1⊢\perp$. Therefore continuous flat functors are exactly models of the displayed coherent theory. The Diaconescu equivalence for geometric morphisms now identifiesnaturally in $\mathcal{E}$. This proves the asserted classifying topos identification.

Since the site is subcanonical, the generic local ring is the representable sheafIts ring operations are induced by polynomial substitution. Its infinitesimal subobject is likewise representable:Products of representables correspond to tensor products of their rings. For every finitely presented $A$, exponential adjunction and the Yoneda lemma giveEvery element of this ring has a unique expression $a+b\epsilon$, because $1,\epsilon$ form an $A$-basis. This holds even when $A$ has nilpotents or zero divisors. The expression is natural under every ring homomorphism $A\to B$, which sends its two coefficients to their images. The canonical morphism $R\times R\to R^D$, transposed from $(a,b,d)\mapsto a+bd$, is consequently a bijection at every stage $C_A$, with a natural inverse extracting the coefficients. ThereforeThis proves the Kock-Lawvere axiom for the generic local ring. The uniqueness is an internal sheaf assertion tested at all finitely presented rings, including the ring with its generic square-zero element; restricting the test to fields would lose that element and would not prove uniqueness.

The general adjoint functor theorem says that, for a functor $U:\mathcal{A}\to \mathcal{B}$ between locally small categories, with $\mathcal{A}$ a complete category, $U$ has a left adjoint if and only if it preserves small limits and satisfies the solution-set condition. The solution-set condition means that for each $B\in \mathcal{B}$ there is a set of arrows $b_i:B\to U{A}_i$ such that every $b:B\to UA$ factors as $U(a)b_i$ for some $i$ and some $a:A_i\to A$.

Here is the smallness argument behind the theorem. A complete category that is locally small and has a weakly initial set $(W_i)$ has an initial object. Form the product in a category $W={\prod }_i{W}_i$. It is weakly initial, since a projection $W\to W_i$ followed by a chosen arrow $W_i\to X$ gives an arrow $W\to X$. Take the simultaneous equalizer $e:E\to W$ of all endomorphisms of $W$ with $1_W$; the family is a set by local smallness. The object $E$ is also weakly initial. For parallel arrows $a,b:E\rightrightarrows X$, take their equalizer $j:Y\to E$ and a map $t:W\to Y$. Since $ejt$ is an endomorphism of $W$, the defining property of $e$ gives $ejte=e$. Cancellation of the monomorphism $e$ gives $jte=1_E$. Thus the monomorphism $j$ has a right inverse and is an isomorphism, so $a=b$. Existence of arrows out of $E$ was already established, proving the initial-object lemma for complete categories with a weakly initial set.

If $U$ preserves small categorical limits, the comma category $(B\downarrow U)$ is complete: take the limiting object in $\mathcal{A}$, and use preservation to assemble the arrows from $B$. It is locally small, and the solution-set condition provides a weakly initial set. Its initial object is an arrow ${\eta }_B:B\to ULB$ with a natural bijectionFor a morphism $v:B\to B^{\prime }$, define $Lv$ by $U(Lv){\eta }_B={\eta }_{B^{\prime }}v$. Uniqueness proves functoriality and naturality, giving a left adjoint $L⊣U$. Conversely, a right adjoint preserves categorical limits, because applying the adjunction converts a limiting cone over a diagram into the corresponding limiting cone of hom-sets. Its adjunction unit at $B$ is a singleton solution set. This proves both directions of the general adjoint functor theorem.

Apply it to the inclusion $J:\mathbf{KHaus}\hookrightarrow \mathbf{Top}$, where $\mathbf{KHaus}$ is the full subcategory of compact Hausdorff spaces. A small product in a category of compact Hausdorff spaces is again compact Hausdorff by the Tychonoff theorem. An equalizer of two continuous functions into a Hausdorff space is closed, hence compact Hausdorff. Products and equalizers construct all small categorical limits, and the inclusion preserves these constructions. Both categories are locally small.

Fix a topological space $X$ and a continuous function $f:X\to K$ into a compact Hausdorff space. The closure $K_f=\overline{f(X)}$ is compact Hausdorff and has a dense subset of cardinality at most $|X|$. The permitted cardinal estimate givesFor $X=\varnothing$, use $K_f=\varnothing$. Choose a set of representatives of all compact Hausdorff topologies on underlying sets of cardinality at most this bound, and all continuous functions from $X$ to those representatives. Every $f$ factors through one of them, using the inclusion $K_f\hookrightarrow K$. This is the solution-set condition. Consequently the inclusion of compact Hausdorff spaces has a left adjoint, the compact Hausdorff reflection:The universal arrow $X\to RX$ has dense image: factoring it through its closed image and applying uniqueness supplies an inverse to that image inclusion. It need not be an embedding when $X$ is not sufficiently separated; the assertion is a compact Hausdorff reflection, rather than a claim that every topological space embeds in a compact Hausdorff space.

For completeness, the alternative Special adjoint functor theorem can be proved from the same initial-object argument. Its limit form assumes that $\mathcal{A}$ is complete, locally small and well-powered, with a small cogenerating family $(Q_i{)}_{i\in I}$; then $U:\mathcal{A}\to \mathcal{B}$, with $\mathcal{B}$ locally small, has a left adjoint precisely when it preserves small limits. To prove sufficiency, start with $b:B\to UA$ and intersect all subobjects $M\hookrightarrow A$ through which $b$ factors after applying $U$. Well-poweredness makes this a small intersection of subobjects, and preservation of limits gives a smallest supporting object $(M,b_M)$. For maps $s,t:M\rightrightarrows Q_i$, equality $U(s)b_M=U(t)b_M$ makes their equalizer another supporting subobject. Minimality forces that equalizer to be invertible, so $s=t$. Thereforeis an injective function. The evaluation embedding into cogenerator products embeds $M$ in ${\prod }_i{Q}_i^{S_i}$, where $S_i$ is the realized subset of the fixed set $\mathcal{B}(B,U{Q}_i)$. There are only a set of choices of these subsets, only a set of subobjects of each resulting product, and only a set of arrows from $B$ to each $U$-image. They give a solution set, so the already proved general adjoint functor theorem applies. Necessity is again preservation of limits by a right adjoint. For $\mathbf{KHaus}$, monomorphisms are embeddings onto closed subspaces, and $[0,1]$ is a coseparator because continuous functions to it separate points. Thus the Special adjoint functor theorem gives the same reflection directly.

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.

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.

Here Cartesian category means a category with all finite limits. Let $m:Y\hookrightarrow X$ be a monomorphism, and let ${\epsilon }_X\hookrightarrow X\times PX$ be universal membership for a power object $PX$. Pull it back to $Y\times PX$ and then regard the result as a subobject of $X\times PX$. Its unique classifier is an endomorphism $r:PX\to PX$: on a parameterized subobject it intersects with $Y$. Intersecting twice has the same result as intersecting once, so uniqueness of classification gives $r^2=r$.

Take the equalizer $q:Q\hookrightarrow PX$ of $r$ and the identity, and restrict membership to $Y\times Q$. For any subobject $S\hookrightarrow Y\times Z$, its composite with $m\times 1_Z$ is a subobject of $X\times Z$, hence is uniquely classified by $s:Z\to PX$. Since it is contained in $Y\times Z$, its intersection with that subobject changes nothing, so $rs=s$ and $s$ factors uniquely through $Q$. Conversely every map $Z\to Q$ gives the original kind of family by pulling back the restricted membership. These inverse operations givenaturally in $Z$. Thus $Q$ is a power object of a subobject, even if no power objects other than $PX$ were assumed.

Now consider the slice category $\mathcal{E}/A$, which has finite limits formed by ambient pullbacks and has terminal object $1_A:A\to A$. The constant family $A\times X\to A$ has slice power object $A\times PX\to A$: for $Z\to A$,and a classifier $Z\to PX$ is equivalently the map $Z\to A\times PX$ over $A$. Every family $p:X\to A$ is a subobject of this constant family through its monic graph $(p,1_X):X\hookrightarrow A\times X$. Apply the first construction inside the slice. Consequently every slice of a Cartesian category with power objects again has power objects.