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A covariant functor into a Grothendieck topos is flat when its tensor extension from presheaves preserves finite limits. For set values, its category of elements with arrows carrying source elements to target elements is cofiltered; equivalently the functor is a filtered colimit of covariant representables. If the indexing category has finite limits, flatness is equivalent to preservation of finite limits. For a site, cover-to-joint-epimorphism continuity adds the condition needed by the Diaconescu equivalence for geometric morphisms.

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.

Let $\mathbf{FinPos}$ be a small skeleton of finite partially ordered sets with all weak order-preserving functions, and let $\mathbf{FinSPos}$ consist of finite strict partial orders with all strict order-preserving functions. Include the empty structure, because no inhabitation axiom has been imposed. The two classifiers areThese are covariant functor categories, or presheaves on the opposite finite-model categories; reversing this variance would give the wrong generic models. In each case the generic carrier sends a finite model to its underlying set, with the generic relation given by its order at that stage. These are the classifying topos of partial orders and the classifying topos of strict partial orders.

We justify the finite-model presentations internally, so an identification of set-based model categories alone is not being mistaken for a classifying property. A finite weak-order presentation takes a finite graph of required inequalities, closes it transitively and reflexively, and identifies vertices in mutually reachable cycles; this produces a finite poset representing that positive diagram. For the strict theory, a transitive cycle makes the presentation contradictory by irreflexivity; otherwise transitive closure produces a finite strict poset. Homomorphisms from these presentations into any model are exactly the tuples satisfying the specified finite relations.

More explicitly, an internal model $M$ gives $F_M(P)=\mathrm{Hom}(P,M)$ on the opposite finite-model category. Its finite realizations form an internally filtered system. It is nonempty through the empty presentation. Two realizations combine through a disjoint-union presentation. To equalize parallel presentation maps whose composites in $M$ agree, quotient their target by the finitely many required identifications and close the relation transitively. In the weak case, collapse the resulting mutually comparable blocks; antisymmetry in $M$ guarantees that the realization factors through this quotient. In the strict case, either the quotient is a finite strict poset and gives the required factorization, or it has a cycle, making the supposed realization impossible. These constructions prove the flatness conditions without deciding equality of arbitrary elements of $M$.

Conversely a flat functor gives an internal filtered colimit of finite models with the indicated carrier and relation. Each finite axiom remains true under that filtered presentation. One-point presentations recover every element; two-point comparison presentations recover every relation; and finite identifications recover equality. These constructions are inverse: a realization of any finite diagram in the colimit factors locally through a finite stage, and two such realizations agreeing in the colimit agree after a further stage. Thus internal models in every Grothendieck topos correspond to flat functors on the opposite finite-model category. The presheaf case of the Diaconescu equivalence for geometric morphisms gives the displayed classifiers. This direct finite-presentation argument is the relevant safeguard in identifying presheaf classifiers; agreement of set-model categories by itself would not suffice, as emphasized in Research on theories of presheaf type.

The reflexification of a strict partial order interprets the weak theory byReflexivity is supplied by the equality disjunct. For transitivity, distribute over the two disjunctions: a pair of strict comparisons composes by strict transitivity, and any equality substitutes into the other comparison. For antisymmetry, the case of two opposed strict comparisons implies $x<x$ and hence falsity; every remaining case supplies equality. All these are coherent deductions, so the interpretation works in an arbitrary topos rather than only in classical sets.

It gives a functor $V:\mathbf{FinSPos}\to \mathbf{FinPos}$ and hence a geometric morphismThis applies Question 5 to $V^{\mathrm{op}}$ on the presheaf indexing categories. Every finite weak poset is the reflexification of its classical strict part, so every target indexing object is in the image. Question 5(ii) therefore makes $g$ surjective. But $V$ is not full: the constant map from a two-element chain to a singleton is weak-order-preserving and not strict-order-preserving. Question 5(iii) consequently proves the morphism is a surjection, not an injection. The strict part of a weak order would require an inequality predicate, so the classical equivalence of weak and strict orders on sets does not supply an inverse geometric interpretation.

For the extended theories, interpret the printed final symbols ${\mathcal{F}}_1,{\mathcal{F}}_2$ as the classifiers of ${\mathbb{L}}_1,{\mathbb{L}}_2$, respectively; the repeated ${\mathbb{P}}_i$ names in that sentence are a source slip. Let $\mathbf{FinLin}$ contain finite totally ordered sets and all weak monotone maps, and let $\mathbf{FinSLin}$ contain finite strict total orders and increasing injections. Again include the empty order. Then both linear-order classifiers are presheaf toposes:For the weak-total model, two finite nondecreasing realized tuples can locally be merged into a single nondecreasing tuple: compare their first remaining entries using totality, take an entry known to be no larger than the other, and continue. This finite procedure uses only finite totality disjunctions and permits duplicate values; it does not require decidable equality. To equalize maps between finite chains, collapse the convex intervals forced equal by their realization; antisymmetry guarantees that all values in such intervals agree. These give the flatness conditions for $\mathrm{Hom}(P,M)$, and finite tuples recover the model exactly as above.

For a strict-total model, trichotomy merges two finite increasing tuples, identifying entries when the equality alternative holds. Every strict map out of a finite chain is injective: if two indices differ, their images are strictly comparable and cannot be equal. Consequently parallel increasing injections with the same realized tuple are already equal, supplying the equalization condition. The same reconstruction proves the classifying topos of strict linear orders claim; in the weak case it proves the classifying topos of linear orders claim. In particular the latter proof has not silently replaced weak totality by decidable strict trichotomy.

Finally, reflexification sends a strict-total model to a weak-total one. Indeed pulling back weak totality under the interpretation givesexactly the strict trichotomy axiom. Thus the original morphism restricts to the corresponding subtoposes, giving the commutative squareThe vertical embeddings are induced by the full inclusions of finite chains among finite partial orders, and their images classify precisely the added axioms. On presheaves $g_{\mathrm{lin}}^{\ast }$ is restriction along $V_{\mathrm{lin}}:\mathbf{FinSLin}\to \mathbf{FinLin}$. The same object-surjectivity and two-element-collapse argument show that the restricted morphism is also surjective and not an embedding. The exact pullback of the added totality axiom further identifies the square as the corresponding base change of subtoposes.

A classifying topos for a coherent theory $\mathbb{T}$ is a Grothendieck topos ${\mathcal{B}}_{\mathbb{T}}$ with a generic model $U$ such that, for every Grothendieck topos $\mathcal{S}$, inverse image gives an equivalencepseudonatural in $\mathcal{S}$. The equivalence includes model homomorphisms and geometric transformations, not merely a bijection between isomorphism classes of set-based models.

Construct the coherent syntactic category ${\mathcal{C}}_{\mathbb{T}}$. Its objects are coherent formulas in finite contexts, modulo provable equivalence, and an arrow is a provably total single-valued coherent relation between their contexts. Composition existentially quantifies the intermediate tuple. Context concatenation, conjunction and equality give finite limits; existential quantification gives stable images; finite disjunction gives stable finite unions. Thus it is a small coherent category, after choosing representatives.

Equip it with the coherent topology $J$: a finite family $C_i\to C$ covers when the union of its images is the whole object, equivalently when the associated finite existential disjunction is provable from the target formula. Pullback stability and closure of covers under composition follow from the corresponding properties of images and unions. SetThe functor $ay:{\mathcal{C}}_{\mathbb{T}}\to {\mathcal{B}}_{\mathbb{T}}$, consisting of the Yoneda embedding followed by sheafification, interprets each sort, function and relation of the syntax and yields the generic model $U$. It preserves finite limits, images and finite unions: syntactic covers become epimorphic families after sheafification, and the syntactic image factorization thus becomes the image factorization of the corresponding arrow. All the axioms consequently hold in $U$.

Here is the classifying argument. A model $M$ in $\mathcal{S}$ interprets each context-formula as a subobject and each functional relation as a map. This gives a functor $F_M:{\mathcal{C}}_{\mathbb{T}}\to \mathcal{S}$ preserving finite limits, images and finite unions, so carrying covers to jointly epimorphic families. Conversely such an interpretation functor recovers exactly a model and all its formula interpretations. Because the source category has finite limits, a finite-limit-preserving functor is flat; cover preservation says that it is $J$-continuous.

For explicitness, the associated inverse image is the flat tensor extensionwhere a set of sections indexes a coproduct and the coend identifies the two actions of every arrow. Flatness makes this extension preserve finite limits. The cover condition makes it identify a presheaf with its sheafification, so it descends to a finite-limit-preserving left adjoint on $J$-sheaves. Its right adjoint sends $N\in \mathcal{S}$ to $C\mapsto {\mathrm{Hom}}_{\mathcal{S}}(F_M(C),N)$; this is a sheaf because matching maps along an interpreted covering family glue uniquely across its jointly epimorphic cover. This is the Diaconescu equivalence for geometric morphisms, with its actual functors displayed.

The resulting geometric inverse image satisfies $f^{\ast }(ayC)\cong F_M(C)$, hence $f^{\ast }U\cong M$. Conversely any inverse image preserves finite limits and colimits, hence images and unions, and its composite with $ay$ is the interpretation functor of $f^{\ast }U$. Since the sheafified representables generate the topos, these constructions recover the geometric morphism up to its unique appropriate isomorphism. Natural transformations of interpretation functors correspond to the model homomorphisms and geometric transformations. This proves the required equivalence and the construction's classifying property.

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.

Internal Horn models are finite-limit-preserving functors out of the Cartesian syntactic category. Since that category has finite limits, these are exactly flat functors; the presheaf Diaconescu equivalence for geometric morphisms proves the classifying property. Equivalently the classifier is the covariant functor category on finitely presented set-based models.