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A preadditive category has an abelian-group structure on every hom-set and bilinear composition. One equivalent definition of an additive category is a preadditive category with a zero object and finite products in a category; the following argument proves that these products in a category are automatically coproducts in a category. An abelian category is an additive category with kernels and cokernels in which every monomorphism is a kernel and every epimorphism is a cokernel. Equivalently, for every arrow the canonical map from its coimage to its image is invertible.

Let $P={\prod }_{i=1}^n{A}_i$, with projections $p_i$. Define ${\iota }_i:A_i\to P$ by $p_j{\iota }_i={\delta }_{ji}$, where the diagonal entry is the identity and the others are zero. Since its composites with all projections are identities,Given $f_i:A_i\to B$, the map $f={\sum }_i{f}_i{p}_i$ satisfies $f{\iota }_i=f_i$. If another map has those composites, multiplying the displayed identity by that map forces it to equal $f$. Thus $P$ is also a coproduct in a category. The empty case is the zero object. Reversing arrows proves that a finite coproduct in a category is likewise a product in a category. Hence finite products in a category and coproducts in a category are canonically biproducts in an additive category.

For the category of abelian groups, let $(A_i,D_{ij})$ be a directed diagram, with no injectivity assumption on its transition maps. Its set colimit consists of classes $[i,a]$, withDirectedness proves transitivity of this relation. Define addition by moving representatives to a common upper index:Moving all indices involved to one further upper bound proves independence of representatives and of $k$. Define $-[i,a]=[i,-a]$ and use the common class of the zeros as zero. The abelian-group laws hold at a common index, so descend to the quotient. Every canonical map $A_i\to C$ is a homomorphism, and a compatible cocone induces the homomorphism $[i,a]\mapsto f_i(a)$. This is the unique extension. Moreover the formula for addition is forced by those canonical homomorphisms, proving that the underlying-set functor creates directed colimits. This establishes directed colimits created by the abelian-group forgetful functor explicitly.

If a cocone ${\nu }_i:A_i\to B$ has injective legs, the induced map $C\to B$ is injective. Indeed, equal images of $[i,a]$ and $[j,b]$ give, at a common index $k$,Injectivity of ${\nu }_k$ gives equal representatives at $k$. Since monomorphisms of abelian groups are exactly injective homomorphisms, we have proved that the category of abelian groups is a finitary abelian category in the directed-union sense of the question.

Now let $\mathcal{A}$ be complete, cocomplete and finitary. Index by finite subsets $J\subseteq I$ and put $S_J={⨁}_{j\in J}{A}_j$. For $J\subseteq K$, insert the additional zero coordinates. A compatible cocone on these finite biproducts is exactly a family of maps from every $A_i$, so their directed colimit is $S={\coprod }_{i\in I}{A}_i$.

Each finite biproduct has a map $k_J:S_J\to P={\prod }_{i\in I}{A}_i$ using its coordinates in $J$ and zeros outside $J$. Projection to the finite set of coordinates in $J$ is a retraction, so $k_J$ is a split monomorphism. These maps form a cocone of monomorphisms. The finitary property makes its induced map a monic arrow, and that induced map is exactly the canonical infinite identity matrix:This proves the coproduct-to-product comparison in a finitary abelian category. The PDF's use of finite products in a category in the hint is valid because these are the same finite biproducts.

If $\mathcal{A}$ is also a cofinitary abelian category, apply this argument in its opposite category. The same canonical comparison $j$ is then an epic arrow in $\mathcal{A}$. In an abelian category an arrow that is both a monomorphism and an epimorphism is invertible, so $j$ is an isomorphism.

For countably many copies of $A$, put $S={\coprod }_{n\ge 0}A$ and let $q=j^{-1}\Delta :A\to S$, where $\Delta$ is the product in a category diagonal. Let ${\iota }_n$ denote the coproduct in a category injections, let $s:S\to S$ shift them by $s{\iota }_n={\iota }_{n+1}$, and let $\nabla :S\to A$ be the codiagonal. Comparing every product in a category coordinate givesSince $j$ is a monic arrow, $q={\iota }_0+sq$. Also $\nabla {\iota }_0=1_A$ and $\nabla s=\nabla$. Therefore, for $z=\nabla q$,Hom-sets are abelian groups, so cancellation gives $1_A=0$. Any arrow to or from $A$ is consequently zero by composition with its identity. Thus every object is both initial and terminal, andThis is the countable shift-and-fold argument behind countable biproducts in an additive category force triviality.

Assume $F$ is full and faithful and its image is closed under regular quotients. Its unit is invertible, hence monic. Factor the counit using the hypothesis on $\mathcal{D}$:with $e$ regular epic and $m$ monic. Closure gives $I\cong F{A}^{\prime }$ for some $A^{\prime }$. A right adjoint preserves monomorphisms: postcomposition with $Gm$ on $\mathcal{C}(X,-)$ corresponds under adjunction to postcomposition with $m$ on $\mathcal{D}(FX,-)$, which is injective. Thus $Gm$ is monic.

The triangle identity and invertibility of ${\eta }_{GB}$ give $G{\epsilon }_B=({\eta }_{GB}{)}^{-1}$, so $GmGe$ is invertible. A monomorphism whose composite with another arrow is an isomorphism is itself split epic and hence an isomorphism; therefore both $Gm$ and $Ge$ are isomorphisms. The arrow $e$ is between objects in the essential image of the full and faithful $F$. After choosing $I\cong F{A}^{\prime }$, write it as $Fh$. Naturality of the invertible unit identifies $GFh$ with $h$, so $Ge$ invertible implies $h$ and hence $e$ invertible. Now ${\epsilon }_B=me$ is monic. This proves $\boxed{\text{(ii)}\Rightarrow \text{(i)}}$.

Assume both unit and counit are monic. The unit criterion makes $F$ faithful. The triangle identity is ${\epsilon }_{FA}F{\eta }_A=1_{FA}$. Since ${\epsilon }_{FA}$ is monic, this split epimorphism is an isomorphism: cancelling it in ${\epsilon }_{FA}(F{\eta }_A{\epsilon }_{FA})={\epsilon }_{FA}$ gives $F{\eta }_A{\epsilon }_{FA}=1$. Thus $F{\eta }_A$ is invertible. As $\mathcal{C}$ is balanced, the reflection result in the root solution makes ${\eta }_A$ invertible. Hence $F$ is full and faithful.

Let $q:FA\to B$ be a regular epimorphism. Its adjunct factorization is $q={\epsilon }_{B}Fh$ for $h:A\to GB$. A regular epimorphism is extremal: if a coequalizer $q$ factors through a monomorphism $m$, its other factor also coequalizes the defining pair, descends through $q$, and gives a right inverse for $m$; monicity then makes $m$ invertible. Apply this to the monic ${\epsilon }_B$. More explicitly, if $q$ coequalizes $r,s$, monicity gives $Fhr=Fhs$, so there is $d:B\to FGB$ with $dq=Fh$. Then ${\epsilon }_{B}dq=q$ implies ${\epsilon }_{B}d=1$, and monicity gives $d{\epsilon }_B=1$. Therefore $B\cong FGB$. The image is closed under regular quotients, proving $\boxed{\text{(i)}\Rightarrow \text{(ii)}}$.

Assume $L_{1}F$ is left exact for every additive right exact $F$. For $0\to K\to P\to X\to 0$ with $P$ projective, left exactness makes $L_{1}F(K)\to L_{1}F(P)=0$ a monomorphism. Hence $L_{1}F(K)=0$. Dimension shifting then gives $L_{2}F(X)=0$. As $X,F$ are arbitrary, (iii)$\Rightarrow$(ii), completingThe argument also recovers projectivity of every such kernel in a category directly from the universal first-derived-functor criterion.

An abelian category is an additive category with kernels in a category and cokernels in a category in which the canonical comparison from coimage to image of every morphism is an isomorphism. This is the standard image-coimage form of the definition, equivalently expressed by normality of every monomorphism and epimorphism. Additivity means abelian-group hom-sets with bilinear composition, a zero object and finite biproducts. A sequence is exact at an object when the image of the incoming morphism is the kernel in a category of the outgoing one.

For $f:X\to Y$, construct $q:X\to \mathrm{coim}f=\mathrm{coker}(\ker f)$ and $j:\mathrm{im}f=\ker (\mathrm{coker}f)\to Y$. The universal properties give $f=j\bar{f}q$: first $f$ kills its kernel in a category and factors through $q$; then the resulting map is killed by $\mathrm{coker}f$ because $q$ is epic, so it factors through $j$. The comparison $\bar{f}$ is invertible by the abelian axiom. Cokernels in a category are epic and kernels in a category monic, directly by uniqueness in their universal properties, so $e=\bar{f}q$ is epic and $j$ monic. Thus $\boxed{f=je\text{ is an epimorphism-monomorphism factorization}}$.

In a short exact sequence $0\to X\stackrel{i}{\to }Y\stackrel{p}{\to }Z\to 0$, we have $i=\ker p$ and $p=\mathrm{coker}i$, up to the canonical isomorphisms. If $r:Y\to X$ retracts $i$, the map $1_Y-ir$ kills $i$ and factors as $sp$ for some $s:Z\to Y$. Composing with $p$ gives $psp=p$, so epicity gives $ps=1_Z$. Conversely if $ps=1_Z$, then $1_Y-sp$ is killed by $p$ and factors as $ir$. Composing with $i$ gives $iri=i$, so monicity gives $ri=1_X$. ConsequentlyIn that case $Y\cong X\oplus Z$.

For the derived-functor requests, use additive right exact functors, as customary in abelian homological algebra. An abelian category with enough projectives permits an augmented projective resolution $\cdots \to P_2\to P_1\to P_0\to X\to 0$, obtained by repeatedly taking an epimorphism from a projective onto the previous kernel in a category. Apply $F$ and defineProjective lifts construct maps of resolutions; the comparison and chain-homotopy results make the construction independent of choices and functorial. For a short exact sequence, the horseshoe construction produces a degreewise split short exact sequence of resolutions, hence of their images under additive $F$. Its homology objects give the long exact sequenceThis supplies the requested sketch; detailed comparison and functoriality proofs are not needed here.

A projective object $X$ has a resolution concentrated in degree zero, so $L_{1}F(X)=0$ for every such $F$. Conversely choose $0\to K\stackrel{i}{\to }P\to X\to 0$ with $P$ projective, and take $F=\mathcal{A}(-,K)$ with target ${\mathbf{Ab}}^{\mathrm{op}}$. Precomposition makes this covariant to the opposite category. The usual left exactness of contravariant Hom, which follows directly from the cokernel in a category universal property, says exactly that this $F$ is right exact in that opposite target. If $L_{1}F(X)=0$, the long exact sequence and $L_{1}F(P)=0$ make $F(K)\to F(P)$ monic in ${\mathbf{Ab}}^{\mathrm{op}}$. Equivalently,is epic, hence surjective, in abelian groups. The identity of $K$ has a preimage, giving a retraction of $i$. The sequence splits, and $X$ is a direct summand of $P$, hence projective: a lifting problem for $X$ extends along the projection $P\to X$, is solved for $P$, and restricts along the summand inclusion. ThereforeThis is projectivity detected by all first left derived functors. The following parts prove the three requested equivalent formulations of a hereditary abelian category with enough projectives.

The equations $pi=1_A$ give $(ip)i=i$, so $i$ equalizes $ip$ and $1_B$. If $h:X\to B$ satisfies $(ip)h=h$, put $k=ph:X\to A$. Then $ik=iph=h$. If $k^{\prime }$ is another such factorization, applying $p$ gives $k^{\prime }=pi{k}^{\prime }=ph=k$. This provesIn particular the inclusion in a retract in a category is a monomorphism.

The Freyd general adjoint functor theorem is as follows. Let $G:\mathcal{D}\to \mathcal{C}$, with $\mathcal{D}$ a complete category which is locally small. Then $G$ has a left adjoint if and only if it preserves all small categorical limits and satisfies the solution-set condition: for each $c\in \mathcal{C}$, there is a set-indexed familysuch that every $u:c\to Gd$ can be written $u=G(h)u_i$ for some $i$ and some $h:d_i\to d$. The target category need not be complete. We prove all the existence steps.

First establish the initial-object lemma for complete categories with a weakly initial set. Let $\mathcal{E}$ be locally small and complete, with a weakly initial set $\{w_i\}$, meaning every object receives a map from at least one $w_i$. Its categorical product $w={\prod }_i{w}_i$ is weakly initial: compose a categorical product projection with the map from the appropriate $w_i$. Local smallness makes $\mathcal{E}(w,w)$ a set. Let $e:v\to w$ be the simultaneous equaliser of every endomorphism of $w$ and the identity. Explicitly it is the equaliser of the two arrowswith components $k$ and $1_w$. Thus $ke=e$ for every endomorphism $k$, and $e$ is monic.

Every object $x$ receives a morphism from $v$, by composing $e$ with a map $w\to x$. To prove uniqueness, take parallel maps $a,b:v\to x$ and their equaliser $m:y\to v$. Weak initiality of $w$ gives $t:w\to y$. The composite $emt$ is an endomorphism of $w$, so $(emt)e=e$. Cancelling the monomorphism $e$ gives $mte=1_v$. Hence $m$ is both monic and a split epimorphism, and is therefore invertible. Since $am=bm$, this forces $a=b$. We have proved $v$ is initial, without using a class-sized categorical product or equaliser.

For sufficiency in the theorem, fix $c$ and consider the comma category $(c\downarrow G)$. Its objects are pairs $(d,u:c\to Gd)$; a morphism $h:(d,u)\to (d^{\prime },u^{\prime })$ satisfies $G(h)u=u^{\prime }$. This category is locally small because each hom-set is a subset of $\mathcal{D}(d,d^{\prime })$. It is complete: for a small diagram of such pairs, take the underlying categorical limit $d$ in $\mathcal{D}$. The arrows from $c$ form a cone to its image under $G$, and preservation of the categorical limit gives a unique $u:c\to Gd$ inducing them. The same universal property gives all mediators in the comma category. The solution-set family is a weakly initial set in this comma category, so the lemma supplies an initial pairFor $f:c\to c^{\prime }$, initiality defines a unique $Ff:Fc\to F{c}^{\prime }$ satisfying $G(Ff){\eta }_c={\eta }_{c^{\prime }}f$. Uniqueness proves the identity and composition laws, making $F$ a functor. Initiality also gives bijectionsThe definition of $Ff$ proves naturality in $c$, and postcomposition proves naturality in $d$. These bijections are the required adjunction $F⊣G$.

For necessity, if $F⊣G$, the single arrow ${\eta }_c:c\to GFc$ is a solution set: the transpose of any $u:c\to Gd$ gives its factorization through ${\eta }_c$. If $d$ is a limiting cone vertex of a small diagram $D_j$ in $\mathcal{D}$, then, naturally in $c$,These are the canonical cone-factorization bijections, so $Gd$ with its image projections is a categorical limit of $G{D}_j$. Thus $G$ preserves all small categorical limits, completing both directions of the theorem.

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

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.

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.

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 balanced category is one in which every morphism that is both a monomorphism and an epimorphism is an isomorphism. A strong monomorphism $m:A\to B$ is a monomorphism right orthogonal to all epimorphisms: whenever $e:X\to Y$ is epic and $a:X\to A$, $b:Y\to B$ satisfy $ma=be$, there is a unique $d:Y\to A$ with $de=a$ and $md=b$. A regular monomorphism is an equalizer of some parallel pair of morphisms. Its equalizer property supplies the indicated lift after epic cancellation, so every regular monomorphism is strong.

If every monomorphism is strong and $m:A\to B$ is also epic, apply strongness to the square whose left and right sides are both $m$, with horizontal arrows $1_A$ and $1_B$. The lift $d:B\to A$ satisfies $dm=1_A$ and $md=1_B$. Thus $m$ is invertible and the category is balanced.

Conversely, suppose the category is balanced and has pullbacks in a category. Given the lifting square above, form $P=A{\times }_{B}Y$ with projections $p:P\to Y$, $q:P\to A$. The pair $(a,e)$ gives $h:X\to P$ with $ph=e$, $qh=a$. The pullback of the monomorphism $m$ is monic, so $p$ is monic. It is also epic: $rp=sp$ implies $re=se$, hence $r=s$ because $e$ is epic. Balancedness makes $p$ invertible. Now $d=q{p}^{-1}$ satisfies both required equations, and monicity of $m$ proves uniqueness. Hence in a balanced category with pullbacks, every monomorphism is strong.

For the finite example, choose a strong monomorphism $m:A\to B$ that is not regular. It is not an isomorphism, since an isomorphism is the equalizer of an identical pair. It is not epic either, since an epic strong monomorphism is invertible by the first argument. Consequently there are distinct arrows $r,s:B\rightrightarrows D$ with $rm=sm$. Since $m$ is not the equalizer of this pair, there is a map $h:X\to B$ with $rh=sh$ that does not factor through $m$. Monicity rules out failure of uniqueness, so failure of existence is the only possible failure of its equalizer property.

Define ${\mathcal{C}}_0$ to have four formally distinct objects $a,b,x,d$ and exactly the following six nonidentity arrows:Their only nontrivial composites areThere are no longer composable strings of nonidentity arrows, so these rules define an associative category. Sending $a,b,x,d$ to $A,B,X,D$, respectively, and the six arrows to $m,h,r,s,rm,rh$, defines a functor. It is a faithful functor: every hom-set has at most one arrow except ${\mathcal{C}}_0(b,d)=\{r_0,s_0\}$, and their images are distinct. Faithfulness does not require the four image objects to be distinct.

The map $m_0$ is monic, by direct inspection of arrows into $a$. Neither $m_0$ nor $h_0$ is epic, since the distinct pair $r_0,s_0$ agrees after either. Every other nonidentity arrow has target $d$ and is epic, because there is only one arrow out of $d$. A lifting square against $m_0$ whose left side is one of these epimorphisms would require an arrow $d\to b$, which does not exist. Squares whose left side is an identity have the evident unique lift. Thus $m_0$ is a strong monomorphism.

It is not a regular monomorphism. The only distinct parallel pair out of $b$ is $r_0,s_0$, and $h_0$ equalizes it but has no factorization through $m_0$, since there is no arrow $x\to a$. An identical pair has $1_b$ as its equalizer and cannot have the noninvertible $m_0$ as its equalizer. Therefore

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.

Suppose $f$ is a surjective geometric morphism, so $f^{\ast }$ is faithful. First, if a monomorphism $m:S\hookrightarrow X$ becomes an isomorphism, then its characteristic map ${\chi }_m$ and the constantly true map become equal after $f^{\ast }$: by part (i), the pulled-back subobject is the whole object, and equivalently $f^{\ast }{\chi }_m=f^{\ast }{\top }_X$ because $f^{\ast }m$ is epic and these maps agree after precomposition with it. Faithfulness gives ${\chi }_m={\top }_X$, so $m$ was already invertible.

Now let $a,b:X\to {\Omega }_{\mathcal{E}}$ satisfy $\lambda a=\lambda b$. Adjunction and part (i) identify their pulled-back subobjects $f^{\ast }A=f^{\ast }B$. Each inclusion $A\cap B\hookrightarrow A,B$ consequently becomes an isomorphism. By the preceding argument it already is an isomorphism, so $A=B$ and $a=b$. This works for every test object $X$, proving

Take two subobjects $A,B\hookrightarrow X$ whose inverse images coincide. Their characteristic maps $a,b:X\to {\Omega }_{\mathcal{E}}$ satisfy $\bar{\lambda }{f}^{\ast }a=\bar{\lambda }{f}^{\ast }b$ by part (i). Under the adjunction $f^{\ast }⊣f_{\ast }$, these maps transpose to $\lambda a$ and $\lambda b$, respectively, by naturality of the unit. If $\lambda$ is a monomorphism, then $a=b$ and hence $A=B$. Thus $f^{\ast }$ reflects equality of subobjects.

For parallel arrows $r,s:X\rightrightarrows Y$ with $f^{\ast }r=f^{\ast }s$, their equalizer $K\hookrightarrow X$ pulls back to all of $f^{\ast }X$. Reflection of equality with the whole subobject gives $K=X$, so $r=s$. Therefore the inverse image is faithful, and the geometric morphism is surjective. This is the surjectivity detected by the subobject classifier criterion. For completeness, faithfulness is equivalent here to conservativity: if $f^{\ast }h$ is invertible, the image inclusion of $h$ and its kernel-pair diagonal become whole subobjects; reflection makes $h$ both a regular epimorphism and monic, hence invertible in the topos.