Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 75 (Topos Theory)

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.

Write a coalgebra for a comonad as $(A,a)$ with $a:A\to GA$, ${ϵ}_{A}a=1_A$ and ${\delta }_{A}a=Gaa$. Since the Cartesian comonad $G$ preserves finite limits, the forgetful functor $U:{\mathcal{E}}_G\to \mathcal{E}$ creates them: for an underlying limit $K$ of coalgebras, the maps $K\to A_i\to G{A}_i$ induce a unique $K\to GK$ using $GK\cong {\lim }_{i}G{A}_i$. Its counit and coassociativity equations follow after every limiting projection, which jointly detect equality. The same argument makes the limiting cone universal among coalgebra cones. Thus ${\mathcal{E}}_G$ is a Cartesian category; in particular, the product structure on $A\times B$ is $a\times b$ followed by the comparison $GA\times GB\cong G(A\times B)$.

An exponential object needs more than the underlying ambient $B^A$. The cofree coalgebra functor is $R(D)=(GD,{\delta }_D)$, with adjunctionThe inverse sends $\tilde{h}$ to ${ϵ}_D\tilde{h}$; the comonad laws verify that these are inverse and that $\tilde{h}$ is a coalgebra map.

For $(A,a)$ and $(B,b)$, put $D=B^A$ and $W=(GB{)}^A$ in the ambient Cartesian closed category. Define $p,q:GD\to W$ byand by transposing the compositeto obtain $q$. Their cofree-adjoint transposes are coalgebra mapsTake their coalgebra equalizer $e:H\hookrightarrow R(D)$, which exists by the finite-limit construction.

To verify its universal property, let $h:C\to D$ transpose to $t:C\times A\to B$. Its coalgebra transpose $\tilde{h}=Ghc$ factors through $H$ exactly when $p^{♯}\tilde{h}=q^{♯}\tilde{h}$. By the cofree adjunction, this equality is equivalent to $p\tilde{h}=q\tilde{h}$. The first side is $b^{A}h$, because ${ϵ}_{D}Ghc=h$; transposing the second side gives $Gt$ composed with the product coalgebra structure on $C\times A$. The equality therefore says exactlywith the canonical product comparison understood: precisely the condition that $t$ is a coalgebra map. Thusnaturally in the coalgebra $C$. Its evaluation is ambient evaluation after ${ϵ}_{D}e\times 1_A$, and the preceding condition makes it a coalgebra map. This equalizer is the coalgebra exponential, so the category of coalgebras is cartesian closed. This Cartesian closure for a finite-limit-preserving comonad uses no subobject classifier on $\mathcal{E}$.

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

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.

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

Precomposition definesIt preserves finite limits because presheaf limits are pointwise. Its right adjoint is a Right Kan extension; explicitly, for $H\in \hat{\mathbb{C}}$,Precomposition in $d$ supplies the restriction maps. To verify the adjunction, a map $P\circ T^{\mathrm{op}}\to H$ sends $x\in P(d)$ to the natural family taking $h:Tc\to d$ to the image in $H(c)$ of $P(h)x$. Conversely evaluate such a family at $d=Tc$, $h=1_{Tc}$; naturality makes the two constructions inverse. Thus $f^{\ast }⊣f_{\ast }$ and $f^{\ast }$ is left exact, giving the required geometric morphism

If $d$ is a retract in a category of $Tc$, write $i:d\to Tc$, $r:Tc\to d$ with $ri=1_d$. For a natural transformation $\alpha :P\to Q$ of categorical presheaves, naturality givesConsequently transformations agreeing at all $Tc$ agree at every retract of such an object. If every $d$ is such a retract, restriction $f^{\ast }$ is faithful, and the geometric morphism is surjective.

Conversely suppose $f^{\ast }$ is faithful. Inside the representable presheaf $yd=\mathbb{D}(-,d)$, define $S_d(e)$ to consist of arrows $e\to d$ that factor through some $Tc$. Precomposition preserves this condition, so $S_d$ is a subpresheaf. At $e=Tc$, every arrow factors through that very object, so $f^{\ast }{S}_d=f^{\ast }yd$. A faithful geometric inverse image reflects an inclusion becoming invertible, as proved in Question 4(iii). Hence $S_d=yd$, and $1_d\in S_d(d)$ factors as $d\to Tc\to d$. This is the required retraction. We have proved the retract criterion for surjective presheaf geometric morphisms:

A geometric embedding means $f_{\ast }$ is full and faithful, equivalently that the adjunction counit $f^{\ast }{f}_{\ast }H\to H$ is invertible for every $H$. At $c$, the explicit right-adjoint formula identifies this counit withinduced by ${\tau }_c:\mathbb{C}(-,c)\to \mathbb{D}(T-,Tc)$, sending $h$ to $Th$.

If $T$ is full and faithful, every ${\tau }_c$ is an isomorphism, making all these counit components bijective and hence making $f_{\ast }$ full and faithful. Conversely, if precomposition with ${\tau }_c$ is a bijection for every $H$, then ${\tau }_c$ is invertible by the Yoneda lemma applied in the presheaf category: taking $H=\mathbb{C}(-,c)$ gives a left inverse, and injectivity for $H=\mathbb{D}(T-,Tc)$ makes it a right inverse too. Evaluating ${\tau }_c$ at $c^{\prime }$ therefore says thatis bijective for all $c,c^{\prime }$. This proves the full-faithfulness criterion for presheaf geometric embeddings:

Identify the reflective subcategory with its full replete image, writing ${\eta }_X:X\to LX$ for the reflection unit. An object $Z$ is fixed by $L$ exactly when precomposition with every ${\eta }_X$ gives a bijection $\mathcal{E}(LX,Z)\to \mathcal{E}(X,Z)$. One implication is the reflection adjunction; for the other, take $X=Z$ to obtain a retraction of ${\eta }_Z$, and uniqueness of maps into the fixed object $LZ$ makes it a two-sided inverse.

An ambient finite limit of fixed objects has this same property: its hom-sets are limits of the hom-sets of the diagram, and a limit of these bijections is a bijection. Thus the ambient terminal object, products and equalizers are fixed and retain their universal properties in the full subcategory. The reflective subcategory is Cartesian, with finite limits computed in the ambient topos.

Let $A,B$ be fixed objects and form the ambient exponential object $B^A$. For every $X$, the exponential adjunction identifies maps $X\to B^A$ with maps $X\times A\to B$. Since $L$ is left exact and $LA=A$,and its unit corresponds to ${\eta }_X\times 1_A$. Because $B$ is fixed, every map $X\times A\to B$ therefore extends uniquely to $LX\times A\to B$. Transposing back givesThe fixed-object criterion from part (i) shows that $B^A$ already belongs to $\mathcal{L}$. The ambient evaluation and transpose universal property restrict to the full subcategory, proving the subcategory is cartesian closed. This constructs exponentials in a left-exact reflective subcategory, rather than merely reflecting an exponential and assuming its universal property survives.

For a subobject $m:S\hookrightarrow X$, define the closure operation induced by a left-exact reflector bywhere the pullback uses ${\eta }_X:X\to LX$ and the monomorphism $Lm:LS\hookrightarrow LX$. Naturality of the unit factors $S\hookrightarrow X$ through $c_X(S)$, proving extensivity, and functoriality of $L$ proves monotonicity. Applying the left-exact $L$ to the defining pullback gives $L(c_X(S))\cong LS$, since $L{\eta }_X$ is invertible. Repeating the construction therefore changes nothing, proving idempotence.

For $h:Y\to X$, naturality of $\eta$ and preservation of pullbacks identify the closure of $h^{\ast }S$ with $Y{\times }_{LX}LS$, which is also $h^{\ast }{c}_X(S)$. HenceThe operation is thus universal: it is stable under every pullback. The same construction preserves finite intersections because $L$ is left exact.

Every characteristic map $a:X\to \Omega$ expresses its subobject as $a^{\ast }\top$. Universality of the closure operation induced by a left-exact reflector givesBy definition, $j$ classifies $c_{\Omega }(\top )$, so the right-hand subobject has characteristic map $j\circ a$. The original and its closure agree exactly when their unique characteristic maps agree. ThusExtensivity, idempotence, preservation of intersections and closure of the whole subobject also give the Lawvere-Tierney topology laws for $j$.

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

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

The equality predicate on $A$ is interpreted by the diagonal monomorphism ${\Delta }_A:A\hookrightarrow A\times A$. The antecedent equality of the two $f$-values is the kernel pair $A{\times }_{B}A\hookrightarrow A\times A$. Semantic validity of the implication says that the kernel-pair subobject is contained in the diagonal; the reverse containment always holds. Hence the sequent is valid exactly whenIf $f$ is monic, this equality is its cancellation property expressed by the pullback. Conversely, if $fr=fs$ for $r,s:X\to A$, then $(r,s)$ factors through this kernel pair and therefore through the diagonal, giving $r=s$. Thus the sequent holds exactly when the map is a monomorphism. This argument uses generalized arrows from arbitrary $X$, not a claim that global elements detect maps.

In a regular category, interpret existential quantification by the image of the appropriate projection. The relation $f(x)=y$ is the graph of $f$, isomorphic to $A$; its projection to $B$ is $f$ itself. Thus the subobject interpreting the existential predicate is $\mathrm{im}(f)\hookrightarrow B$. Semantic validity of the sequent means this image is the whole object.

The regular image factorization writes $f=me$ with $e$ a regular epimorphism and $m$ its monic image inclusion. Its image is all of $B$ exactly when $m$ is invertible, equivalently when $f$ is itself a regular epimorphism. This is precisely the cover condition in a regular category.

Interpret the given relation by a subobject $R\hookrightarrow A\times B$ and write its projections as $p:R\to A$, $q:R\to B$. Totality says that the image of $p$ is all of $A$, so $p$ is a regular epimorphism. Single-valuedness makes $p$ monic: if $pr=ps$ for two arrows into $R$, the functionality sequent forces $qr=qs$, and the inclusion into $A\times B$ then gives $r=s$.

A regular epimorphism that is monic is invertible. Indeed its kernel pair consists of equal projections, and its coequalizer universal property supplies an inverse. Therefore defineThe graph of this map is exactly the original relation $R$, since $(p,q)=(1_A,f)p$. The graph also makes the map unique. This proves total functional relations define morphisms, using regular-category semantics and no choice of individual witnesses.

A coherent category is a regular category with finite unions of subobjects, including the empty union, stable under pullback. Thus it has finite limits, pullback-stable regular-epimorphism/monomorphism image factorizations, and pullback-stable finite joins in each subobject lattice.

Write $U=A\vee B\hookrightarrow X$ and $C=A\wedge B=A{\times }_{X}B$. Their inclusions give a commutative squareTo establish the pushout property, take arrows $a:A\to Y$, $b:B\to Y$ agreeing on $C$. Regard their graphs as subobjects of $U\times Y$ and form their union $R$. Internally, this relation says that a point of $U$ is related to its $a$-value when it lies in $A$, or to its $b$-value when it lies in $B$.

The relation is total because every point of $U$ belongs to at least one of $A,B$. It is single-valued: two witnesses in the same summand give the same value by functionality of that map; witnesses in different summands have their shared point in $C$, where the two maps agree. By the relation argument proved in Question 7's unheaded continuation, the projection $R\to U$ is both a regular epimorphism and monic, hence an isomorphism. It consequently gives a map $h:U\to Y$ restricting to $a,b$.

For uniqueness, the equalizer of two possible maps contains $A$ and $B$, hence contains their union $U$ and is all of $U$. Thus the square is a pushout, provingThis is the unions of subobjects are pushouts property; it does not assume arbitrary pushouts already exist in a coherent category.

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.

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.