Small presentations and saturated reflection equivalences

Written by GPT-6.1 Sol (OpenAI), October 2026. Public domain (CC0).

A reflection identifies both a family of local objects and a class of maps that become invertible. Accessibility lets us describe its local objects using a small set of tests. Conversely, the small-test construction determines its entire inverted class by explicit operations on maps. The uniqueness quotient in that construction is essential to this second assertion.

Work in fixed universes, with locally small categories, small diagrams and set-indexed choice. Let \(C\) be locally presentable and \(D\subseteq C\) a full replete reflective subcategory. Write \(L\dashv i\), with inclusion \(i\), unit \(\eta\), and endofunctor \(T=iL\). Fullness and repleteness mean that every map between local objects is included, and every object isomorphic to a local object is local.

Descent through replacement objects, Sections 4–6 proves the actual reflection and projector identities. Presentable probes, Sections 1–4 supplies small full presenting categories, individual ranks and larger regular bounds. Small orthogonality, Sections 1–5 constructs the accessible reflector of a small set of maps. We use those proved arguments at their stated hypotheses. The morphism-and-object orthogonality setting and related deduction rules are discussed in [AHS].

An object \(y\) is orthogonal to \(f:a\to b\) if restriction \(C(b,y)\to C(a,y)\) is bijective. Write \(S^\perp\) for the objects orthogonal to every member of a class \(S\).

1. Accessible reflection and the colimits of local objects

The unit \(\eta_y\) is invertible precisely when \(y\in D\). For local \(y\), its inverse is the counit, using the actual adjunction triangles. Conversely an invertible unit identifies \(y\) with \(T(y)\in D\). We will also use the reflection bijection

For any small diagram \(d:J\to D\), the local colimit exists and is

Indeed maps from this expression to \(y\in D\) correspond by adjunction to compatible maps from the \(i(d_j)\) to \(i(y)\), hence by fullness to its colimit cocones in \(D\). Its leg from \(d_j\) is \(L(c_j)\varepsilon_{d_j}^{-1}\), where \(c_j:i(d_j)\to\operatorname{colim}_C i d\) is the original leg and \(\varepsilon:Li\to1_D\) is the counit. This proves the actual universal property, including the empty diagram. The same Hom calculation proves that \(L\) preserves every small colimit in \(C\).

Choose an infinite regular \(\lambda\) for which \(C\) is locally \(\lambda\)-presentable and \(T\) preserves \(\lambda\)-filtered colimits. Such a choice exists if \(T\) is accessible: increase its accessibility bound and a local-presentability bound together, using Presentable probes, Section 4. A \(\lambda\)-filtered category is also filtered at either smaller bound.

Proposition 1.1. The inclusion creates \(\lambda\)-filtered colimits, and \(D\) is locally \(\lambda\)-presentable.

Proof: the colimit comparison. For a \(\lambda\)-filtered diagram of local objects, put \(c=\operatorname{colim}_C i(d_j)\). The comparison from \(\operatorname{colim}_C T(i(d_j))\) to \(T(c)\) is invertible by the hypothesis on \(T\). The colimit of the invertible units of its entries identifies the first object with \(c\). The composite is \(\eta_c\), since on every original leg this is exactly naturality of \(\eta\). Thus \(\eta_c\) is invertible and \(c\) is local. Its original cocone has the same universal property in the full subcategory. This proves creation, with the actual comparison.

Proof: presentable generators. Let \(B\) be a small full category of \(\lambda\)-presentable objects presenting \(C\). For \(b\in B\), the adjunction identifies \(D(Lb,-)\) with \(C(b,i-)\). The created filtered colimits and presentability of \(b\) therefore make \(Lb\) \(\lambda\)-presentable in \(D\). Every \(i(d)\) has a small \(\lambda\)-filtered presentation by objects of \(B\). Apply \(L\), its colimit preservation, and \(L(i(d))\simeq d\). This presents every \(d\) as a \(\lambda\)-filtered colimit of the small family \(L(B)\). Equation (1.2) supplied all small colimits. These are precisely the defining conditions. \(\square\)

These arguments also prove the following equivalence without assuming accessibility of \(D\) in advance: \(T\) is accessible if and only if \(i\) preserves sufficiently filtered colimits. For the reverse direction, use (1.2) to obtain colimits in \(D\) and compose the colimit-preserving \(L\) with \(i\). For the forward direction, Proposition 1.1 proves the claimed preservation and local presentability. The assertion is about the inclusion functor, rather than just the abstract category \(D\).

2. A small set of unit maps presents the whole reflection

Keep this \(\lambda\) and \(B\), and form the small set

Theorem 2.1. The local objects are exactly \(S_B^\perp\).

Proof: the easy inclusion. A local object is orthogonal to every unit by (1.1), hence to this set.

Proof: extending from the probes. Suppose \(y\in S_B^\perp\). Write an arbitrary \(x\) as a small \(\lambda\)-filtered colimit of a diagram \(b_j\in B\), with legs \(c_j:b_j\to x\). Preservation by \(T\) makes \(T(x)\), with legs \(T(c_j)\), the colimit of \(T(b_j)\). The unit \(\eta_x\) is the map of these colimits induced by the \(\eta_{b_j}\), by naturality.

Given \(h:x\to y\), each \(hc_j\) extends uniquely to \(v_j:T(b_j)\to y\). For an arrow \(a:b_j\to b_k\) of the presenting diagram, the maps \(v_kT(a)\) and \(v_j\) have equal composites with \(\eta_{b_j}\), because

Injectivity of the test for \(b_j\) makes those maps equal. Hence the family is compatible and has a unique colimit factor \(v:T(x)\to y\). It satisfies \(v\eta_x=h\). Conversely any such factor restricts to the same unique \(v_j\); uniqueness in the colimit proves uniqueness of \(v\). Thus \(y\) is orthogonal to every \(\eta_x\), not just to the probe units.

Take \(x=y\). There is an \(r:T(y)\to y\) with \(r\eta_y=1_y\). The maps \(\eta_yr\) and \(1_{T(y)}\) agree after \(\eta_y\); their target is local, so (1.1) makes them equal. Thus \(\eta_y\) is invertible and \(y\in D\). \(\square\)

The endpoints in (2.1) need not both be \(\lambda\)-presentable in \(C\). The object \(Lb\) is \(\lambda\)-presentable in \(D\), but the inclusion has not been shown to preserve presentable objects at that same bound. Since (2.1) is small, the individual-rank theorem supplies a larger common regular \(\mu\) making all \(b,T(b)\) \(\mu\)-presentable in \(C\). Raise the local-presentability bound of \(C\) as well. The small-orthogonality construction then applies with that bound, and its reflector is uniquely isomorphic to \(L\), preserving the unit.

Consequently a full replete subcategory of a locally presentable category is a small orthogonality class if and only if it is reflective with accessible reflection endofunctor. The forward implication is the proved small-orthogonality construction; the reverse implication is Theorem 2.1. Equivalently one may require the reflective inclusion to be accessible, by Section 1. This is a small-set statement; it does not assert that arbitrary proper classes of tests can be replaced by a set.

3. Detect every inverted map by local objects

For any reflection, define

Proposition 3.1. A map belongs to \(W_L\) exactly when every local object is orthogonal to it.

Proof. Adjunction identifies its restriction map into \(i(y)\) with restriction along \(L(f)\) into \(y\), for every \(y\in D\). An invertible \(L(f)\) therefore gives bijections. Conversely suppose all these maps are bijective. Put \(g=L(f):La\to Lb\). Surjectivity with target \(La\) gives \(r:Lb\to La\) with \(rg=1_{La}\). The composites \((gr)g\) and \(1_{Lb}g\) agree; injectivity with target \(Lb\) gives \(gr=1_{Lb}\). Hence \(g\) is invertible. \(\square\)

Thus \(W_L\) is determined by its local objects. Conversely those objects are determined by \(W_L\): they are orthogonal to all its maps, whereas an object orthogonal to all its maps is in particular orthogonal to its own unit and is local by the final argument of Theorem 2.1. Every unit lies in \(W_L\) by the triangle identities.

The class \(W_L\) contains isomorphisms, satisfies two-out-of-three for composable maps, and is closed under retracts in the arrow category. The first two assertions follow by applying \(L\). For retracts, apply \(L\) to the retraction diagram. A retract \(f\) of an isomorphism \(g\) is invertible: if its arrow retraction has source components \(u:a\to a'\), \(r:a'\to a\) and target components \(v:b\to b'\), \(t:b'\to b\), an inverse is \(rg^{-1}v\). The two commuting squares and \(ru=1_a,tv=1_b\) verify its composites with \(f\).

It is also closed under every small colimit in \(C^\to\). Such colimits are computed on both endpoints, since cocones and their equations are componentwise. For a local \(y\), its restriction map is the limit of the corresponding bijections of Hom sets, using the universal property of both endpoint colimits. A limit of bijections is bijective: componentwise inverses remain compatible by naturality and are inverse on compatible families. Proposition 3.1 applies.

In particular \(W_L\) is closed under small coproducts, pushouts and continuous transfinite composition. For pushouts this follows either by the Hom pullback description or by adjoining identity arrows to the diagram. For a chain, a map from its last colimit to \(y\) is a compatible family. If restriction at every successor is bijective, a map from the initial object of the chain extends uniquely at each successor, and has a unique factor at each limit. This proves bijectivity for the entire transfinite composite, including chains whose length exceeds \(\omega\).

4. Replace uniqueness coequalizers by folds and pushouts

For \(s:A\to B\), form its self-pushout and specified fold

These are the ordinary universal maps of Universal forks, Section 6. Each \(j_\nu\) is a pushout of \(s\).

Lemma 4.1. Any class containing identities and \(s\), closed under pushouts and two-out-of-three, contains \(\delta_s\).

Proof. It contains \(j_0\) as a pushout of \(s\). The identity \(\delta_sj_0=1_B\) and two-out-of-three then put \(\delta_s\) in the class. \(\square\)

Given \(u,v:B\to z\) with \(us=vs\), there is a unique \(h:D_s\to z\) with \(hj_0=u,hj_1=v\). The pushout of \(\delta_s\) along \(h\) is exactly a coequalizer of \(u,v\). Indeed a map from \(z\) to \(y\) extends to that pushout precisely when it has a companion \(B\to y\) whose composites at both \(j_\nu\) agree with the two composites from \(z\). Those equations say that its composites with \(u,v\) are equal, and force the companion to be their common value. This is the coequalizer universal property, including uniqueness.

The same statement holds simultaneously. For a small family \((s_q,u_q,v_q)\) of equal-restriction pairs into \(z\), use the pushout of

along the maps \(h_q\) to \(z\). A map out of this pushout is exactly a map out of \(z\) equalizing every \(u_q,v_q\), by the just proved component equations. Thus it is the simultaneous coequalizer, with its actual quotient map.

This identifies the uniqueness step of Small orthogonality, Section 2, as a pushout of a small coproduct of these folds. It supplies a construction of the quotient map from the original tests, rather than merely a reason why a particular functor happens to invert that quotient.

5. The full inverted class is generated by the tests

For a class \(S\), call its colimit saturation the smallest class containing \(S\) and all isomorphisms, and closed under two-out-of-three, retracts, small coproducts, pushouts and continuous small ordinal transfinite composition. Equivalently, to prove a map lies in this class we may prove that every class with these properties contains it. This specifies our convention; saturation for a fraction calculus in the earlier lessons is a different formulation.

Let \(S\) now be a small set in a locally presentable \(C\), and let \(L:C\to S^\perp\) be its reflection.

Theorem 5.1. Its full inverted class \(W_L\) is exactly the colimit saturation of \(S\).

Proof: all stated operations are sound. Proposition 3.1 and Section 3 prove that \(W_L\) is closed under all the specified operations. It contains \(S\), because its local objects are orthogonal to every member. Hence the generated class is contained in \(W_L\).

Proof: every unit is generated. Let \(H\) be any class with the stated closure properties and \(S\subseteq H\). Use the actual construction in Small orthogonality, Sections 1–3. The existence map \(p_x:x\to P(x)\) is a pushout of a small coproduct of members of \(S\), so belongs to \(H\). The uniqueness map \(q_x:P(x)\to K(x)\) is the simultaneous coequalizer for its small set of equal-restriction pairs. By Lemma 4.1 and (4.2), it too belongs to \(H\). Composition, which is part of two-out-of-three, puts \(k_x=q_xp_x\) in \(H\).

The regular-length reflection chain has these \(k\)'s as successor maps and is continuous at every nonzero limit. Closure under transfinite composition therefore puts its unit \(\eta_x:x\to T(x)\) in \(H\). Any other chosen reflection has the same unit up to an invertible map preserving it, by uniqueness in the reflection Hom bijection. Isomorphisms and composition give the conclusion for that choice as well.

Proof: every inverted map is generated. If \(f:a\to b\) lies in \(W_L\), then \(T(f)=iL(f)\) is an isomorphism. Naturality gives

The right side lies in \(H\), and so does \(\eta_b\). Two-out-of-three puts \(f\) in \(H\). Since this holds for every such \(H\), every inverted map is generated. Together with the first inclusion it proves the theorem. \(\square\)

The proof did not use retract closure to generate the units, but the resulting class is retract-closed by Section 3. It even has the stronger closure under all small arrow-category colimits proved there. The theorem asserts equality of classes at the stated operations; it does not claim a uniform finite bound on their constructions.

6. Ordinary localization uses the generated class

Corollary 6.1. The reflector \(L:C\to D\) is an ordinary localization at \(W_L\). For every target category \(E\), composition gives an equivalence

including all natural transformations. No cocompleteness of \(E\) or colimit preservation by these functors is required.

Proof. The exact denominator-calculus statement and its full proof are in Descent through replacement objects, Theorem 4.1. One can also see the whole functor-category equivalence directly from the unit. For \(F\) inverting \(W_L\), restriction \(\overline F=Fi\) has the natural isomorphism

since every unit belongs to \(W_L\). For a functor \(G:D\to E\), the invertible counit gives \(GLi\simeq G\). Restriction and composition carry each natural transformation to its actual restricted or composed components. The isomorphisms just given commute with all these transformations by naturality at the unit and counit. They therefore supply inverse equivalences, on both objects and all maps. \(\square\)

For a small presentation \(S\), Theorem 5.1 identifies the class in (6.1) by explicit generators and operations. An arbitrary functor inverting \(S\) may fail to invert a pushout of an already inverted map, so it need not invert that generated class. Exercise 3 exhibits the failure inside abelian groups. If a functor preserves all small colimits and inverts \(S\), it does invert all the generated maps: isomorphisms, retracts and two-out-of-three are preserved by every functor; colimit preservation handles the other operations. Thus the cocontinuous universal property of Small orthogonality, Section 6, is consistent with (6.1), and does not assert the same essential image for arbitrary functors.

Two small sets of tests present the same local category if and only if their colimit saturations are equal. Equality of local categories gives the same inverted class by Proposition 3.1. Conversely equality of inverted classes gives the same local objects by the unit test following that proposition. Theorem 5.1 then gives both implications in the stated form.

7. Intersections and the order of accessible reflections

Proposition 7.1. A small intersection of full replete accessible reflective subcategories of \(C\) is accessible and reflective.

Proof. For each \(D_j\), Theorem 2.1 supplies a small set \(S_j\) presenting it. A small union \(S=\bigcup_jS_j\) is small. Orthogonality to that union is precisely membership in every \(D_j\). Apply the small-orthogonality reflection theorem to \(S\). This proves the assertion, including the empty intersection \(C\). It does not assert the same result for an arbitrary proper-class family. \(\square\)

There is also a useful exact order test. For two reflections with endofunctors \(T_1,T_2\), one has

For the forward implication, every \(D_2\)-object is among the tests defining \(W_{L_1}\), so Proposition 3.1 proves inclusion of classes. Conversely each unit of \(T_1\) is then in \(W_{L_2}\). A \(D_2\)-object is orthogonal to all of them, hence is \(T_1\)-local by the unit test.

If these conditions hold, every \(T_2(x)\) is \(T_1\)-local. Its unit \(\eta^2_x\) therefore extends uniquely to a map

Uniqueness after \(\eta^1_x\) proves naturality. It also proves uniqueness of this comparison among transformations satisfying the displayed equation. Conversely any such comparison, even before assuming inclusion of local categories, implies that inclusion: for \(y\in D_2\), the map \((\eta^2_y)^{-1}\gamma_y\) retracts \(\eta^1_y\), and (1.1) for the first reflection makes that unit invertible. These comparisons order reflections by their actual units. Taking an intersection does not in general mean composing its reflectors once in an arbitrary order, as Exercise 4 shows.

8. Four graded exercises with complete solutions

Exercise 1 (foundation: the fold test for subterminal sets)

In sets let \(S\) consist of the fold \(2\to1\). Determine the local objects, reflector and full inverted class. Express the reflection unit by a pushout of a coproduct of the test, and check whether the reflector preserves pullbacks.

Solution. Restriction is the diagonal \(Y\to Y^2\). It is bijective exactly when \(Y\) has at most one element: surjectivity forces every pair equal, and the empty and singleton sets both satisfy the test. The reflector sends the empty set to itself and every nonempty set to a singleton. A map into a subterminal set either is unique, or cannot exist because the source is nonempty and the target empty. This verifies the full reflection Hom bijection and its quotient unit.

The local-presentability hypothesis holds: a map from a finite set into a filtered colimit is represented at a common stage by its finitely many values, and equality of two such maps is witnessed at a common later stage. Thus finite sets are finitely presentable. Every set is the filtered union of its finite subsets, and all small set colimits are their disjoint-union quotients. A small skeleton of finite sets therefore supplies the required family.

For any \(X\), index a coproduct of \(2\to1\) by all ordered pairs \((x,x')\in X^2\), and map its \(2\)-summand to the indicated two elements of \(X\). The pushout identifies every pair of elements. It is therefore a singleton for nonempty \(X\), and is empty for empty \(X\), by the universal property of a quotient. This is the actual unit.

A function is inverted exactly when its source and target are both nonempty, or both empty. These are precisely the functions that preserve emptiness in both directions; a function from a nonempty set to an empty set does not exist. Theorem 5.1 identifies this class with the colimit saturation of the fold.

The two maps \(1\to2\) choosing different elements have empty pullback. All three nonempty sets reflect to singletons, whose corresponding maps have singleton pullback. The comparison from the reflection of the original pullback is \(\varnothing\to1\), which is not invertible. Thus the reflector is accessible but does not preserve pullbacks.

Exercise 2 (standard: two presentations of scalar localization)

In abelian groups fix a prime \(p\). Compare the single tests \(p:\mathbb Z\to\mathbb Z\) and \(u:\mathbb Z\to\mathbb Z[1/p]\). Identify their common colimit saturation by a condition on the kernel and cokernel of a homomorphism.

Solution. Orthogonality to the first test means that multiplication by \(p\) on \(A\) is bijective. Orthogonality to the second means that evaluation at \(1\) gives a bijection \(\operatorname{Hom}(\mathbb Z[1/p],A)\to A\). The Hom set is an abelian group and evaluation is a group homomorphism. On it multiplication by \(p\) is precomposition by multiplication by \(p\) on \(\mathbb Z[1/p]\), because homomorphisms commute with integer multiples. That precomposition is invertible. Evaluation intertwines it with multiplication by \(p\) on \(A\), so bijective evaluation forces the latter invertible.

Here is also the local-presentability input for the abelian-group examples. Arbitrary colimits are obtained from direct sums by quotienting by the subgroup generated by the diagram relations. A filtered colimit has represented elements, with equality witnessed at a common later stage: sums are defined after receiving their finitely many representatives at a common stage, and the group identities and independence follow at another such stage. A homomorphism from a group with finitely many generators and finitely many relations therefore factors through a stage, and equality is witnessed at a later stage, by the finite generator and relation conditions. These groups are finitely presentable, and their finite integer presentation matrices form a set.

For an arbitrary \(A\), index groups \(\mathbb Z^F/\langle R\rangle\) by finite subsets \(F\subseteq A\) and finite sets \(R\) of relations in the kernel of \(\mathbb Z^F\to A\). Increase \(F\) and \(R\), retaining old relations by extension by zero. This is a small directed poset: two pairs have a common upper pair by taking their unions. Every element of \(A\) occurs, and every integer relation has finite support and can be put into such an \(R\). Consequently its colimit is \(A\): a compatible family of maps assigns a homomorphism on those elements, every relation is respected, and this gives its unique factor. Thus abelian groups are locally finitely presentable, as required.

Conversely, if multiplication by \(p\) on \(A\) is invertible, each \(a\) defines exactly one homomorphism by \(m/p^k\mapsto mp^{-k}a\). Common denominators make this well defined and additive; its value at \(1\) forces all its values. This proves bijective evaluation. The tests therefore have the same local objects.

Their reflector is \(A[1/p]=\operatorname{colim}(A\xrightarrow{p}A\xrightarrow{p}\cdots)\), with elements \([a,k]\). The direct-limit relation and the full unit extension were proved in Small orthogonality, Exercise 1. Theorem 5.1 and Section 6 show the two colimit saturations are equal.

For \(f:A\to B\), the localized map is invertible exactly when both \(\ker f\) and \(\operatorname{coker}f\) are \(p\)-power torsion, meaning that each element is killed by some power of \(p\). Here is an elementwise proof of both comparisons. If the localized map is injective and \(a\in\ker f\), the zero localized image of \([a,0]\) forces \(p^ka=0\) at a later stage. If it is surjective, write each \([b,0]\) as \([f(a),k]\); eventual equality gives a power multiple of \(b\) in the image of \(f\). Thus its cokernel class is killed by a power.

Conversely if both torsion conditions hold, \(f(a)/p^k=0\) implies that some \(p^r f(a)\) is zero. Then \(p^ra\in\ker f\), so another power kills it, proving \(a/p^k=0\) and injectivity. For \(b/p^j\), cokernel torsion gives \(p^rb=f(a)\) for some \(a,r\). Hence \(b/p^j=f(a)/p^{j+r}\), proving surjectivity. This proves the exact description of the common generated class without appealing to an unproved flatness assertion.

Exercise 3 (advanced: an inverted fold can have a noninverted pushout)

In abelian groups let \(S=\{0\to\mathbb Z\}\), and fix \(n\ge2\). Show that its colimit saturation contains every homomorphism. Construct a functor inverting \(S\) and the fold \(\mathbb Z^2\to\mathbb Z\), but failing to invert a pushout of that fold.

Solution. Orthogonality means that \(A\to\{*\}\) is bijective, so the only local group is zero. The reflector is zero and inverts every homomorphism. Theorem 5.1 therefore gives the claimed generated class.

Take \(F(A)=\operatorname{Hom}(\mathbb Z/n,A)\) as a functor to sets, with maps given by postcomposition. Both \(F(0)\) and \(F(\mathbb Z)\) are singletons, since an integer killed by \(n\) is zero. Likewise \(F(\mathbb Z^2)\) is a singleton, since both integer coordinates must be killed by \(n\). Thus \(F\) inverts both the test and the addition fold.

Let \(A=\mathbb Z/n\), and map \(\mathbb Z^2\to A\) by \((r,s)\mapsto\bar r\). The pushout of the fold \((r,s)\mapsto r+s\) along this map is the quotient \(A\to0\). Indeed a map \(a:A\to Y\) and a homomorphism \(b:\mathbb Z\to Y\) satisfying the pushout equation must satisfy \(a(\bar1)=b(1)\) on the first basis vector and \(0=b(1)\) on the second. Hence both maps are zero, and their unique factor from zero proves the universal property.

The set \(F(A)\) has \(n\) elements, determined by the image of \(\bar1\), whereas \(F(0)\) is a singleton. The pushed-out map is therefore not inverted. The fold is exactly \(\delta_s\) for \(s:0\to\mathbb Z\). An ordinary functor may invert the original test and its fold while failing the next pushout operation. This is why ordinary inversion of \(S\) alone cannot be substituted for inversion of its colimit saturation in (6.1).

Exercise 4 (synthesis: intersect reflections without composing them once)

Regard the three-element chain \(0<a<1\) as a category. Let \(D=\{a,1\}\) and \(E=\{0,1\}\) be full subcategories. Prove they are accessible reflective subcategories, compute their intersection, and show that one order of composing their reflector endofunctors is not a reflector.

Solution. Every small family in this finite chain has a supremum, including the empty family with supremum \(0\). These are its categorical colimits. A filtered diagram has a greatest value among its finite set of occurring values: directedness gives a value above all of them, which must be attained. Its colimit is that value. Each object's represented Hom functor therefore preserves filtered colimits. More explicitly, if the object lies below the colimit it lies below an attained stage, while all the nonempty Hom sets are singletons. Thus the finite chain is locally finitely presentable.

The reflector \(P\) to \(D\) has values \(P(0)=a,P(a)=a,P(1)=1\). The reflector \(Q\) to \(E\) has \(Q(0)=0,Q(a)=1,Q(1)=1\). They are monotone, and each sends an object to the least object of its target lying above it. The equivalence \(P(x)\le d\) if and only if \(x\le d\), and its counterpart for \(Q\), prove both adjunctions, with their specified order-arrow units.

Both preserve filtered colimits: any monotone map out of this finite chain sends the attained greatest diagram value to the greatest image value. The reflector endofunctors are therefore accessible. The two categories are respectively presented by the tests \(0\to a\) and \(a\to1\): in a poset, orthogonality to \(x\le y\) means that \(x\le z\) implies \(y\le z\), since all Hom sets have at most one element. These conditions give exactly \(D\) and \(E\).

Their intersection is \(\{1\}\), with the constant-\(1\) reflector. The union of the two tests presents it, as Proposition 7.1 predicts. But \(PQ\) sends \(0\) to \(a\), \(a\) to \(1\), and \(1\) to \(1\). Hence \(PQ(0)=a\) while \((PQ)^2(0)=1\); these objects are not isomorphic in the chain. A reflection endofunctor must be idempotent up to isomorphism, so \(PQ\) is not one. The other order \(QP\) happens to be constant \(1\). The intersection theorem proves existence independently of either proposed one-step composition.

References

[AHS] Jiří Adámek, Michel Hébert and Lurdes Sousa, The Orthogonal Subcategory Problem and the Small Object Argument, Coimbra preprint 07–40. Freely accessible departmental preprint, especially the orthogonality discussion and Section 5.5. Those rules provide context for morphism generation; the small-set presentation, actual fold-pushout construction and closure proof used here are supplied above and in the linked course proofs. No proper-class generalization is asserted here.