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A geometric formula is built from atomic formulas, including equality, using finite conjunctions, arbitrary set-indexed disjunctions and existential quantification in a finite variable context. Truth is the empty conjunction and falsity the empty disjunction. A geometric theory is a set of sequents $\varphi {⊢}_{\vec{x}}\psi$ between such formulas over a many-sorted signature. The context supplies the universal force of an axiom; unrestricted universal quantification, implication and negation are not formula constructors in this fragment.

These choices are exactly suited to inverse image functors of geometric morphisms. Finite-limit preservation handles equality and conjunction; colimit and image preservation handle disjunction and existential quantification. Therefore an inverse image carries an internal model of a geometric theory to another model.

Construct the geometric syntactic category ${\mathcal{C}}_{\mathbb{T}}$ as follows. Objects are formulas in context $\{\vec{x}.\varphi \}$, modulo provable renaming and equivalence. A morphism to $\{\vec{y}.\psi \}$ is an equivalence class of formulas $\theta (\vec{x},\vec{y})$ that are provably total and single-valued:The last notation abbreviates componentwise equality. Identity is equality of the context variables; composition is existential conjunction over the intermediate tuple. The category has finite limits, formed through conjunction and equality.

Give it the geometric syntactic topology $J_{\mathbb{T}}$. A family of arrows represented by ${\theta }_i({\vec{y}}_i,\vec{x})$ into $\{\vec{x}.\varphi \}$ covers whenPullback stability is substitution, and transitivity comes from distributing existential conjunction through the covering disjunctions. Thus the generated sieves define a Grothendieck topology. Empty covers impose the interpretation of falsity as the initial object.

The classifying topos isIts universal model $U_{\mathbb{T}}$ assigns a sort $A$ the sheafified representable of $\{x^A.\top \}$, functions their definable graph morphisms and relations their definable subobjects. The syntactic covering conditions make the axioms valid in this model.

For the universal property, interpreting formulas in a model $M$ in a Grothendieck topos $\mathcal{F}$ gives a finite-limit-preserving, $J_{\mathbb{T}}$-continuous functor ${\mathcal{C}}_{\mathbb{T}}\to \mathcal{F}$: covers go to jointly epimorphic families precisely because the relevant sequents hold. The Diaconescu equivalence for geometric morphisms associates to this functor a geometric morphism $\mathcal{F}\to \mathbf{Set}[\mathbb{T}]$. Conversely, pulling $U_{\mathbb{T}}$ back along a geometric morphism gives a model. The constructions on models and their homomorphisms are inverse up to natural isomorphism, yieldingnaturally in $\mathcal{F}$. This proves the required classifying property, not merely classification of set-valued models.

Yes, with the Horn/Cartesian-fragment convention. A Horn theory uses positive finite-conjunction sequents and has a Cartesian syntactic category ${\mathcal{C}}_{\mathbb{T}}^{\mathrm{cart}}$ with finite limits. Provably uniquely witnessed existential formulas can also be admitted without changing the finite-limit nature of its semantics. Its internal models are precisely finite-limit-preserving functors from that category.

For a category with finite limits, flat functors into a Grothendieck topos are exactly finite-limit-preserving functors. In sets, the reason is concrete: the category of elements of a covariant left-exact functor is cofiltered. Its terminal object supplies nonemptiness, products supply cones over pairs of elements, and equalizers supply equalizing cones over parallel arrows. Conversely a cofiltered category of elements expresses the functor as a filtered colimit of covariant representables; filtered colimits of sets commute with finite limits. The internal version uses the same cone conditions locally.

The presheaf form of the Diaconescu equivalence for geometric morphisms therefore identifies geometric morphisms intowith internal Horn models, naturally in the domain topos. This is the presheaf classifier of a Horn theory.

Equivalently, using a small skeleton of the category $\mathcal{A}$ of finitely presented set-based Horn models, the classifier is $[\mathcal{A},\mathbf{Set}]$, with covariant functors. The duality $\mathcal{A}\simeq ({\mathcal{C}}_{\mathbb{T}}^{\mathrm{cart}}{)}^{\mathrm{op}}$ sends a formula to the model generated by its tuple subject to its finite constraints. The presheaf claim does not extend merely because a general geometric or regular theory has no written disjunctions: arbitrary existential witnesses are not uniquely witnessed Horn data.

For the covariant version of the Yoneda embedding, put $Y(A)=\mathcal{C}(A,-)$. A morphism $u:A\to B$ induces the natural transformation $Y(u):Y(B)\to Y(A)$ whose component sends $v:B\to X$ to $vu$. Thus $Y$ is a functor from the opposite category ${\mathcal{C}}^{\mathrm{op}}$ to the functor category $[\mathcal{C},\mathbf{Set}]$. The locally small category hypothesis makes each value a set.

The Yoneda lemma says that, for a functor $F:\mathcal{C}\to \mathbf{Set}$, evaluation at the identity morphism gives a natural bijectionGiven $a\in F(A)$, define ${\theta }_X^a(v)=F(v)(a)$. For $w:X\to X^{\prime }$, the functor law gives $F(w){\theta }_X^a(v)=F(wv)(a)={\theta }_{X^{\prime }}^a(wv)$, proving naturality. Conversely, naturality of $\theta$ at $v:A\to X$ gives ${\theta }_X(v)=F(v){\theta }_A(1_A)$. These two constructions are inverse, because $F(1_A)(a)=a$.

The bijection is natural in both variables. A natural transformation $\beta :F\to H$ sends the element $a$ to ${\beta }_A(a)$, matching postcomposition by $\beta$. For $u:A\to B$, precomposing with $Y(u)$ sends $\theta$ to a transformation from $Y(B)$ and its distinguished element is ${\theta }_B(u)=F(u)(a)$. This is exactly the required dependence on $A$.

Taking $F=Y(B)$ gives $\mathrm{Nat}(Y(A),Y(B))\cong \mathcal{C}(B,A)$. Consequently the covariant Yoneda embedding is full and faithful, with the reversal of arrows accounted for by its opposite category domain.

The limit form of the Special adjoint functor theorem is as follows. Let $\mathcal{C}$ be a locally small category, a complete category, and a well-powered category, and suppose it has a small cogenerating family $(Q_i{)}_{i\in I}$. Let $\mathcal{D}$ be a locally small category. Then a functor $G:\mathcal{C}\to \mathcal{D}$ has a left adjoint if and only if it preserves all small categorical limits. The dual exchanges completeness, well-poweredness and cogenerators for cocompleteness, well-copoweredness and generators, and characterizes functors with a right adjoint.

Here a cogenerating set distinguishes unequal parallel maps by postcomposition into one of the $Q_i$. A well-powered category has a set of subobjects of each fixed object. Completeness means existence of all small categorical limits. We use the initial-object lemma for complete categories with a weakly initial set: a complete category that is locally small has an initial object exactly when it has a weakly initial set. A proof is given in Question 5(c). We also use the fact that when $G$ preserves small categorical limits, the comma category $(B\downarrow G)$ is complete, with its limits constructed in $\mathcal{C}$; local smallness is inherited from $\mathcal{C}$. Finally, having an initial object in $(B\downarrow G)$ for each $B$ is the universal arrow from an object to a functor characterization of a left adjoint.

Assume $G$ preserves small categorical limits, and fix $B\in \mathcal{D}$. We will construct a weakly initial set in $(B\downarrow G)$, rather than assume a solution set. Start with an arbitrary object $(C,x)$, where $x:B\to GC$. Call a subobject $m:S\hookrightarrow C$ supporting if $x=G(m)x_S$ for some $x_S:B\to GS$. There is at least one, namely $1_C$. Because $G$ preserves pullbacks in a category, it preserves monomorphisms: the self-pullback characterization of a monic map is preserved. Consequently each $x_S$ is unique.

There is a set of supporting subobjects by well-poweredness. Their intersection of subobjects exists by completeness; write it $m_0:C_0\hookrightarrow C$. Applying $G$ to this intersection categorical limit gives a unique $x_0:B\to G{C}_0$ lifting all the $x_S$, and hence $x=G(m_0)x_0$. Minimality says that any subobject $n:R\hookrightarrow C_0$ supporting $x_0$ is invertible: $m_{0}n$ supports $x$, so $m_0$ factors through $m_{0}n$, giving a right inverse to the monic $n$.

If $u,v:C_0\to Q_i$ satisfy $G(u)x_0=G(v)x_0$, their equalizer supports $x_0$, since $G$ preserves that equalizer. It is therefore an isomorphism, and $u=v$. We have obtained an injective functionLet $J_i$ be its image, a subset of the fixed set $S_i=\mathcal{D}(B,G{Q}_i)$. The evaluation embedding into cogenerator products is a monomorphismIndeed each $s\in J_i$ corresponds to a unique map $C_0\to Q_i$, and the cogenerating set property makes the resulting family jointly monic. Notice that we use the realized subsets $J_i$, not all of $S_i$: there need not be a map $C_0\to Q_i$ available for an unused index.

There is a set of tuples $J=(J_i)$, since $I$ and every $S_i$ are sets. For each resulting $P_J$, choose representatives of its subobjects; these form a set by well-poweredness. For every representative $R$, include every map $y:B\to GR$, a set by local smallness of $\mathcal{D}$. All resulting pairs $(R,y)$ form a set. Our given $(C,x)$ receives a map from such a pair: transport $x_0$ along the isomorphism between $C_0$ and the chosen representative, then compose its inverse with $m_0$. Thus these pairs form a weakly initial set in $(B\downarrow G)$. This is the cogenerator bound for comma-category solution sets.

The initial-object lemma for complete categories with a weakly initial set gives an initial object $(LB,{\eta }_B)$. For $h:B\to B^{\prime }$, define $Lh$ by the unique equationUniqueness proves the functor laws and gives the natural bijectionsThus $L⊣G$. Conversely, a right adjoint preserves small categorical limits: its adjunction identifies maps into the proposed limiting object with compatible families of maps into the diagram. This proves both directions of the limit form of the special adjoint functor theorem. Applying the argument to the opposite categories proves the dual statement. Choices of representatives and adjoint objects are understood in the usual ambient-universe convention for large categories.

The comma category $(X\downarrow G)$ has objects $(A,x)$ with $A\in \mathcal{C}$ and a function $x:X\to G(A)$. Its morphisms $(A,x)\to (B,y)$ are maps $f:A\to B$ in $\mathcal{C}$ satisfying $G(f)x=y$. The identity morphism and composition in a category are inherited from $\mathcal{C}$; the functor laws make the defining equations stable under both.

For a singleton $1$, its map $1\to G(A)$ is just an element $a\in G(A)$. Thus $(1\downarrow G)$ is the covariant category of elements of $G$, with arrows $f:(A,a)\to (B,b)$ satisfying $G(f)(a)=b$.

A monad $\mathbb{T}=(T,\eta ,\mu )$ consists of an endofunctor $T:\mathcal{C}\to \mathcal{C}$ and natural transformations $\eta :1_{\mathcal{C}}\Rightarrow T$ and $\mu :T^2\Rightarrow T$ satisfyingAn algebra for a monad is $(A,a)$ with $a:TA\to A$ satisfying $a{\eta }_A=1_A$ and $a{\mu }_A=aT(a)$. A morphism of algebras for a monad $h:(A,a)\to (B,b)$ obeys $ha=bT(h)$. The functor laws show that identities and composites obey this equation, giving the Eilenberg-Moore category ${\mathcal{C}}^{\mathbb{T}}$.

For an adjunction $F⊣G$, let $\eta$ be its adjunction unit and $\epsilon$ its adjunction counit. The monad induced by an adjunction isThe two unit laws are $G{\epsilon }_{FA}{\eta }_{GFA}=1_{GFA}$ and $G({\epsilon }_{FA}F{\eta }_A)=1_{GFA}$, respectively the two triangle identities for an adjunction.

For associativity, naturality of the adjunction counit at ${\epsilon }_{FA}:FGFA\to FA$ saysApply $G$ to obtain ${\mu }_A{\mu }_{TA}={\mu }_{A}T{\mu }_A$. The composites defining $\eta$ and $\mu$ are natural transformations, so all monad axioms have been checked, not only their underlying objectwise types.

A split coequalizer consists of $f,g:A\rightrightarrows B$, $q:B\to Q$, $s:Q\to B$ and $t:B\to A$ with $qf=qg$, $qs=1_Q$, $ft=1_B$ and $gt=sq$. If $hf=hg$, then $h=hft=hgt=hsq$. Thus $hs$ factors $h$ through $q$, and the factor is unique because $q$ has the right inverse $s$. This proves the coequalizer property directly. Every functor preserves the diagram, since all these equations are preserved.

For idempotent splitting through a coequalizer, first suppose $e=ir$ with $ri=1_Q$. Then $re=r$. Any $h:E\to Z$ satisfying $he=h$ factors as $h=(hi)r$, uniquely since $r$ is a split epimorphism. Hence $r$ coequalizes $(e,1_E)$.

Conversely, let $q:E\to Q$ coequalize $(e,1_E)$. Since $e^2=e$, the arrow $e:E\to E$ itself equalizes the pair, so there is a unique $i:Q\to E$ with $iq=e$. Then $qiq=qe=q$, and every coequalizer is an epimorphism, so $qi=1_Q$. Thus $e=iq$ is a splitting of an idempotent morphism.

Construct pointwise limits in a functor category. For $H:\mathcal{J}\to [\mathcal{D},\mathcal{C}]$, choose at each object $d$ a categorical limit $L(d)$ of $j\mapsto H_j(d)$, with projections $p_j(d)$. For $u:d\to d^{\prime }$, the family $H_j(u)p_j(d)$ is compatible; define $L(u)$ uniquely byThe universal property gives $L(1_d)=1_{L(d)}$ and $L(vu)=L(v)L(u)$, since those equalities hold after every projection. Thus $L$ is a functor, and each $p_j:L\Rightarrow H_j$ is a natural transformation.

For any categorical cone $(M,(a_j))$, the pointwise categorical limits give unique maps $a(d):M(d)\to L(d)$. To check naturality, compose $L(u)a(d)$ and $a(d^{\prime })M(u)$ with every $p_j(d^{\prime })$; both become $H_j(u)a_j(d)=a_j(d^{\prime })M(u)$. The projections distinguish arrows into their categorical limit, so the two maps agree. Componentwise uniqueness gives uniqueness of the natural transformation $a$. This proves that $(L,p_j)$ really is the required categorical limit, rather than merely a family of objectwise candidates.

A specified categorical limit categorical cone in ${\mathcal{C}}^{\mathrm{ob}\mathcal{D}}$ fixes these objectwise vertices and projections. The displayed equation forces every arrow $L(u)$, and the argument forces every categorical cone factorization. Hence the forgetful functor uniquely lifts that categorical cone and is a limit-creating functor. The construction only takes small categorical limits in $\mathcal{C}$; it does not require $\mathcal{D}$ to be small. As usual, the functor categories are understood in a universe where their collections of transformations are meaningful.

The currying adjunction for small categories uses the bijectionIt sends $H$ to the functor $d\mapsto H(d,-)$, with an arrow $u:d\to d^{\prime }$ inducing the natural transformation whose $c$-component is $H(u,1_c)$. Conversely, for $K:\mathcal{D}\to [\mathcal{C},\mathcal{E}]$, define its uncurried functor by $(d,c)\mapsto K(d)(c)$ andNaturality of $K(u)$ allows the two factors to be interchanged in the appropriate order, giving functoriality. The constructions are inverse and natural in $\mathcal{D}$ and $\mathcal{E}$. Hence $-\times \mathcal{C}$ is a left adjoint to $[\mathcal{C},-]$ on the category of small categories.

The unit sends $d$ to the functor $c\mapsto (d,c)$ and $u$ to the transformation $(u,1_c)$. The counit is evaluation $[\mathcal{C},\mathcal{E}]\times \mathcal{C}\to \mathcal{E}$: $(H,c)\mapsto H(c)$ and $(\alpha ,v)\mapsto H^{\prime }(v){\alpha }_c={\alpha }_{c^{\prime }}H(v)$.

For the monad $(T,\eta ,\mu )$, an algebra for a monad is $(A,a:TA\to A)$ with $a{\eta }_A=1_A$ and $aT(a)=a{\mu }_A$. A morphism of algebras for a monad $f:(A,a)\to (B,b)$ satisfies $fa=bTf$. These objects and arrows form the Eilenberg-Moore category ${\mathcal{C}}^T$.

The free algebra functor sends $A$ to $(TA,{\mu }_A)$ and $f$ to $Tf$. The monad identities verify the algebra laws. For the forgetful functor $U$, the free adjunction iswith inverse $v\mapsto bTv$. The algebra law makes the latter an monad algebra morphism. The identities $h{\eta }_A\mapsto bT(h{\eta }_A)=h{\mu }_{A}T{\eta }_A=h$ and $bTv{\eta }_A=b{\eta }_{B}v=v$ prove the bijection.

In the category of adjunctions inducing a fixed monad, objects are adjunctions $F⊣G$ with their induced monad identified with $T$. A morphism to $F^{\prime }⊣G^{\prime }$ is a functor $H$ between the right-hand categories satisfying $G^{\prime }H=G$, $HF=F^{\prime }$ and compatibility with units and counits. With these strict identifications, defineThe triangle identities give the unit algebra law; naturality of $\epsilon$ at ${\epsilon }_B$ gives the multiplication law. Naturality at $f$ makes $Gf$ an monad algebra morphism. Moreover $UK=G$, $KF=F^T$ because $G{\epsilon }_{FA}={\mu }_A$, and $K{\epsilon }_B$ is the free-adjunction counit at $K(B)$. Hence $K$ is a morphism into the Eilenberg-Moore adjunction.

For any other such $H$, its underlying object at $B$ must be $GB$. Counit compatibility forces its algebra action to be $G{\epsilon }_B$, and the forgetful functor, which is a faithful functor, forces $H(f)=Gf$. Thus $H=K$. This proves terminality of the Eilenberg-Moore adjunction. If adjunctions are specified only up to coherent isomorphisms, the same argument gives uniqueness up to the corresponding compatible natural isomorphism.